LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

Functions/Subroutines  
DOUBLE PRECISION function  dlamch (CMACH) 
DLAMCH  
DOUBLE PRECISION function  dlamc3 (A, B) 
DLAMC3  
subroutine  dlamc1 (BETA, T, RND, IEEE1) 
DLAMC1  
subroutine  dlamc2 (BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX) 
DLAMC2  
program  __dlamchtst.f__ 
DLAMCHTST  
DOUBLE PRECISION function  dsecnd () 
DSECND Using ETIME  
program  __dsecndtst.f__ 
DSECNDTST  
subroutine  ilaver (VERS_MAJOR, VERS_MINOR, VERS_PATCH) 
ILAVER returns the LAPACK version.  
program  __lapack_version.f__ 
LAPACK_VERSION  
LOGICAL function  lsame (CA, CB) 
LSAME  
program  __lsametst.f__ 
LSAMETST  
REAL function  second () 
SECOND Using ETIME  
REAL function  slamch (CMACH) 
SLAMCH  
REAL function  slamc3 (A, B) 
SLAMC3  
subroutine  slamc1 (BETA, T, RND, IEEE1) 
SLAMC1  
subroutine  slamc2 (BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX) 
SLAMC2  
program  tstiee 
TSTIEE  
LOGICAL function  disnan (DIN) 
DISNAN tests input for NaN.  
subroutine  dlabad (SMALL, LARGE) 
DLABAD  
subroutine  dlacpy (UPLO, M, N, A, LDA, B, LDB) 
DLACPY copies all or part of one twodimensional array to another.  
subroutine  dladiv (A, B, C, D, P, Q) 
DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.  
subroutine  dlae2 (A, B, C, RT1, RT2) 
DLAE2 computes the eigenvalues of a 2by2 symmetric matrix.  
subroutine  dlaebz (IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO) 
DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.  
subroutine  dlaev2 (A, B, C, RT1, RT2, CS1, SN1) 
DLAEV2 computes the eigenvalues and eigenvectors of a 2by2 symmetric/Hermitian matrix.  
subroutine  dlagts (JOB, N, A, B, C, D, IN, Y, TOL, INFO) 
DLAGTS solves the system of equations (TλI)x = y or (TλI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.  
LOGICAL function  dlaisnan (DIN1, DIN2) 
DLAISNAN tests input for NaN by comparing two arguments for inequality.  
INTEGER function  dlaneg (N, D, LLD, SIGMA, PIVMIN, R) 
DLANEG computes the Sturm count.  
DOUBLE PRECISION function  dlanst (NORM, N, D, E) 
DLANST returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.  
DOUBLE PRECISION function  dlapy2 (X, Y) 
DLAPY2 returns sqrt(x2+y2).  
DOUBLE PRECISION function  dlapy3 (X, Y, Z) 
DLAPY3 returns sqrt(x2+y2+z2).  
subroutine  dlarnv (IDIST, ISEED, N, X) 
DLARNV returns a vector of random numbers from a uniform or normal distribution.  
subroutine  dlarra (N, D, E, E2, SPLTOL, TNRM, NSPLIT, ISPLIT, INFO) 
DLARRA computes the splitting points with the specified threshold.  
subroutine  dlarrb (N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO) 
DLARRB provides limited bisection to locate eigenvalues for more accuracy.  
subroutine  dlarrc (JOBT, N, VL, VU, D, E, PIVMIN, EIGCNT, LCNT, RCNT, INFO) 
DLARRC computes the number of eigenvalues of the symmetric tridiagonal matrix.  
subroutine  dlarrd (RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO) 
DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.  
subroutine  dlarre (RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO) 
DLARRE given the tridiagonal matrix T, sets small offdiagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.  
subroutine  dlarrf (N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK, INFO) 
DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.  
subroutine  dlarrj (N, D, E2, IFIRST, ILAST, RTOL, OFFSET, W, WERR, WORK, IWORK, PIVMIN, SPDIAM, INFO) 
DLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.  
subroutine  dlarrk (N, IW, GL, GU, D, E2, PIVMIN, RELTOL, W, WERR, INFO) 
DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.  
subroutine  dlarrr (N, D, E, INFO) 
DLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.  
subroutine  dlartg (F, G, CS, SN, R) 
DLARTG generates a plane rotation with real cosine and real sine.  
subroutine  dlartgp (F, G, CS, SN, R) 
DLARTGP generates a plane rotation so that the diagonal is nonnegative.  
subroutine  dlaruv (ISEED, N, X) 
DLARUV returns a vector of n random real numbers from a uniform distribution.  
subroutine  dlas2 (F, G, H, SSMIN, SSMAX) 
DLAS2 computes singular values of a 2by2 triangular matrix.  
subroutine  dlascl (TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO) 
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.  
subroutine  dlasd0 (N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK, WORK, INFO) 
DLASD0 computes the singular values of a real upper bidiagonal nbym matrix B with diagonal d and offdiagonal e. Used by sbdsdc.  
subroutine  dlasd1 (NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO) 
DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.  
subroutine  dlasd2 (NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO) 
DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.  
subroutine  dlasd3 (NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO) 
DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.  
subroutine  dlasd4 (N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO) 
DLASD4 computes the square root of the ith updated eigenvalue of a positive symmetric rankone modification to a positive diagonal matrix. Used by sbdsdc.  
subroutine  dlasd5 (I, D, Z, DELTA, RHO, DSIGMA, WORK) 
DLASD5 computes the square root of the ith eigenvalue of a positive symmetric rankone modification of a 2by2 diagonal matrix. Used by sbdsdc.  
subroutine  dlasd6 (ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO) 
DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.  
subroutine  dlasd7 (ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO) 
DLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.  
subroutine  dlasd8 (ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, DSIGMA, WORK, INFO) 
DLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.  
subroutine  dlasda (ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO) 
DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.  
subroutine  dlasdq (UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO) 
DLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.  
subroutine  dlasdt (N, LVL, ND, INODE, NDIML, NDIMR, MSUB) 
DLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.  
subroutine  dlaset (UPLO, M, N, ALPHA, BETA, A, LDA) 
DLASET initializes the offdiagonal elements and the diagonal elements of a matrix to given values.  
subroutine  dlasr (SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA) 
DLASR applies a sequence of plane rotations to a general rectangular matrix.  
subroutine  dlassq (N, X, INCX, SCALE, SUMSQ) 
DLASSQ updates a sum of squares represented in scaled form.  
subroutine  dlasv2 (F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL) 
DLASV2 computes the singular value decomposition of a 2by2 triangular matrix.  
INTEGER function  ieeeck (ISPEC, ZERO, ONE) 
IEEECK  
INTEGER function  iladlc (M, N, A, LDA) 
ILADLC scans a matrix for its last nonzero column.  
INTEGER function  iladlr (M, N, A, LDA) 
ILADLR scans a matrix for its last nonzero row.  
INTEGER function  ilaenv (ISPEC, NAME, OPTS, N1, N2, N3, N4) 
ILAENV  
INTEGER function  iparmq (ISPEC, NAME, OPTS, N, ILO, IHI, LWORK) 
IPARMQ  
LOGICAL function  lsamen (N, CA, CB) 
LSAMEN  
LOGICAL function  sisnan (SIN) 
SISNAN tests input for NaN.  
subroutine  slabad (SMALL, LARGE) 
SLABAD  
subroutine  slacpy (UPLO, M, N, A, LDA, B, LDB) 
SLACPY copies all or part of one twodimensional array to another.  
subroutine  sladiv (A, B, C, D, P, Q) 
SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.  
subroutine  slae2 (A, B, C, RT1, RT2) 
SLAE2 computes the eigenvalues of a 2by2 symmetric matrix.  
subroutine  slaebz (IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL, RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT, NAB, WORK, IWORK, INFO) 
SLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.  
subroutine  slaev2 (A, B, C, RT1, RT2, CS1, SN1) 
SLAEV2 computes the eigenvalues and eigenvectors of a 2by2 symmetric/Hermitian matrix.  
subroutine  slag2d (M, N, SA, LDSA, A, LDA, INFO) 
SLAG2D converts a single precision matrix to a double precision matrix.  
subroutine  slagts (JOB, N, A, B, C, D, IN, Y, TOL, INFO) 
SLAGTS solves the system of equations (TλI)x = y or (TλI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.  
LOGICAL function  slaisnan (SIN1, SIN2) 
SLAISNAN tests input for NaN by comparing two arguments for inequality.  
INTEGER function  slaneg (N, D, LLD, SIGMA, PIVMIN, R) 
SLANEG computes the Sturm count.  
REAL function  slanst (NORM, N, D, E) 
SLANST returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.  
REAL function  slapy2 (X, Y) 
SLAPY2 returns sqrt(x2+y2).  
REAL function  slapy3 (X, Y, Z) 
SLAPY3 returns sqrt(x2+y2+z2).  
subroutine  slarnv (IDIST, ISEED, N, X) 
SLARNV returns a vector of random numbers from a uniform or normal distribution.  
subroutine  slarra (N, D, E, E2, SPLTOL, TNRM, NSPLIT, ISPLIT, INFO) 
SLARRA computes the splitting points with the specified threshold.  
subroutine  slarrb (N, D, LLD, IFIRST, ILAST, RTOL1, RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK, PIVMIN, SPDIAM, TWIST, INFO) 
SLARRB provides limited bisection to locate eigenvalues for more accuracy.  
subroutine  slarrc (JOBT, N, VL, VU, D, E, PIVMIN, EIGCNT, LCNT, RCNT, INFO) 
SLARRC computes the number of eigenvalues of the symmetric tridiagonal matrix.  
subroutine  slarrd (RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO) 
SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.  
subroutine  slarre (RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO) 
SLARRE given the tridiagonal matrix T, sets small offdiagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.  
subroutine  slarrf (N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK, INFO) 
SLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.  
subroutine  slarrj (N, D, E2, IFIRST, ILAST, RTOL, OFFSET, W, WERR, WORK, IWORK, PIVMIN, SPDIAM, INFO) 
SLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.  
subroutine  slarrk (N, IW, GL, GU, D, E2, PIVMIN, RELTOL, W, WERR, INFO) 
SLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.  
subroutine  slarrr (N, D, E, INFO) 
SLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.  
subroutine  slartg (F, G, CS, SN, R) 
SLARTG generates a plane rotation with real cosine and real sine.  
subroutine  slartgp (F, G, CS, SN, R) 
SLARTGP generates a plane rotation so that the diagonal is nonnegative.  
subroutine  slaruv (ISEED, N, X) 
SLARUV returns a vector of n random real numbers from a uniform distribution.  
subroutine  slas2 (F, G, H, SSMIN, SSMAX) 
SLAS2 computes singular values of a 2by2 triangular matrix.  
subroutine  slascl (TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO) 
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.  
subroutine  slasd0 (N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK, WORK, INFO) 
SLASD0 computes the singular values of a real upper bidiagonal nbym matrix B with diagonal d and offdiagonal e. Used by sbdsdc.  
subroutine  slasd1 (NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO) 
SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.  
subroutine  slasd2 (NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO) 
SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.  
subroutine  slasd3 (NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO) 
SLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.  
subroutine  slasd4 (N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO) 
SLASD4 computes the square root of the ith updated eigenvalue of a positive symmetric rankone modification to a positive diagonal matrix. Used by sbdsdc.  
subroutine  slasd5 (I, D, Z, DELTA, RHO, DSIGMA, WORK) 
SLASD5 computes the square root of the ith eigenvalue of a positive symmetric rankone modification of a 2by2 diagonal matrix. Used by sbdsdc.  
subroutine  slasd6 (ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO) 
SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.  
subroutine  slasd7 (ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO) 
SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.  
subroutine  slasd8 (ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDDIFR, DSIGMA, WORK, INFO) 
SLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.  
subroutine  slasda (ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK, IWORK, INFO) 
SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.  
subroutine  slasdq (UPLO, SQRE, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO) 
SLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.  
subroutine  slasdt (N, LVL, ND, INODE, NDIML, NDIMR, MSUB) 
SLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.  
subroutine  slaset (UPLO, M, N, ALPHA, BETA, A, LDA) 
SLASET initializes the offdiagonal elements and the diagonal elements of a matrix to given values.  
subroutine  slasr (SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA) 
SLASR applies a sequence of plane rotations to a general rectangular matrix.  
subroutine  slassq (N, X, INCX, SCALE, SUMSQ) 
SLASSQ updates a sum of squares represented in scaled form.  
subroutine  slasv2 (F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL) 
SLASV2 computes the singular value decomposition of a 2by2 triangular matrix.  
subroutine  xerbla (SRNAME, INFO) 
XERBLA  
subroutine  xerbla_array (SRNAME_ARRAY, SRNAME_LEN, INFO) 
XERBLA_ARRAY 
This is the group of Other Auxiliary routines
program __dlamchtst.f__  (  ) 
DLAMCHTST
Definition at line 35 of file dlamchtst.f.
program __dsecndtst.f__  (  ) 
DSECNDTST
Definition at line 37 of file dsecndtst.f.
program __lapack_version.f__  (  ) 
LAPACK_VERSION
Definition at line 32 of file LAPACK_version.f.
program __lsametst.f__  (  ) 
LSAMETST
Definition at line 36 of file lsametst.f.
LOGICAL function disnan  (  double precision  DIN  ) 
DISNAN tests input for NaN.
Download DISNAN + dependencies [TGZ] [ZIP] [TXT]DISNAN returns .TRUE. if its argument is NaN, and .FALSE. otherwise. To be replaced by the Fortran 2003 intrinsic in the future.
[in]  DIN  DIN is DOUBLE PRECISION Input to test for NaN. 
Definition at line 60 of file disnan.f.
subroutine dlabad  (  double precision  SMALL, 
double precision  LARGE  
) 
DLABAD
Download DLABAD + dependencies [TGZ] [ZIP] [TXT]DLABAD takes as input the values computed by DLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by DLAMCH. This subroutine is needed because DLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray.
[in,out]  SMALL  SMALL is DOUBLE PRECISION On entry, the underflow threshold as computed by DLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of SMALL, otherwise unchanged. 
[in,out]  LARGE  LARGE is DOUBLE PRECISION On entry, the overflow threshold as computed by DLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of LARGE, otherwise unchanged. 
subroutine dlacpy  (  character  UPLO, 
integer  M,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB  
) 
DLACPY copies all or part of one twodimensional array to another.
Download DLACPY + dependencies [TGZ] [ZIP] [TXT]DLACPY copies all or part of a twodimensional matrix A to another matrix B.
[in]  UPLO  UPLO is CHARACTER*1 Specifies the part of the matrix A to be copied to B. = 'U': Upper triangular part = 'L': Lower triangular part Otherwise: All of the matrix A 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. If UPLO = 'U', only the upper triangle or trapezoid is accessed; if UPLO = 'L', only the lower triangle or trapezoid is accessed. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[out]  B  B is DOUBLE PRECISION array, dimension (LDB,N) On exit, B = A in the locations specified by UPLO. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). 
Definition at line 104 of file dlacpy.f.
subroutine dladiv  (  double precision  A, 
double precision  B,  
double precision  C,  
double precision  D,  
double precision  P,  
double precision  Q  
) 
DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
Download DLADIV + dependencies [TGZ] [ZIP] [TXT]DLADIV performs complex division in real arithmetic a + i*b p + i*q =  c + i*d The algorithm is due to Robert L. Smith and can be found in D. Knuth, The art of Computer Programming, Vol.2, p.195
[in]  A  A is DOUBLE PRECISION 
[in]  B  B is DOUBLE PRECISION 
[in]  C  C is DOUBLE PRECISION 
[in]  D  D is DOUBLE PRECISION The scalars a, b, c, and d in the above expression. 
[out]  P  P is DOUBLE PRECISION 
[out]  Q  Q is DOUBLE PRECISION The scalars p and q in the above expression. 
Definition at line 91 of file dladiv.f.
subroutine dlae2  (  double precision  A, 
double precision  B,  
double precision  C,  
double precision  RT1,  
double precision  RT2  
) 
DLAE2 computes the eigenvalues of a 2by2 symmetric matrix.
Download DLAE2 + dependencies [TGZ] [ZIP] [TXT]DLAE2 computes the eigenvalues of a 2by2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, and RT2 is the eigenvalue of smaller absolute value.
[in]  A  A is DOUBLE PRECISION The (1,1) element of the 2by2 matrix. 
[in]  B  B is DOUBLE PRECISION The (1,2) and (2,1) elements of the 2by2 matrix. 
[in]  C  C is DOUBLE PRECISION The (2,2) element of the 2by2 matrix. 
[out]  RT1  RT1 is DOUBLE PRECISION The eigenvalue of larger absolute value. 
[out]  RT2  RT2 is DOUBLE PRECISION The eigenvalue of smaller absolute value. 
RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*CB*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.
Definition at line 103 of file dlae2.f.
subroutine dlaebz  (  integer  IJOB, 
integer  NITMAX,  
integer  N,  
integer  MMAX,  
integer  MINP,  
integer  NBMIN,  
double precision  ABSTOL,  
double precision  RELTOL,  
double precision  PIVMIN,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( * )  E2,  
integer, dimension( * )  NVAL,  
double precision, dimension( mmax, * )  AB,  
double precision, dimension( * )  C,  
integer  MOUT,  
integer, dimension( mmax, * )  NAB,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.
Download DLAEBZ + dependencies [TGZ] [ZIP] [TXT]DLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases halfopen intervals, i.e., of the form (a,b] , which includes b but not a . To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966 Note: the arguments are, in general, *not* checked for unreasonable values.
[in]  IJOB  IJOB is INTEGER Specifies what is to be done: = 1: Compute NAB for the initial intervals. = 2: Perform bisection iteration to find eigenvalues of T. = 3: Perform bisection iteration to invert N(w), i.e., to find a point which has a specified number of eigenvalues of T to its left. Other values will cause DLAEBZ to return with INFO=1. 
[in]  NITMAX  NITMAX is INTEGER The maximum number of "levels" of bisection to be performed, i.e., an interval of width W will not be made smaller than 2^(NITMAX) * W. If not all intervals have converged after NITMAX iterations, then INFO is set to the number of nonconverged intervals. 
[in]  N  N is INTEGER The dimension n of the tridiagonal matrix T. It must be at least 1. 
[in]  MMAX  MMAX is INTEGER The maximum number of intervals. If more than MMAX intervals are generated, then DLAEBZ will quit with INFO=MMAX+1. 
[in]  MINP  MINP is INTEGER The initial number of intervals. It may not be greater than MMAX. 
[in]  NBMIN  NBMIN is INTEGER The smallest number of intervals that should be processed using a vector loop. If zero, then only the scalar loop will be used. 
[in]  ABSTOL  ABSTOL is DOUBLE PRECISION The minimum (absolute) width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. This must be at least zero. 
[in]  RELTOL  RELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum absolute value of a "pivot" in the Sturm sequence loop. This must be at least max e(j)**2*safe_min and at least safe_min, where safe_min is at least the smallest number that can divide one without overflow. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T. 
[in]  E  E is DOUBLE PRECISION array, dimension (N) The offdiagonal elements of the tridiagonal matrix T in positions 1 through N1. E(N) is arbitrary. 
[in]  E2  E2 is DOUBLE PRECISION array, dimension (N) The squares of the offdiagonal elements of the tridiagonal matrix T. E2(N) is ignored. 
[in,out]  NVAL  NVAL is INTEGER array, dimension (MINP) If IJOB=1 or 2, not referenced. If IJOB=3, the desired values of N(w). The elements of NVAL will be reordered to correspond with the intervals in AB. Thus, NVAL(j) on output will not, in general be the same as NVAL(j) on input, but it will correspond with the interval (AB(j,1),AB(j,2)] on output. 
[in,out]  AB  AB is DOUBLE PRECISION array, dimension (MMAX,2) The endpoints of the intervals. AB(j,1) is a(j), the left endpoint of the jth interval, and AB(j,2) is b(j), the right endpoint of the jth interval. The input intervals will, in general, be modified, split, and reordered by the calculation. 
[in,out]  C  C is DOUBLE PRECISION array, dimension (MMAX) If IJOB=1, ignored. If IJOB=2, workspace. If IJOB=3, then on input C(j) should be initialized to the first search point in the binary search. 
[out]  MOUT  MOUT is INTEGER If IJOB=1, the number of eigenvalues in the intervals. If IJOB=2 or 3, the number of intervals output. If IJOB=3, MOUT will equal MINP. 
[in,out]  NAB  NAB is INTEGER array, dimension (MMAX,2) If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). If IJOB=2, then on input, NAB(i,j) should be set. It must satisfy the condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with IJOB=1. On output, NAB(i,j) will contain max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input interval that the output interval (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2). If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless N(w) > NVAL(i) for all search points w , in which case NAB(i,1) will not be modified, i.e., the output value will be the same as the input value (modulo reorderings  see NVAL and AB), or unless N(w) < NVAL(i) for all search points w , in which case NAB(i,2) will not be modified. Normally, NAB should be set to some distinctive value(s) before DLAEBZ is called. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (MMAX) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (MMAX) Workspace. 
[out]  INFO  INFO is INTEGER = 0: All intervals converged. = 1MMAX: The last INFO intervals did not converge. = MMAX+1: More than MMAX intervals were generated. 
This routine is intended to be called only by other LAPACK routines, thus the interface is less userfriendly. It is intended for two purposes: (a) finding eigenvalues. In this case, DLAEBZ should have one or more initial intervals set up in AB, and DLAEBZ should be called with IJOB=1. This sets up NAB, and also counts the eigenvalues. Intervals with no eigenvalues would usually be thrown out at this point. Also, if not all the eigenvalues in an interval i are desired, NAB(i,1) can be increased or NAB(i,2) decreased. For example, set NAB(i,1)=NAB(i,2)1 to get the largest eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX no smaller than the value of MOUT returned by the call with IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1 through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the tolerance specified by ABSTOL and RELTOL. (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l). In this case, start with a Gershgorin interval (a,b). Set up AB to contain 2 search intervals, both initially (a,b). One NVAL element should contain f1 and the other should contain l , while C should contain a and b, resp. NAB(i,1) should be 1 and NAB(i,2) should be N+1, to flag an error if the desired interval does not lie in (a,b). DLAEBZ is then called with IJOB=3. On exit, if w(f1) < w(f), then one of the intervals  j  will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f1, while if, to the specified tolerance, w(fk)=...=w(f+r), k > 0 and r >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=fk and N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and w(lr)=...=w(l+k) are handled similarly.
Definition at line 318 of file dlaebz.f.
subroutine dlaev2  (  double precision  A, 
double precision  B,  
double precision  C,  
double precision  RT1,  
double precision  RT2,  
double precision  CS1,  
double precision  SN1  
) 
DLAEV2 computes the eigenvalues and eigenvectors of a 2by2 symmetric/Hermitian matrix.
Download DLAEV2 + dependencies [TGZ] [ZIP] [TXT]DLAEV2 computes the eigendecomposition of a 2by2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition [ CS1 SN1 ] [ A B ] [ CS1 SN1 ] = [ RT1 0 ] [SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
[in]  A  A is DOUBLE PRECISION The (1,1) element of the 2by2 matrix. 
[in]  B  B is DOUBLE PRECISION The (1,2) element and the conjugate of the (2,1) element of the 2by2 matrix. 
[in]  C  C is DOUBLE PRECISION The (2,2) element of the 2by2 matrix. 
[out]  RT1  RT1 is DOUBLE PRECISION The eigenvalue of larger absolute value. 
[out]  RT2  RT2 is DOUBLE PRECISION The eigenvalue of smaller absolute value. 
[out]  CS1  CS1 is DOUBLE PRECISION 
[out]  SN1  SN1 is DOUBLE PRECISION The vector (CS1, SN1) is a unit right eigenvector for RT1. 
RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*CB*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. CS1 and SN1 are accurate to a few ulps barring over/underflow. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.
Definition at line 121 of file dlaev2.f.
subroutine dlagts  (  integer  JOB, 
integer  N,  
double precision, dimension( * )  A,  
double precision, dimension( * )  B,  
double precision, dimension( * )  C,  
double precision, dimension( * )  D,  
integer, dimension( * )  IN,  
double precision, dimension( * )  Y,  
double precision  TOL,  
integer  INFO  
) 
DLAGTS solves the system of equations (TλI)x = y or (TλI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.
Download DLAGTS + dependencies [TGZ] [ZIP] [TXT]DLAGTS may be used to solve one of the systems of equations (T  lambda*I)*x = y or (T  lambda*I)**T*x = y, where T is an n by n tridiagonal matrix, for x, following the factorization of (T  lambda*I) as (T  lambda*I) = P*L*U , by routine DLAGTF. The choice of equation to be solved is controlled by the argument JOB, and in each case there is an option to perturb zero or very small diagonal elements of U, this option being intended for use in applications such as inverse iteration.
[in]  JOB  JOB is INTEGER Specifies the job to be performed by DLAGTS as follows: = 1: The equations (T  lambda*I)x = y are to be solved, but diagonal elements of U are not to be perturbed. = 1: The equations (T  lambda*I)x = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. = 2: The equations (T  lambda*I)**Tx = y are to be solved, but diagonal elements of U are not to be perturbed. = 2: The equations (T  lambda*I)**Tx = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. 
[in]  N  N is INTEGER The order of the matrix T. 
[in]  A  A is DOUBLE PRECISION array, dimension (N) On entry, A must contain the diagonal elements of U as returned from DLAGTF. 
[in]  B  B is DOUBLE PRECISION array, dimension (N1) On entry, B must contain the first superdiagonal elements of U as returned from DLAGTF. 
[in]  C  C is DOUBLE PRECISION array, dimension (N1) On entry, C must contain the subdiagonal elements of L as returned from DLAGTF. 
[in]  D  D is DOUBLE PRECISION array, dimension (N2) On entry, D must contain the second superdiagonal elements of U as returned from DLAGTF. 
[in]  IN  IN is INTEGER array, dimension (N) On entry, IN must contain details of the matrix P as returned from DLAGTF. 
[in,out]  Y  Y is DOUBLE PRECISION array, dimension (N) On entry, the right hand side vector y. On exit, Y is overwritten by the solution vector x. 
[in,out]  TOL  TOL is DOUBLE PRECISION On entry, with JOB .lt. 0, TOL should be the minimum perturbation to be made to very small diagonal elements of U. TOL should normally be chosen as about eps*norm(U), where eps is the relative machine precision, but if TOL is supplied as nonpositive, then it is reset to eps*max( abs( u(i,j) ) ). If JOB .gt. 0 then TOL is not referenced. On exit, TOL is changed as described above, only if TOL is nonpositive on entry. Otherwise TOL is unchanged. 
[out]  INFO  INFO is INTEGER = 0 : successful exit .lt. 0: if INFO = i, the ith argument had an illegal value .gt. 0: overflow would occur when computing the INFO(th) element of the solution vector x. This can only occur when JOB is supplied as positive and either means that a diagonal element of U is very small, or that the elements of the righthand side vector y are very large. 
Definition at line 162 of file dlagts.f.
LOGICAL function dlaisnan  (  double precision  DIN1, 
double precision  DIN2  
) 
DLAISNAN tests input for NaN by comparing two arguments for inequality.
Download DLAISNAN + dependencies [TGZ] [ZIP] [TXT]This routine is not for general use. It exists solely to avoid overoptimization in DISNAN. DLAISNAN checks for NaNs by comparing its two arguments for inequality. NaN is the only floatingpoint value where NaN != NaN returns .TRUE. To check for NaNs, pass the same variable as both arguments. A compiler must assume that the two arguments are not the same variable, and the test will not be optimized away. Interprocedural or wholeprogram optimization may delete this test. The ISNAN functions will be replaced by the correct Fortran 03 intrinsic once the intrinsic is widely available.
[in]  DIN1  DIN1 is DOUBLE PRECISION 
[in]  DIN2  DIN2 is DOUBLE PRECISION Two numbers to compare for inequality. 
Definition at line 75 of file dlaisnan.f.
subroutine dlamc1  (  integer  BETA, 
integer  T,  
logical  RND,  
logical  IEEE1  
) 
DLAMC1
Purpose:
DLAMC1 determines the machine parameters given by BETA, T, RND, and IEEE1.
[out]  BETA  The base of the machine. 
[out]  T  The number of ( BETA ) digits in the mantissa. 
[out]  RND  Specifies whether proper rounding ( RND = .TRUE. ) or chopping ( RND = .FALSE. ) occurs in addition. This may not be a reliable guide to the way in which the machine performs its arithmetic. 
[out]  IEEE1  Specifies whether rounding appears to be done in the IEEE 'round to nearest' style. 
Further Details
The routine is based on the routine ENVRON by Malcolm and incorporates suggestions by Gentleman and Marovich. See Malcolm M. A. (1972) Algorithms to reveal properties of floatingpoint arithmetic. Comms. of the ACM, 15, 949951. Gentleman W. M. and Marovich S. B. (1974) More on algorithms that reveal properties of floating point arithmetic units. Comms. of the ACM, 17, 276277.
Definition at line 206 of file dlamchf77.f.
subroutine dlamc2  (  integer  BETA, 
integer  T,  
logical  RND,  
double precision  EPS,  
integer  EMIN,  
double precision  RMIN,  
integer  EMAX,  
double precision  RMAX  
) 
DLAMC2
Purpose:
DLAMC2 determines the machine parameters specified in its argument list.
[out]  BETA  The base of the machine. 
[out]  T  The number of ( BETA ) digits in the mantissa. 
[out]  RND  Specifies whether proper rounding ( RND = .TRUE. ) or chopping ( RND = .FALSE. ) occurs in addition. This may not be a reliable guide to the way in which the machine performs its arithmetic. 
[out]  EPS  The smallest positive number such that fl( 1.0  EPS ) .LT. 1.0, where fl denotes the computed value. 
[out]  EMIN  The minimum exponent before (gradual) underflow occurs. 
[out]  RMIN  The smallest normalized number for the machine, given by BASE**( EMIN  1 ), where BASE is the floating point value of BETA. 
[out]  EMAX  The maximum exponent before overflow occurs. 
[out]  RMAX  The largest positive number for the machine, given by BASE**EMAX * ( 1  EPS ), where BASE is the floating point value of BETA. 
Further Details
The computation of EPS is based on a routine PARANOIA by W. Kahan of the University of California at Berkeley.
Definition at line 419 of file dlamchf77.f.
DOUBLE PRECISION function dlamc3  (  double precision  A, 
double precision  B  
) 
DLAMC3
Purpose:
DLAMC3 is intended to force A and B to be stored prior to doing the addition of A and B , for use in situations where optimizers might hold one of these in a register.
[in]  A  A is a DOUBLE PRECISION 
[in]  B  B is a DOUBLE PRECISION The values A and B. 
Definition at line 168 of file dlamch.f.
DOUBLE PRECISION function dlamch  (  character  CMACH  ) 
DLAMCH
DLAMCHF77 deprecated
DLAMCH determines double precision machine parameters.
[in]  CMACH  Specifies the value to be returned by DLAMCH: = 'E' or 'e', DLAMCH := eps = 'S' or 's , DLAMCH := sfmin = 'B' or 'b', DLAMCH := base = 'P' or 'p', DLAMCH := eps*base = 'N' or 'n', DLAMCH := t = 'R' or 'r', DLAMCH := rnd = 'M' or 'm', DLAMCH := emin = 'U' or 'u', DLAMCH := rmin = 'L' or 'l', DLAMCH := emax = 'O' or 'o', DLAMCH := rmax where eps = relative machine precision sfmin = safe minimum, such that 1/sfmin does not overflow base = base of the machine prec = eps*base t = number of (base) digits in the mantissa rnd = 1.0 when rounding occurs in addition, 0.0 otherwise emin = minimum exponent before (gradual) underflow rmin = underflow threshold  base**(emin1) emax = largest exponent before overflow rmax = overflow threshold  (base**emax)*(1eps) 
DLAMCHF77 determines double precision machine parameters.
[in]  CMACH  Specifies the value to be returned by DLAMCH: = 'E' or 'e', DLAMCH := eps = 'S' or 's , DLAMCH := sfmin = 'B' or 'b', DLAMCH := base = 'P' or 'p', DLAMCH := eps*base = 'N' or 'n', DLAMCH := t = 'R' or 'r', DLAMCH := rnd = 'M' or 'm', DLAMCH := emin = 'U' or 'u', DLAMCH := rmin = 'L' or 'l', DLAMCH := emax = 'O' or 'o', DLAMCH := rmax where eps = relative machine precision sfmin = safe minimum, such that 1/sfmin does not overflow base = base of the machine prec = eps*base t = number of (base) digits in the mantissa rnd = 1.0 when rounding occurs in addition, 0.0 otherwise emin = minimum exponent before (gradual) underflow rmin = underflow threshold  base**(emin1) emax = largest exponent before overflow rmax = overflow threshold  (base**emax)*(1eps) 
Definition at line 64 of file dlamch.f.
INTEGER function dlaneg  (  integer  N, 
double precision, dimension( * )  D,  
double precision, dimension( * )  LLD,  
double precision  SIGMA,  
double precision  PIVMIN,  
integer  R  
) 
DLANEG computes the Sturm count.
Download DLANEG + dependencies [TGZ] [ZIP] [TXT]DLANEG computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T  sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from DLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is nonzero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE754 Features in Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)
[in]  N  N is INTEGER The order of the matrix. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D. 
[in]  LLD  LLD is DOUBLE PRECISION array, dimension (N1) The (N1) elements L(i)*L(i)*D(i). 
[in]  SIGMA  SIGMA is DOUBLE PRECISION Shift amount in T  sigma I = L D L^T. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence. May be used when zero pivots are encountered on nonIEEE754 architectures. 
[in]  R  R is INTEGER The twist index for the twisted factorization that is used for the negcount. 
Definition at line 119 of file dlaneg.f.
DOUBLE PRECISION function dlanst  (  character  NORM, 
integer  N,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E  
) 
DLANST returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
Download DLANST + dependencies [TGZ] [ZIP] [TXT]DLANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A.
DLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
[in]  NORM  NORM is CHARACTER*1 Specifies the value to be returned in DLANST as described above. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. When N = 0, DLANST is set to zero. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The diagonal elements of A. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The (n1) subdiagonal or superdiagonal elements of A. 
Definition at line 101 of file dlanst.f.
DOUBLE PRECISION function dlapy2  (  double precision  X, 
double precision  Y  
) 
DLAPY2 returns sqrt(x2+y2).
Download DLAPY2 + dependencies [TGZ] [ZIP] [TXT]DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary overflow.
[in]  X  X is DOUBLE PRECISION 
[in]  Y  Y is DOUBLE PRECISION X and Y specify the values x and y. 
Definition at line 64 of file dlapy2.f.
DOUBLE PRECISION function dlapy3  (  double precision  X, 
double precision  Y,  
double precision  Z  
) 
DLAPY3 returns sqrt(x2+y2+z2).
Download DLAPY3 + dependencies [TGZ] [ZIP] [TXT]DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause unnecessary overflow.
[in]  X  X is DOUBLE PRECISION 
[in]  Y  Y is DOUBLE PRECISION 
[in]  Z  Z is DOUBLE PRECISION X, Y and Z specify the values x, y and z. 
Definition at line 69 of file dlapy3.f.
subroutine dlarnv  (  integer  IDIST, 
integer, dimension( 4 )  ISEED,  
integer  N,  
double precision, dimension( * )  X  
) 
DLARNV returns a vector of random numbers from a uniform or normal distribution.
Download DLARNV + dependencies [TGZ] [ZIP] [TXT]DLARNV returns a vector of n random real numbers from a uniform or normal distribution.
[in]  IDIST  IDIST is INTEGER Specifies the distribution of the random numbers: = 1: uniform (0,1) = 2: uniform (1,1) = 3: normal (0,1) 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated. 
[in]  N  N is INTEGER The number of random numbers to be generated. 
[out]  X  X is DOUBLE PRECISION array, dimension (N) The generated random numbers. 
This routine calls the auxiliary routine DLARUV to generate random real numbers from a uniform (0,1) distribution, in batches of up to 128 using vectorisable code. The BoxMuller method is used to transform numbers from a uniform to a normal distribution.
Definition at line 98 of file dlarnv.f.
subroutine dlarra  (  integer  N, 
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( * )  E2,  
double precision  SPLTOL,  
double precision  TNRM,  
integer  NSPLIT,  
integer, dimension( * )  ISPLIT,  
integer  INFO  
) 
DLARRA computes the splitting points with the specified threshold.
Download DLARRA + dependencies [TGZ] [ZIP] [TXT]Compute the splitting points with threshold SPLTOL. DLARRA sets any "small" offdiagonal elements to zero.
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T. 
[in,out]  E  E is DOUBLE PRECISION array, dimension (N) On entry, the first (N1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, are set to zero, the other entries of E are untouched. 
[in,out]  E2  E2 is DOUBLE PRECISION array, dimension (N) On entry, the first (N1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zero 
[in]  SPLTOL  SPLTOL is DOUBLE PRECISION The threshold for splitting. Two criteria can be used: SPLTOL<0 : criterion based on absolute offdiagonal value SPLTOL>0 : criterion that preserves relative accuracy 
[in]  TNRM  TNRM is DOUBLE PRECISION The norm of the matrix. 
[out]  NSPLIT  NSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N. 
[out]  ISPLIT  ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLITth consists of rows/columns ISPLIT(NSPLIT1)+1 through ISPLIT(NSPLIT)=N. 
[out]  INFO  INFO is INTEGER = 0: successful exit 
Definition at line 136 of file dlarra.f.
subroutine dlarrb  (  integer  N, 
double precision, dimension( * )  D,  
double precision, dimension( * )  LLD,  
integer  IFIRST,  
integer  ILAST,  
double precision  RTOL1,  
double precision  RTOL2,  
integer  OFFSET,  
double precision, dimension( * )  W,  
double precision, dimension( * )  WGAP,  
double precision, dimension( * )  WERR,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
double precision  PIVMIN,  
double precision  SPDIAM,  
integer  TWIST,  
integer  INFO  
) 
DLARRB provides limited bisection to locate eigenvalues for more accuracy.
Download DLARRB + dependencies [TGZ] [ZIP] [TXT]Given the relatively robust representation(RRR) L D L^T, DLARRB does "limited" bisection to refine the eigenvalues of L D L^T, W( IFIRSTOFFSET ) through W( ILASTOFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses and their gaps are input in WERR and WGAP, respectively. During bisection, intervals [left, right] are maintained by storing their midpoints and semiwidths in the arrays W and WERR respectively.
[in]  N  N is INTEGER The order of the matrix. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D. 
[in]  LLD  LLD is DOUBLE PRECISION array, dimension (N1) The (N1) elements L(i)*L(i)*D(i). 
[in]  IFIRST  IFIRST is INTEGER The index of the first eigenvalue to be computed. 
[in]  ILAST  ILAST is INTEGER The index of the last eigenvalue to be computed. 
[in]  RTOL1  RTOL1 is DOUBLE PRECISION 
[in]  RTOL2  RTOL2 is DOUBLE PRECISION Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHTLEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(LEFT,RIGHT) ) where GAP is the (estimated) distance to the nearest eigenvalue. 
[in]  OFFSET  OFFSET is INTEGER Offset for the arrays W, WGAP and WERR, i.e., the IFIRSTOFFSET through ILASTOFFSET elements of these arrays are to be used. 
[in,out]  W  W is DOUBLE PRECISION array, dimension (N) On input, W( IFIRSTOFFSET ) through W( ILASTOFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST throug ILAST. On output, these estimates are refined. 
[in,out]  WGAP  WGAP is DOUBLE PRECISION array, dimension (N1) On input, the (estimated) gaps between consecutive eigenvalues of L D L^T, i.e., WGAP(IOFFSET) is the gap between eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST then WGAP(IFIRSTOFFSET) must be set to ZERO. On output, these gaps are refined. 
[in,out]  WERR  WERR is DOUBLE PRECISION array, dimension (N) On input, WERR( IFIRSTOFFSET ) through WERR( ILASTOFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (2*N) Workspace. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence. 
[in]  SPDIAM  SPDIAM is DOUBLE PRECISION The spectral diameter of the matrix. 
[in]  TWIST  TWIST is INTEGER The twist index for the twisted factorization that is used for the negcount. TWIST = N: Compute negcount from L D L^T  LAMBDA I = L+ D+ L+^T TWIST = 1: Compute negcount from L D L^T  LAMBDA I = U D U^T TWIST = R: Compute negcount from L D L^T  LAMBDA I = N(r) D(r) N(r) 
[out]  INFO  INFO is INTEGER Error flag. 
Definition at line 195 of file dlarrb.f.
subroutine dlarrc  (  character  JOBT, 
integer  N,  
double precision  VL,  
double precision  VU,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision  PIVMIN,  
integer  EIGCNT,  
integer  LCNT,  
integer  RCNT,  
integer  INFO  
) 
DLARRC computes the number of eigenvalues of the symmetric tridiagonal matrix.
Download DLARRC + dependencies [TGZ] [ZIP] [TXT]Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.
[in]  JOBT  JOBT is CHARACTER*1 = 'T': Compute Sturm count for matrix T. = 'L': Compute Sturm count for matrix L D L^T. 
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in]  VL  VL is DOUBLE PRECISION 
[in]  VU  VU is DOUBLE PRECISION The lower and upper bounds for the eigenvalues. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) JOBT = 'T': The N diagonal elements of the tridiagonal matrix T. JOBT = 'L': The N diagonal elements of the diagonal matrix D. 
[in]  E  E is DOUBLE PRECISION array, dimension (N) JOBT = 'T': The N1 offdiagonal elements of the matrix T. JOBT = 'L': The N1 offdiagonal elements of the matrix L. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence for T. 
[out]  EIGCNT  EIGCNT is INTEGER The number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] 
[out]  LCNT  LCNT is INTEGER 
[out]  RCNT  RCNT is INTEGER The left and right negcounts of the interval. 
[out]  INFO  INFO is INTEGER 
Definition at line 136 of file dlarrc.f.
subroutine dlarrd  (  character  RANGE, 
character  ORDER,  
integer  N,  
double precision  VL,  
double precision  VU,  
integer  IL,  
integer  IU,  
double precision, dimension( * )  GERS,  
double precision  RELTOL,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( * )  E2,  
double precision  PIVMIN,  
integer  NSPLIT,  
integer, dimension( * )  ISPLIT,  
integer  M,  
double precision, dimension( * )  W,  
double precision, dimension( * )  WERR,  
double precision  WL,  
double precision  WU,  
integer, dimension( * )  IBLOCK,  
integer, dimension( * )  INDEXW,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
Download DLARRD + dependencies [TGZ] [ZIP] [TXT]DLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. The user may ask for all eigenvalues, all eigenvalues in the halfopen interval (VL, VU], or the ILth through IUth eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
[in]  RANGE  RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the halfopen interval (VL, VU] will be found. = 'I': ("Index") the ILth through IUth eigenvalues (of the entire matrix) will be found. 
[in]  ORDER  ORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by splitoff block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest. 
[in]  N  N is INTEGER The order of the tridiagonal matrix T. N >= 0. 
[in]  VL  VL is DOUBLE PRECISION 
[in]  VU  VU is DOUBLE PRECISION If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'. 
[in]  IL  IL is INTEGER 
[in]  IU  IU is INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. 
[in]  GERS  GERS is DOUBLE PRECISION array, dimension (2*N) The N Gerschgorin intervals (the ith Gerschgorin interval is (GERS(2*i1), GERS(2*i)). 
[in]  RELTOL  RELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. 
[in]  E  E is DOUBLE PRECISION array, dimension (N1) The (n1) offdiagonal elements of the tridiagonal matrix T. 
[in]  E2  E2 is DOUBLE PRECISION array, dimension (N1) The (n1) squared offdiagonal elements of the tridiagonal matrix T. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence for T. 
[in]  NSPLIT  NSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N. 
[in]  ISPLIT  ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLITth consists of rows/columns ISPLIT(NSPLIT1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.) 
[out]  M  M is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.) 
[out]  W  W is DOUBLE PRECISION array, dimension (N) On exit, the first M elements of W will contain the eigenvalue approximations. DLARRD computes an interval I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue approximation is given as the interval midpoint W(j)= ( a_j + b_j)/2. The corresponding error is bounded by WERR(j) = abs( a_j  b_j)/2 
[out]  WERR  WERR is DOUBLE PRECISION array, dimension (N) The error bound on the corresponding eigenvalue approximation in W. 
[out]  WL  WL is DOUBLE PRECISION 
[out]  WU  WU is DOUBLE PRECISION The interval (WL, WU] contains all the wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin bounds on the spectrum. If RANGE='I', then WL and WU are computed by DLAEBZ from the index range specified. 
[out]  IBLOCK  IBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (DLARRD may use the remaining NM elements as workspace.) 
[out]  INDEXW  INDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= j and IBLOCK(i)=k imply that the ith eigenvalue W(i) is the jth eigenvalue in block k. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (4*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (3*N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1IL Cause: nonmonotonic arithmetic, causing the Sturm sequence to be nonmonotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floatingpoint arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again. 
FUDGE DOUBLE PRECISION, default = 2 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution.
Definition at line 319 of file dlarrd.f.
subroutine dlarre  (  character  RANGE, 
integer  N,  
double precision  VL,  
double precision  VU,  
integer  IL,  
integer  IU,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( * )  E2,  
double precision  RTOL1,  
double precision  RTOL2,  
double precision  SPLTOL,  
integer  NSPLIT,  
integer, dimension( * )  ISPLIT,  
integer  M,  
double precision, dimension( * )  W,  
double precision, dimension( * )  WERR,  
double precision, dimension( * )  WGAP,  
integer, dimension( * )  IBLOCK,  
integer, dimension( * )  INDEXW,  
double precision, dimension( * )  GERS,  
double precision  PIVMIN,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DLARRE given the tridiagonal matrix T, sets small offdiagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
Download DLARRE + dependencies [TGZ] [ZIP] [TXT]To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, DLARRE sets any "small" offdiagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i  sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by DSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to conpute all and then discard any unwanted one. As an added benefit, DLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.
[in]  RANGE  RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the halfopen interval (VL, VU] will be found. = 'I': ("Index") the ILth through IUth eigenvalues (of the entire matrix) will be found. 
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in,out]  VL  VL is DOUBLE PRECISION 
[in,out]  VU  VU is DOUBLE PRECISION If RANGE='V', the lower and upper bounds for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', DLARRE computes bounds on the desired part of the spectrum. 
[in]  IL  IL is INTEGER 
[in]  IU  IU is INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N. 
[in,out]  D  D is DOUBLE PRECISION array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T. On exit, the N diagonal elements of the diagonal matrices D_i. 
[in,out]  E  E is DOUBLE PRECISION array, dimension (N) On entry, the first (N1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, E contains the subdiagonal elements of the unit bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, contain the base points sigma_i on output. 
[in,out]  E2  E2 is DOUBLE PRECISION array, dimension (N) On entry, the first (N1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zero 
[in]  RTOL1  RTOL1 is DOUBLE PRECISION 
[in]  RTOL2  RTOL2 is DOUBLE PRECISION Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHTLEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(LEFT,RIGHT) ) 
[in]  SPLTOL  SPLTOL is DOUBLE PRECISION The threshold for splitting. 
[out]  NSPLIT  NSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N. 
[out]  ISPLIT  ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLITth consists of rows/columns ISPLIT(NSPLIT1)+1 through ISPLIT(NSPLIT)=N. 
[out]  M  M is INTEGER The total number of eigenvalues (of all L_i D_i L_i^T) found. 
[out]  W  W is DOUBLE PRECISION array, dimension (N) The first M elements contain the eigenvalues. The eigenvalues of each of the blocks, L_i D_i L_i^T, are sorted in ascending order ( DLARRE may use the remaining NM elements as workspace). 
[out]  WERR  WERR is DOUBLE PRECISION array, dimension (N) The error bound on the corresponding eigenvalue in W. 
[out]  WGAP  WGAP is DOUBLE PRECISION array, dimension (N) The separation from the right neighbor eigenvalue in W. The gap is only with respect to the eigenvalues of the same block as each block has its own representation tree. Exception: at the right end of a block we store the left gap 
[out]  IBLOCK  IBLOCK is INTEGER array, dimension (N) The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc. 
[out]  INDEXW  INDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the ith eigenvalue W(i) is the 10th eigenvalue in block 2 
[out]  GERS  GERS is DOUBLE PRECISION array, dimension (2*N) The N Gerschgorin intervals (the ith Gerschgorin interval is (GERS(2*i1), GERS(2*i)). 
[out]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence for T. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (6*N) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (5*N) Workspace. 
[out]  INFO  INFO is INTEGER = 0: successful exit > 0: A problem occured in DLARRE. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =1: Problem in DLARRD. = 2: No base representation could be found in MAXTRY iterations. Increasing MAXTRY and recompilation might be a remedy. =3: Problem in DLARRB when computing the refined root representation for DLASQ2. =4: Problem in DLARRB when preforming bisection on the desired part of the spectrum. =5: Problem in DLASQ2. =6: Problem in DLASQ2. 
The base representations are required to suffer very little element growth and consequently define all their eigenvalues to high relative accuracy.
Definition at line 295 of file dlarre.f.
subroutine dlarrf  (  integer  N, 
double precision, dimension( * )  D,  
double precision, dimension( * )  L,  
double precision, dimension( * )  LD,  
integer  CLSTRT,  
integer  CLEND,  
double precision, dimension( * )  W,  
double precision, dimension( * )  WGAP,  
double precision, dimension( * )  WERR,  
double precision  SPDIAM,  
double precision  CLGAPL,  
double precision  CLGAPR,  
double precision  PIVMIN,  
double precision  SIGMA,  
double precision, dimension( * )  DPLUS,  
double precision, dimension( * )  LPLUS,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
Download DLARRF + dependencies [TGZ] [ZIP] [TXT]Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), DLARRF finds a new relatively robust representation L D L^T  SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
[in]  N  N is INTEGER The order of the matrix (subblock, if the matrix splitted). 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D. 
[in]  L  L is DOUBLE PRECISION array, dimension (N1) The (N1) subdiagonal elements of the unit bidiagonal matrix L. 
[in]  LD  LD is DOUBLE PRECISION array, dimension (N1) The (N1) elements L(i)*D(i). 
[in]  CLSTRT  CLSTRT is INTEGER The index of the first eigenvalue in the cluster. 
[in]  CLEND  CLEND is INTEGER The index of the last eigenvalue in the cluster. 
[in]  W  W is DOUBLE PRECISION array, dimension dimension is >= (CLENDCLSTRT+1) The eigenvalue APPROXIMATIONS of L D L^T in ascending order. W( CLSTRT ) through W( CLEND ) form the cluster of relatively close eigenalues. 
[in,out]  WGAP  WGAP is DOUBLE PRECISION array, dimension dimension is >= (CLENDCLSTRT+1) The separation from the right neighbor eigenvalue in W. 
[in]  WERR  WERR is DOUBLE PRECISION array, dimension dimension is >= (CLENDCLSTRT+1) WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in W 
[in]  SPDIAM  SPDIAM is DOUBLE PRECISION estimate of the spectral diameter obtained from the Gerschgorin intervals 
[in]  CLGAPL  CLGAPL is DOUBLE PRECISION 
[in]  CLGAPR  CLGAPR is DOUBLE PRECISION absolute gap on each end of the cluster. Set by the calling routine to protect against shifts too close to eigenvalues outside the cluster. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence. 
[out]  SIGMA  SIGMA is DOUBLE PRECISION The shift used to form L(+) D(+) L(+)^T. 
[out]  DPLUS  DPLUS is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the diagonal matrix D(+). 
[out]  LPLUS  LPLUS is DOUBLE PRECISION array, dimension (N1) The first (N1) elements of LPLUS contain the subdiagonal elements of the unit bidiagonal matrix L(+). 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N) Workspace. 
[out]  INFO  INFO is INTEGER Signals processing OK (=0) or failure (=1) 
Definition at line 191 of file dlarrf.f.
subroutine dlarrj  (  integer  N, 
double precision, dimension( * )  D,  
double precision, dimension( * )  E2,  
integer  IFIRST,  
integer  ILAST,  
double precision  RTOL,  
integer  OFFSET,  
double precision, dimension( * )  W,  
double precision, dimension( * )  WERR,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
double precision  PIVMIN,  
double precision  SPDIAM,  
integer  INFO  
) 
DLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.
Download DLARRJ + dependencies [TGZ] [ZIP] [TXT]Given the initial eigenvalue approximations of T, DLARRJ does bisection to refine the eigenvalues of T, W( IFIRSTOFFSET ) through W( ILASTOFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses in WERR. During bisection, intervals [left, right] are maintained by storing their midpoints and semiwidths in the arrays W and WERR respectively.
[in]  N  N is INTEGER The order of the matrix. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The N diagonal elements of T. 
[in]  E2  E2 is DOUBLE PRECISION array, dimension (N1) The Squares of the (N1) subdiagonal elements of T. 
[in]  IFIRST  IFIRST is INTEGER The index of the first eigenvalue to be computed. 
[in]  ILAST  ILAST is INTEGER The index of the last eigenvalue to be computed. 
[in]  RTOL  RTOL is DOUBLE PRECISION Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHTLEFT.LT.RTOL*MAX(LEFT,RIGHT). 
[in]  OFFSET  OFFSET is INTEGER Offset for the arrays W and WERR, i.e., the IFIRSTOFFSET through ILASTOFFSET elements of these arrays are to be used. 
[in,out]  W  W is DOUBLE PRECISION array, dimension (N) On input, W( IFIRSTOFFSET ) through W( ILASTOFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined. 
[in,out]  WERR  WERR is DOUBLE PRECISION array, dimension (N) On input, WERR( IFIRSTOFFSET ) through WERR( ILASTOFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (2*N) Workspace. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot in the Sturm sequence for T. 
[in]  SPDIAM  SPDIAM is DOUBLE PRECISION The spectral diameter of T. 
[out]  INFO  INFO is INTEGER Error flag. 
Definition at line 167 of file dlarrj.f.
subroutine dlarrk  (  integer  N, 
integer  IW,  
double precision  GL,  
double precision  GU,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E2,  
double precision  PIVMIN,  
double precision  RELTOL,  
double precision  W,  
double precision  WERR,  
integer  INFO  
) 
DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.
Download DLARRK + dependencies [TGZ] [ZIP] [TXT]DLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
[in]  N  N is INTEGER The order of the tridiagonal matrix T. N >= 0. 
[in]  IW  IW is INTEGER The index of the eigenvalues to be returned. 
[in]  GL  GL is DOUBLE PRECISION 
[in]  GU  GU is DOUBLE PRECISION An upper and a lower bound on the eigenvalue. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T. 
[in]  E2  E2 is DOUBLE PRECISION array, dimension (N1) The (n1) squared offdiagonal elements of the tridiagonal matrix T. 
[in]  PIVMIN  PIVMIN is DOUBLE PRECISION The minimum pivot allowed in the Sturm sequence for T. 
[in]  RELTOL  RELTOL is DOUBLE PRECISION The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. 
[out]  W  W is DOUBLE PRECISION 
[out]  WERR  WERR is DOUBLE PRECISION The error bound on the corresponding eigenvalue approximation in W. 
[out]  INFO  INFO is INTEGER = 0: Eigenvalue converged = 1: Eigenvalue did NOT converge 
FUDGE DOUBLE PRECISION, default = 2 A "fudge factor" to widen the Gershgorin intervals.
Definition at line 145 of file dlarrk.f.
subroutine dlarrr  (  integer  N, 
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
integer  INFO  
) 
DLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
Download DLARRR + dependencies [TGZ] [ZIP] [TXT]Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) The N diagonal elements of the tridiagonal matrix T. 
[in,out]  E  E is DOUBLE PRECISION array, dimension (N) On entry, the first (N1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) is set to ZERO. 
[out]  INFO  INFO is INTEGER INFO = 0(default) : the matrix warrants computations preserving relative accuracy. INFO = 1 : the matrix warrants computations guaranteeing only absolute accuracy. 
Definition at line 95 of file dlarrr.f.
subroutine dlartg  (  double precision  F, 
double precision  G,  
double precision  CS,  
double precision  SN,  
double precision  R  
) 
DLARTG generates a plane rotation with real cosine and real sine.
Download DLARTG + dependencies [TGZ] [ZIP] [TXT]DLARTG generate a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the BLAS1 routine DROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any floating point operations (saves work in DBDSQR when there are zeros on the diagonal). If F exceeds G in magnitude, CS will be positive.
[in]  F  F is DOUBLE PRECISION The first component of vector to be rotated. 
[in]  G  G is DOUBLE PRECISION The second component of vector to be rotated. 
[out]  CS  CS is DOUBLE PRECISION The cosine of the rotation. 
[out]  SN  SN is DOUBLE PRECISION The sine of the rotation. 
[out]  R  R is DOUBLE PRECISION The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH. 
Definition at line 98 of file dlartg.f.
subroutine dlartgp  (  double precision  F, 
double precision  G,  
double precision  CS,  
double precision  SN,  
double precision  R  
) 
DLARTGP generates a plane rotation so that the diagonal is nonnegative.
Download DLARTGP + dependencies [TGZ] [ZIP] [TXT]DLARTGP generates a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the Level 1 BLAS routine DROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=(+/)1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=(+/)1. The sign is chosen so that R >= 0.
[in]  F  F is DOUBLE PRECISION The first component of vector to be rotated. 
[in]  G  G is DOUBLE PRECISION The second component of vector to be rotated. 
[out]  CS  CS is DOUBLE PRECISION The cosine of the rotation. 
[out]  SN  SN is DOUBLE PRECISION The sine of the rotation. 
[out]  R  R is DOUBLE PRECISION The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH. 
Definition at line 96 of file dlartgp.f.
subroutine dlaruv  (  integer, dimension( 4 )  ISEED, 
integer  N,  
double precision, dimension( n )  X  
) 
DLARUV returns a vector of n random real numbers from a uniform distribution.
Download DLARUV + dependencies [TGZ] [ZIP] [TXT]DLARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). This is an auxiliary routine called by DLARNV and ZLARNV.
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated. 
[in]  N  N is INTEGER The number of random numbers to be generated. N <= 128. 
[out]  X  X is DOUBLE PRECISION array, dimension (N) The generated random numbers. 
This routine uses a multiplicative congruential method with modulus 2**48 and multiplier 33952834046453 (see G.S.Fishman, 'Multiplicative congruential random number generators with modulus 2**b: an exhaustive analysis for b = 32 and a partial analysis for b = 48', Math. Comp. 189, pp 331344, 1990). 48bit integers are stored in 4 integer array elements with 12 bits per element. Hence the routine is portable across machines with integers of 32 bits or more.
Definition at line 96 of file dlaruv.f.
subroutine dlas2  (  double precision  F, 
double precision  G,  
double precision  H,  
double precision  SSMIN,  
double precision  SSMAX  
) 
DLAS2 computes singular values of a 2by2 triangular matrix.
Download DLAS2 + dependencies [TGZ] [ZIP] [TXT]DLAS2 computes the singular values of the 2by2 matrix [ F G ] [ 0 H ]. On return, SSMIN is the smaller singular value and SSMAX is the larger singular value.
[in]  F  F is DOUBLE PRECISION The (1,1) element of the 2by2 matrix. 
[in]  G  G is DOUBLE PRECISION The (1,2) element of the 2by2 matrix. 
[in]  H  H is DOUBLE PRECISION The (2,2) element of the 2by2 matrix. 
[out]  SSMIN  SSMIN is DOUBLE PRECISION The smaller singular value. 
[out]  SSMAX  SSMAX is DOUBLE PRECISION The larger singular value. 
Barring over/underflow, all output quantities are correct to within a few units in the last place (ulps), even in the absence of a guard digit in addition/subtraction. In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows, or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.
Definition at line 108 of file dlas2.f.
subroutine dlascl  (  character  TYPE, 
integer  KL,  
integer  KU,  
double precision  CFROM,  
double precision  CTO,  
integer  M,  
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
integer  INFO  
) 
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Download DLASCL + dependencies [TGZ] [ZIP] [TXT]DLASCL multiplies the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.
[in]  TYPE  TYPE is CHARACTER*1 TYPE indices the storage type of the input matrix. = 'G': A is a full matrix. = 'L': A is a lower triangular matrix. = 'U': A is an upper triangular matrix. = 'H': A is an upper Hessenberg matrix. = 'B': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the lower half stored. = 'Q': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the upper half stored. = 'Z': A is a band matrix with lower bandwidth KL and upper bandwidth KU. See DGBTRF for storage details. 
[in]  KL  KL is INTEGER The lower bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'. 
[in]  KU  KU is INTEGER The upper bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'. 
[in]  CFROM  CFROM is DOUBLE PRECISION 
[in]  CTO  CTO is DOUBLE PRECISION The matrix A is multiplied by CTO/CFROM. A(I,J) is computed without over/underflow if the final result CTO*A(I,J)/CFROM can be represented without over/underflow. CFROM must be nonzero. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The matrix to be multiplied by CTO/CFROM. See TYPE for the storage type. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[out]  INFO  INFO is INTEGER 0  successful exit <0  if INFO = i, the ith argument had an illegal value. 
Definition at line 140 of file dlascl.f.
subroutine dlasd0  (  integer  N, 
integer  SQRE,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
integer  SMLSIZ,  
integer, dimension( * )  IWORK,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DLASD0 computes the singular values of a real upper bidiagonal nbym matrix B with diagonal d and offdiagonal e. Used by sbdsdc.
Download DLASD0 + dependencies [TGZ] [ZIP] [TXT]Using a divide and conquer approach, DLASD0 computes the singular value decomposition (SVD) of a real upper bidiagonal NbyM matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes orthogonal matrices U and VT such that B = U * S * VT. The singular values S are overwritten on D. A related subroutine, DLASDA, computes only the singular values, and optionally, the singular vectors in compact form.
[in]  N  N is INTEGER On entry, the row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D. 
[in]  SQRE  SQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N+1; 
[in,out]  D  D is DOUBLE PRECISION array, dimension (N) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values. 
[in]  E  E is DOUBLE PRECISION array, dimension (M1) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed. 
[out]  U  U is DOUBLE PRECISION array, dimension at least (LDQ, N) On exit, U contains the left singular vectors. 
[in]  LDU  LDU is INTEGER On entry, leading dimension of U. 
[out]  VT  VT is DOUBLE PRECISION array, dimension at least (LDVT, M) On exit, VT**T contains the right singular vectors. 
[in]  LDVT  LDVT is INTEGER On entry, leading dimension of VT. 
[in]  SMLSIZ  SMLSIZ is INTEGER On entry, maximum size of the subproblems at the bottom of the computation tree. 
[out]  IWORK  IWORK is INTEGER work array. Dimension must be at least (8 * N) 
[out]  WORK  WORK is DOUBLE PRECISION work array. Dimension must be at least (3 * M**2 + 2 * M) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 152 of file dlasd0.f.
subroutine dlasd1  (  integer  NL, 
integer  NR,  
integer  SQRE,  
double precision, dimension( * )  D,  
double precision  ALPHA,  
double precision  BETA,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  IWORK,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
Download DLASD1 + dependencies [TGZ] [ZIP] [TXT]DLASD1 computes the SVD of an upper bidiagonal NbyM matrix B, where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0. A related subroutine DLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired. DLASD1 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. For each such occurence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLASD2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine DLASD4 (as called by DLASD3). This routine also calculates the singular vectors of the current problem. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE. 
[in,out]  D  D is DOUBLE PRECISION array, dimension (N = NL+NR+1). On entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix. 
[in,out]  ALPHA  ALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row. 
[in,out]  BETA  BETA is DOUBLE PRECISION Contains the offdiagonal element associated with the added row. 
[in,out]  U  U is DOUBLE PRECISION array, dimension(LDU,N) On entry U(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. On exit U contains the left singular vectors of the bidiagonal matrix. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max( 1, N ). 
[in,out]  VT  VT is DOUBLE PRECISION array, dimension(LDVT,M) where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. On exit VT**T contains the right singular vectors of the bidiagonal matrix. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= max( 1, M ). 
[out]  IDXQ  IDXQ is INTEGER array, dimension(N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order. 
[out]  IWORK  IWORK is INTEGER array, dimension( 4 * N ) 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M ) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 204 of file dlasd1.f.
subroutine dlasd2  (  integer  NL, 
integer  NR,  
integer  SQRE,  
integer  K,  
double precision, dimension( * )  D,  
double precision, dimension( * )  Z,  
double precision  ALPHA,  
double precision  BETA,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
double precision, dimension( * )  DSIGMA,  
double precision, dimension( ldu2, * )  U2,  
integer  LDU2,  
double precision, dimension( ldvt2, * )  VT2,  
integer  LDVT2,  
integer, dimension( * )  IDXP,  
integer, dimension( * )  IDX,  
integer, dimension( * )  IDXC,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  COLTYP,  
integer  INFO  
) 
DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
Download DLASD2 + dependencies [TGZ] [ZIP] [TXT]DLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. DLASD2 is called from DLASD1.
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[out]  K  K is INTEGER Contains the dimension of the nondeflated matrix, This is the order of the related secular equation. 1 <= K <=N. 
[in,out]  D  D is DOUBLE PRECISION array, dimension(N) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (NK) updated singular values (those which were deflated) sorted into increasing order. 
[out]  Z  Z is DOUBLE PRECISION array, dimension(N) On exit Z contains the updating row vector in the secular equation. 
[in]  ALPHA  ALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row. 
[in]  BETA  BETA is DOUBLE PRECISION Contains the offdiagonal element associated with the added row. 
[in,out]  U  U is DOUBLE PRECISION array, dimension(LDU,N) On entry U contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (NK) updated left singular vectors (those which were deflated) in its last NK columns. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= N. 
[in,out]  VT  VT is DOUBLE PRECISION array, dimension(LDVT,M) On entry VT**T contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT**T contains the trailing (NK) updated right singular vectors (those which were deflated) in its last NK columns. In case SQRE =1, the last row of VT spans the right null space. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= M. 
[out]  DSIGMA  DSIGMA is DOUBLE PRECISION array, dimension (N) Contains a copy of the diagonal elements (K1 singular values and one zero) in the secular equation. 
[out]  U2  U2 is DOUBLE PRECISION array, dimension(LDU2,N) Contains a copy of the first K1 left singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains nonzero entries only at and above NL; the third contains nonzero entries only below NL+1; and the fourth is dense. 
[in]  LDU2  LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N. 
[out]  VT2  VT2 is DOUBLE PRECISION array, dimension(LDVT2,N) VT2**T contains a copy of the first K right singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains nonzeros only at and before NL +1; the third block contains nonzeros only at and after NL +2. 
[in]  LDVT2  LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= M. 
[out]  IDXP  IDXP is INTEGER array dimension(N) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated Dvalues and IDXP(K+1:N) points to the deflated singular values. 
[out]  IDX  IDX is INTEGER array dimension(N) This will contain the permutation used to sort the contents of D into ascending order. 
[out]  IDXC  IDXC is INTEGER array dimension(N) This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains nonzero entries only at and above NL, the second contains nonzero entries only below NL+2, and the third is dense. 
[in,out]  IDXQ  IDXQ is INTEGER array dimension(N) This contains the permutation which separately sorts the two subproblems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values. 
[out]  COLTYP  COLTYP is INTEGER array dimension(N) As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is: 1 : nonzero in the upper half only 2 : nonzero in the lower half only 3 : dense 4 : deflated On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the Ith type columns. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
Definition at line 268 of file dlasd2.f.
subroutine dlasd3  (  integer  NL, 
integer  NR,  
integer  SQRE,  
integer  K,  
double precision, dimension( * )  D,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( * )  DSIGMA,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldu2, * )  U2,  
integer  LDU2,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
double precision, dimension( ldvt2, * )  VT2,  
integer  LDVT2,  
integer, dimension( * )  IDXC,  
integer, dimension( * )  CTOT,  
double precision, dimension( * )  Z,  
integer  INFO  
) 
DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
Download DLASD3 + dependencies [TGZ] [ZIP] [TXT]DLASD3 finds all the square roots of the roots of the secular equation, as defined by the values in D and Z. It makes the appropriate calls to DLASD4 and then updates the singular vectors by matrix multiplication. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. DLASD3 is called from DLASD1.
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[in]  K  K is INTEGER The size of the secular equation, 1 =< K = < N. 
[out]  D  D is DOUBLE PRECISION array, dimension(K) On exit the square roots of the roots of the secular equation, in ascending order. 
[out]  Q  Q is DOUBLE PRECISION array, dimension at least (LDQ,K). 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= K. 
[in]  DSIGMA  DSIGMA is DOUBLE PRECISION array, dimension(K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. 
[out]  U  U is DOUBLE PRECISION array, dimension (LDU, N) The last N  K columns of this matrix contain the deflated left singular vectors. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= N. 
[in,out]  U2  U2 is DOUBLE PRECISION array, dimension (LDU2, N) The first K columns of this matrix contain the nondeflated left singular vectors for the split problem. 
[in]  LDU2  LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N. 
[out]  VT  VT is DOUBLE PRECISION array, dimension (LDVT, M) The last M  K columns of VT**T contain the deflated right singular vectors. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= N. 
[in,out]  VT2  VT2 is DOUBLE PRECISION array, dimension (LDVT2, N) The first K columns of VT2**T contain the nondeflated right singular vectors for the split problem. 
[in]  LDVT2  LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= N. 
[in]  IDXC  IDXC is INTEGER array, dimension ( N ) The permutation used to arrange the columns of U (and rows of VT) into three groups: the first group contains nonzero entries only at and above (or before) NL +1; the second contains nonzero entries only at and below (or after) NL+2; and the third is dense. The first column of U and the row of VT are treated separately, however. The rows of the singular vectors found by DLASD4 must be likewise permuted before the matrix multiplies can take place. 
[in]  CTOT  CTOT is INTEGER array, dimension ( 4 ) A count of the total number of the various types of columns in U (or rows in VT), as described in IDXC. The fourth column type is any column which has been deflated. 
[in]  Z  Z is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflationadjusted updating row vector. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 224 of file dlasd3.f.
subroutine dlasd4  (  integer  N, 
integer  I,  
double precision, dimension( * )  D,  
double precision, dimension( * )  Z,  
double precision, dimension( * )  DELTA,  
double precision  RHO,  
double precision  SIGMA,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DLASD4 computes the square root of the ith updated eigenvalue of a positive symmetric rankone modification to a positive diagonal matrix. Used by sbdsdc.
Download DLASD4 + dependencies [TGZ] [ZIP] [TXT]This subroutine computes the square root of the Ith updated eigenvalue of a positive symmetric rankone modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rankone modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.
[in]  N  N is INTEGER The length of all arrays. 
[in]  I  I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N. 
[in]  D  D is DOUBLE PRECISION array, dimension ( N ) The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J. 
[in]  Z  Z is DOUBLE PRECISION array, dimension ( N ) The components of the updating vector. 
[out]  DELTA  DELTA is DOUBLE PRECISION array, dimension ( N ) If N .ne. 1, DELTA contains (D(j)  sigma_I) in its jth component. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvectors. 
[in]  RHO  RHO is DOUBLE PRECISION The scalar in the symmetric updating formula. 
[out]  SIGMA  SIGMA is DOUBLE PRECISION The computed sigma_I, the Ith updated eigenvalue. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension ( N ) If N .ne. 1, WORK contains (D(j) + sigma_I) in its jth component. If N = 1, then WORK( 1 ) = 1. 
[out]  INFO  INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed. 
Logical variable ORGATI (originati?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switchfor3poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.
Definition at line 154 of file dlasd4.f.
subroutine dlasd5  (  integer  I, 
double precision, dimension( 2 )  D,  
double precision, dimension( 2 )  Z,  
double precision, dimension( 2 )  DELTA,  
double precision  RHO,  
double precision  DSIGMA,  
double precision, dimension( 2 )  WORK  
) 
DLASD5 computes the square root of the ith eigenvalue of a positive symmetric rankone modification of a 2by2 diagonal matrix. Used by sbdsdc.
Download DLASD5 + dependencies [TGZ] [ZIP] [TXT]This subroutine computes the square root of the Ith eigenvalue of a positive symmetric rankone modification of a 2by2 diagonal matrix diag( D ) * diag( D ) + RHO * Z * transpose(Z) . The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.
[in]  I  I is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2. 
[in]  D  D is DOUBLE PRECISION array, dimension ( 2 ) The original eigenvalues. We assume 0 <= D(1) < D(2). 
[in]  Z  Z is DOUBLE PRECISION array, dimension ( 2 ) The components of the updating vector. 
[out]  DELTA  DELTA is DOUBLE PRECISION array, dimension ( 2 ) Contains (D(j)  sigma_I) in its jth component. The vector DELTA contains the information necessary to construct the eigenvectors. 
[in]  RHO  RHO is DOUBLE PRECISION The scalar in the symmetric updating formula. 
[out]  DSIGMA  DSIGMA is DOUBLE PRECISION The computed sigma_I, the Ith updated eigenvalue. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension ( 2 ) WORK contains (D(j) + sigma_I) in its jth component. 
Definition at line 117 of file dlasd5.f.
subroutine dlasd6  (  integer  ICOMPQ, 
integer  NL,  
integer  NR,  
integer  SQRE,  
double precision, dimension( * )  D,  
double precision, dimension( * )  VF,  
double precision, dimension( * )  VL,  
double precision  ALPHA,  
double precision  BETA,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  PERM,  
integer  GIVPTR,  
integer, dimension( ldgcol, * )  GIVCOL,  
integer  LDGCOL,  
double precision, dimension( ldgnum, * )  GIVNUM,  
integer  LDGNUM,  
double precision, dimension( ldgnum, * )  POLES,  
double precision, dimension( * )  DIFL,  
double precision, dimension( * )  DIFR,  
double precision, dimension( * )  Z,  
integer  K,  
double precision  C,  
double precision  S,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
Download DLASD6 + dependencies [TGZ] [ZIP] [TXT]DLASD6 computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row. This routine is used only for the problem which requires all singular values and optionally singular vector matrices in factored form. B is an NbyM matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine, DLASD1, handles the case in which all singular values and singular vectors of the bidiagonal matrix are desired. DLASD6 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The singular values of B can be computed using D1, D2, the first components of all the right singular vectors of the lower block, and the last components of all the right singular vectors of the upper block. These components are stored and updated in VF and VL, respectively, in DLASD6. Hence U and VT are not explicitly referenced. The singular values are stored in D. The algorithm consists of two stages: The first stage consists of deflating the size of the problem when there are multiple singular values or if there is a zero in the Z vector. For each such occurence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLASD7. The second stage consists of calculating the updated singular values. This is done by finding the roots of the secular equation via the routine DLASD4 (as called by DLASD8). This routine also updates VF and VL and computes the distances between the updated singular values and the old singular values. DLASD6 is called from DLASDA.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well. 
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE. 
[in,out]  D  D is DOUBLE PRECISION array, dimension ( NL+NR+1 ). On entry D(1:NL,1:NL) contains the singular values of the upper block, and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix. 
[in,out]  VF  VF is DOUBLE PRECISION array, dimension ( M ) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix. 
[in,out]  VL  VL is DOUBLE PRECISION array, dimension ( M ) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix. 
[in,out]  ALPHA  ALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row. 
[in,out]  BETA  BETA is DOUBLE PRECISION Contains the offdiagonal element associated with the added row. 
[out]  IDXQ  IDXQ is INTEGER array, dimension ( N ) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order. 
[out]  PERM  PERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0. 
[out]  GIVPTR  GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0. 
[out]  GIVCOL  GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGCOL  LDGCOL is INTEGER leading dimension of GIVCOL, must be at least N. 
[out]  GIVNUM  GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGNUM  LDGNUM is INTEGER The leading dimension of GIVNUM and POLES, must be at least N. 
[out]  POLES  POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) On exit, POLES(1,*) is an array containing the new singular values obtained from solving the secular equation, and POLES(2,*) is an array containing the poles in the secular equation. Not referenced if ICOMPQ = 0. 
[out]  DIFL  DIFL is DOUBLE PRECISION array, dimension ( N ) On exit, DIFL(I) is the distance between Ith updated (undeflated) singular value and the Ith (undeflated) old singular value. 
[out]  DIFR  DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. On exit, DIFR(I, 1) is the distance between Ith updated (undeflated) singular value and the I+1th (undeflated) old singular value. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. See DLASD8 for details on DIFL and DIFR. 
[out]  Z  Z is DOUBLE PRECISION array, dimension ( M ) The first elements of this array contain the components of the deflationadjusted updating row vector. 
[out]  K  K is INTEGER Contains the dimension of the nondeflated matrix, This is the order of the related secular equation. 1 <= K <=N. 
[out]  C  C is DOUBLE PRECISION C contains garbage if SQRE =0 and the Cvalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  S  S is DOUBLE PRECISION S contains garbage if SQRE =0 and the Svalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension ( 4 * M ) 
[out]  IWORK  IWORK is INTEGER array, dimension ( 3 * N ) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 312 of file dlasd6.f.
subroutine dlasd7  (  integer  ICOMPQ, 
integer  NL,  
integer  NR,  
integer  SQRE,  
integer  K,  
double precision, dimension( * )  D,  
double precision, dimension( * )  Z,  
double precision, dimension( * )  ZW,  
double precision, dimension( * )  VF,  
double precision, dimension( * )  VFW,  
double precision, dimension( * )  VL,  
double precision, dimension( * )  VLW,  
double precision  ALPHA,  
double precision  BETA,  
double precision, dimension( * )  DSIGMA,  
integer, dimension( * )  IDX,  
integer, dimension( * )  IDXP,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  PERM,  
integer  GIVPTR,  
integer, dimension( ldgcol, * )  GIVCOL,  
integer  LDGCOL,  
double precision, dimension( ldgnum, * )  GIVNUM,  
integer  LDGNUM,  
double precision  C,  
double precision  S,  
integer  INFO  
) 
DLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.
Download DLASD7 + dependencies [TGZ] [ZIP] [TXT]DLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. DLASD7 is called from DLASD6.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows: = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form. 
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[out]  K  K is INTEGER Contains the dimension of the nondeflated matrix, this is the order of the related secular equation. 1 <= K <=N. 
[in,out]  D  D is DOUBLE PRECISION array, dimension ( N ) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (NK) updated singular values (those which were deflated) sorted into increasing order. 
[out]  Z  Z is DOUBLE PRECISION array, dimension ( M ) On exit Z contains the updating row vector in the secular equation. 
[out]  ZW  ZW is DOUBLE PRECISION array, dimension ( M ) Workspace for Z. 
[in,out]  VF  VF is DOUBLE PRECISION array, dimension ( M ) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix. 
[out]  VFW  VFW is DOUBLE PRECISION array, dimension ( M ) Workspace for VF. 
[in,out]  VL  VL is DOUBLE PRECISION array, dimension ( M ) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix. 
[out]  VLW  VLW is DOUBLE PRECISION array, dimension ( M ) Workspace for VL. 
[in]  ALPHA  ALPHA is DOUBLE PRECISION Contains the diagonal element associated with the added row. 
[in]  BETA  BETA is DOUBLE PRECISION Contains the offdiagonal element associated with the added row. 
[out]  DSIGMA  DSIGMA is DOUBLE PRECISION array, dimension ( N ) Contains a copy of the diagonal elements (K1 singular values and one zero) in the secular equation. 
[out]  IDX  IDX is INTEGER array, dimension ( N ) This will contain the permutation used to sort the contents of D into ascending order. 
[out]  IDXP  IDXP is INTEGER array, dimension ( N ) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated Dvalues and IDXP(K+1:N) points to the deflated singular values. 
[in]  IDXQ  IDXQ is INTEGER array, dimension ( N ) This contains the permutation which separately sorts the two subproblems in D into ascending order. Note that entries in the first half of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values. 
[out]  PERM  PERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each singular block. Not referenced if ICOMPQ = 0. 
[out]  GIVPTR  GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0. 
[out]  GIVCOL  GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGCOL  LDGCOL is INTEGER The leading dimension of GIVCOL, must be at least N. 
[out]  GIVNUM  GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGNUM  LDGNUM is INTEGER The leading dimension of GIVNUM, must be at least N. 
[out]  C  C is DOUBLE PRECISION C contains garbage if SQRE =0 and the Cvalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  S  S is DOUBLE PRECISION S contains garbage if SQRE =0 and the Svalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
Definition at line 278 of file dlasd7.f.
subroutine dlasd8  (  integer  ICOMPQ, 
integer  K,  
double precision, dimension( * )  D,  
double precision, dimension( * )  Z,  
double precision, dimension( * )  VF,  
double precision, dimension( * )  VL,  
double precision, dimension( * )  DIFL,  
double precision, dimension( lddifr, * )  DIFR,  
integer  LDDIFR,  
double precision, dimension( * )  DSIGMA,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.
Download DLASD8 + dependencies [TGZ] [ZIP] [TXT]DLASD8 finds the square roots of the roots of the secular equation, as defined by the values in DSIGMA and Z. It makes the appropriate calls to DLASD4, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA). It also updates the arrays VF and VL, the first and last components of all the right singular vectors of the original bidiagonal matrix. DLASD8 is called from DLASD6.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form in the calling routine: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well. 
[in]  K  K is INTEGER The number of terms in the rational function to be solved by DLASD4. K >= 1. 
[out]  D  D is DOUBLE PRECISION array, dimension ( K ) On output, D contains the updated singular values. 
[in,out]  Z  Z is DOUBLE PRECISION array, dimension ( K ) On entry, the first K elements of this array contain the components of the deflationadjusted updating row vector. On exit, Z is updated. 
[in,out]  VF  VF is DOUBLE PRECISION array, dimension ( K ) On entry, VF contains information passed through DBEDE8. On exit, VF contains the first K components of the first components of all right singular vectors of the bidiagonal matrix. 
[in,out]  VL  VL is DOUBLE PRECISION array, dimension ( K ) On entry, VL contains information passed through DBEDE8. On exit, VL contains the first K components of the last components of all right singular vectors of the bidiagonal matrix. 
[out]  DIFL  DIFL is DOUBLE PRECISION array, dimension ( K ) On exit, DIFL(I) = D(I)  DSIGMA(I). 
[out]  DIFR  DIFR is DOUBLE PRECISION array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I)  DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. 
[in]  LDDIFR  LDDIFR is INTEGER The leading dimension of DIFR, must be at least K. 
[in,out]  DSIGMA  DSIGMA is DOUBLE PRECISION array, dimension ( K ) On entry, the first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. On exit, the elements of DSIGMA may be very slightly altered in value. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension at least 3 * K 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 166 of file dlasd8.f.
subroutine dlasda  (  integer  ICOMPQ, 
integer  SMLSIZ,  
integer  N,  
integer  SQRE,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldu, * )  VT,  
integer, dimension( * )  K,  
double precision, dimension( ldu, * )  DIFL,  
double precision, dimension( ldu, * )  DIFR,  
double precision, dimension( ldu, * )  Z,  
double precision, dimension( ldu, * )  POLES,  
integer, dimension( * )  GIVPTR,  
integer, dimension( ldgcol, * )  GIVCOL,  
integer  LDGCOL,  
integer, dimension( ldgcol, * )  PERM,  
double precision, dimension( ldu, * )  GIVNUM,  
double precision, dimension( * )  C,  
double precision, dimension( * )  S,  
double precision, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.
Download DLASDA + dependencies [TGZ] [ZIP] [TXT]Using a divide and conquer approach, DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal NbyM matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes the singular values in the SVD B = U * S * VT. The orthogonal matrices U and VT are optionally computed in compact form. A related subroutine, DLASD0, computes the singular values and the singular vectors in explicit form.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form. 
[in]  SMLSIZ  SMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree. 
[in]  N  N is INTEGER The row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D. 
[in]  SQRE  SQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N + 1. 
[in,out]  D  D is DOUBLE PRECISION array, dimension ( N ) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values. 
[in]  E  E is DOUBLE PRECISION array, dimension ( M1 ) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed. 
[out]  U  U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left singular vector matrices of all subproblems at the bottom level. 
[in]  LDU  LDU is INTEGER, LDU = > N. The leading dimension of arrays U, VT, DIFL, DIFR, POLES, GIVNUM, and Z. 
[out]  VT  VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right singular vector matrices of all subproblems at the bottom level. 
[out]  K  K is INTEGER array, dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1, on exit, K(I) is the dimension of the Ith secular equation on the computation tree. 
[out]  DIFL  DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ), where NLVL = floor(log_2 (N/SMLSIZ))). 
[out]  DIFR  DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I  1) record distances between singular values on the Ith level and singular values on the (I 1)th level, and DIFR(1:N, 2 * I ) contains the normalizing factors for the right singular vector matrix. See DLASD8 for details. 
[out]  Z  Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. The first K elements of Z(1, I) contain the components of the deflationadjusted updating row vector for subproblems on the Ith level. 
[out]  POLES  POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I  1) and POLES(1, 2*I) contain the new and old singular values involved in the secular equations on the Ith level. 
[out]  GIVPTR  GIVPTR is INTEGER array, dimension ( N ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records the number of Givens rotations performed on the Ith problem on the computation tree. 
[out]  GIVCOL  GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVCOL(1, 2 *I  1) and GIVCOL(1, 2 *I) record the locations of Givens rotations performed on the Ith level on the computation tree. 
[in]  LDGCOL  LDGCOL is INTEGER, LDGCOL = > N. The leading dimension of arrays GIVCOL and PERM. 
[out]  PERM  PERM is INTEGER array, dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records permutations done on the Ith level of the computation tree. 
[out]  GIVNUM  GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVNUM(1, 2 *I  1) and GIVNUM(1, 2 *I) record the C and S values of Givens rotations performed on the Ith level on the computation tree. 
[out]  C  C is DOUBLE PRECISION array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the Ith subproblem is not square, on exit, C( I ) contains the Cvalue of a Givens rotation related to the right null space of the Ith subproblem. 
[out]  S  S is DOUBLE PRECISION array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the Ith subproblem is not square, on exit, S( I ) contains the Svalue of a Givens rotation related to the right null space of the Ith subproblem. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). 
[out]  IWORK  IWORK is INTEGER array. Dimension must be at least (7 * N). 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 273 of file dlasda.f.
subroutine dlasdq  (  character  UPLO, 
integer  SQRE,  
integer  N,  
integer  NCVT,  
integer  NRU,  
integer  NCC,  
double precision, dimension( * )  D,  
double precision, dimension( * )  E,  
double precision, dimension( ldvt, * )  VT,  
integer  LDVT,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldc, * )  C,  
integer  LDC,  
double precision, dimension( * )  WORK,  
integer  INFO  
) 
DLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.
Download DLASDQ + dependencies [TGZ] [ZIP] [TXT]DLASDQ computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired. Letting B denote the input bidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose of P). The singular values S are overwritten on D. The input matrix U is changed to U * Q if desired. The input matrix VT is changed to P**T * VT if desired. The input matrix C is changed to Q**T * C if desired. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm.
[in]  UPLO  UPLO is CHARACTER*1 On entry, UPLO specifies whether the input bidiagonal matrix is upper or lower bidiagonal, and wether it is square are not. UPLO = 'U' or 'u' B is upper bidiagonal. UPLO = 'L' or 'l' B is lower bidiagonal. 
[in]  SQRE  SQRE is INTEGER = 0: then the input matrix is NbyN. = 1: then the input matrix is Nby(N+1) if UPLU = 'U' and (N+1)byN if UPLU = 'L'. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[in]  N  N is INTEGER On entry, N specifies the number of rows and columns in the matrix. N must be at least 0. 
[in]  NCVT  NCVT is INTEGER On entry, NCVT specifies the number of columns of the matrix VT. NCVT must be at least 0. 
[in]  NRU  NRU is INTEGER On entry, NRU specifies the number of rows of the matrix U. NRU must be at least 0. 
[in]  NCC  NCC is INTEGER On entry, NCC specifies the number of columns of the matrix C. NCC must be at least 0. 
[in,out]  D  D is DOUBLE PRECISION array, dimension (N) On entry, D contains the diagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in ascending order. 
[in,out]  E  E is DOUBLE PRECISION array. dimension is (N1) if SQRE = 0 and N if SQRE = 1. On entry, the entries of E contain the offdiagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, E will contain 0. If the algorithm does not converge, D and E will contain the diagonal and superdiagonal entries of a bidiagonal matrix orthogonally equivalent to the one given as input. 
[in,out]  VT  VT is DOUBLE PRECISION array, dimension (LDVT, NCVT) On entry, contains a matrix which on exit has been premultiplied by P**T, dimension NbyNCVT if SQRE = 0 and (N+1)byNCVT if SQRE = 1 (not referenced if NCVT=0). 
[in]  LDVT  LDVT is INTEGER On entry, LDVT specifies the leading dimension of VT as declared in the calling (sub) program. LDVT must be at least 1. If NCVT is nonzero LDVT must also be at least N. 
[in,out]  U  U is DOUBLE PRECISION array, dimension (LDU, N) On entry, contains a matrix which on exit has been postmultiplied by Q, dimension NRUbyN if SQRE = 0 and NRUby(N+1) if SQRE = 1 (not referenced if NRU=0). 
[in]  LDU  LDU is INTEGER On entry, LDU specifies the leading dimension of U as declared in the calling (sub) program. LDU must be at least max( 1, NRU ) . 
[in,out]  C  C is DOUBLE PRECISION array, dimension (LDC, NCC) On entry, contains an NbyNCC matrix which on exit has been premultiplied by Q**T dimension NbyNCC if SQRE = 0 and (N+1)byNCC if SQRE = 1 (not referenced if NCC=0). 
[in]  LDC  LDC is INTEGER On entry, LDC specifies the leading dimension of C as declared in the calling (sub) program. LDC must be at least 1. If NCC is nonzero, LDC must also be at least N. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (4*N) Workspace. Only referenced if one of NCVT, NRU, or NCC is nonzero, and if N is at least 2. 
[out]  INFO  INFO is INTEGER On exit, a value of 0 indicates a successful exit. If INFO < 0, argument number INFO is illegal. If INFO > 0, the algorithm did not converge, and INFO specifies how many superdiagonals did not converge. 
Definition at line 211 of file dlasdq.f.
subroutine dlasdt  (  integer  N, 
integer  LVL,  
integer  ND,  
integer, dimension( * )  INODE,  
integer, dimension( * )  NDIML,  
integer, dimension( * )  NDIMR,  
integer  MSUB  
) 
DLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.
Download DLASDT + dependencies [TGZ] [ZIP] [TXT]DLASDT creates a tree of subproblems for bidiagonal divide and conquer.
[in]  N  N is INTEGER On entry, the number of diagonal elements of the bidiagonal matrix. 
[out]  LVL  LVL is INTEGER On exit, the number of levels on the computation tree. 
[out]  ND  ND is INTEGER On exit, the number of nodes on the tree. 
[out]  INODE  INODE is INTEGER array, dimension ( N ) On exit, centers of subproblems. 
[out]  NDIML  NDIML is INTEGER array, dimension ( N ) On exit, row dimensions of left children. 
[out]  NDIMR  NDIMR is INTEGER array, dimension ( N ) On exit, row dimensions of right children. 
[in]  MSUB  MSUB is INTEGER On entry, the maximum row dimension each subproblem at the bottom of the tree can be of. 
Definition at line 106 of file dlasdt.f.
subroutine dlaset  (  character  UPLO, 
integer  M,  
integer  N,  
double precision  ALPHA,  
double precision  BETA,  
double precision, dimension( lda, * )  A,  
integer  LDA  
) 
DLASET initializes the offdiagonal elements and the diagonal elements of a matrix to given values.
Download DLASET + dependencies [TGZ] [ZIP] [TXT]DLASET initializes an mbyn matrix A to BETA on the diagonal and ALPHA on the offdiagonals.
[in]  UPLO  UPLO is CHARACTER*1 Specifies the part of the matrix A to be set. = 'U': Upper triangular part is set; the strictly lower triangular part of A is not changed. = 'L': Lower triangular part is set; the strictly upper triangular part of A is not changed. Otherwise: All of the matrix A is set. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  ALPHA  ALPHA is DOUBLE PRECISION The constant to which the offdiagonal elements are to be set. 
[in]  BETA  BETA is DOUBLE PRECISION The constant to which the diagonal elements are to be set. 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA,N) On exit, the leading mbyn submatrix of A is set as follows: if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j1, 1<=j<=n, if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n, otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j, and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n). 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
Definition at line 111 of file dlaset.f.
subroutine dlasr  (  character  SIDE, 
character  PIVOT,  
character  DIRECT,  
integer  M,  
integer  N,  
double precision, dimension( * )  C,  
double precision, dimension( * )  S,  
double precision, dimension( lda, * )  A,  
integer  LDA  
) 
DLASR applies a sequence of plane rotations to a general rectangular matrix.
Download DLASR + dependencies [TGZ] [ZIP] [TXT]DLASR applies a sequence of plane rotations to a real matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form A := P*A and when SIDE = 'R', the transformation takes the form A := A*P**T where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and P**T is the transpose of P. When DIRECT = 'F' (Forward sequence), then P = P(z1) * ... * P(2) * P(1) and when DIRECT = 'B' (Backward sequence), then P = P(1) * P(2) * ... * P(z1) where P(k) is a plane rotation matrix defined by the 2by2 rotation R(k) = ( c(k) s(k) ) = ( s(k) c(k) ). When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears as a rank2 modification to the identity matrix in rows and columns k and k+1. When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears in rows and columns 1 and k+1. Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( s(k) c(k) ) where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.
[in]  SIDE  SIDE is CHARACTER*1 Specifies whether the plane rotation matrix P is applied to A on the left or the right. = 'L': Left, compute A := P*A = 'R': Right, compute A:= A*P**T 
[in]  PIVOT  PIVOT is CHARACTER*1 Specifies the plane for which P(k) is a plane rotation matrix. = 'V': Variable pivot, the plane (k,k+1) = 'T': Top pivot, the plane (1,k+1) = 'B': Bottom pivot, the plane (k,z) 
[in]  DIRECT  DIRECT is CHARACTER*1 Specifies whether P is a forward or backward sequence of plane rotations. = 'F': Forward, P = P(z1)*...*P(2)*P(1) = 'B': Backward, P = P(1)*P(2)*...*P(z1) 
[in]  M  M is INTEGER The number of rows of the matrix A. If m <= 1, an immediate return is effected. 
[in]  N  N is INTEGER The number of columns of the matrix A. If n <= 1, an immediate return is effected. 
[in]  C  C is DOUBLE PRECISION array, dimension (M1) if SIDE = 'L' (N1) if SIDE = 'R' The cosines c(k) of the plane rotations. 
[in]  S  S is DOUBLE PRECISION array, dimension (M1) if SIDE = 'L' (N1) if SIDE = 'R' The sines s(k) of the plane rotations. The 2by2 plane rotation part of the matrix P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( s(k) c(k) ). 
[in,out]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The MbyN matrix A. On exit, A is overwritten by P*A if SIDE = 'R' or by A*P**T if SIDE = 'L'. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
Definition at line 200 of file dlasr.f.
subroutine dlassq  (  integer  N, 
double precision, dimension( * )  X,  
integer  INCX,  
double precision  SCALE,  
double precision  SUMSQ  
) 
DLASSQ updates a sum of squares represented in scaled form.
Download DLASSQ + dependencies [TGZ] [ZIP] [TXT]DLASSQ returns the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, where x( i ) = X( 1 + ( i  1 )*INCX ). The value of sumsq is assumed to be nonnegative and scl returns the value scl = max( scale, abs( x( i ) ) ). scale and sumsq must be supplied in SCALE and SUMSQ and scl and smsq are overwritten on SCALE and SUMSQ respectively. The routine makes only one pass through the vector x.
[in]  N  N is INTEGER The number of elements to be used from the vector X. 
[in]  X  X is DOUBLE PRECISION array, dimension (N) The vector for which a scaled sum of squares is computed. x( i ) = X( 1 + ( i  1 )*INCX ), 1 <= i <= n. 
[in]  INCX  INCX is INTEGER The increment between successive values of the vector X. INCX > 0. 
[in,out]  SCALE  SCALE is DOUBLE PRECISION On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares. 
[in,out]  SUMSQ  SUMSQ is DOUBLE PRECISION On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out. 
Definition at line 104 of file dlassq.f.
subroutine dlasv2  (  double precision  F, 
double precision  G,  
double precision  H,  
double precision  SSMIN,  
double precision  SSMAX,  
double precision  SNR,  
double precision  CSR,  
double precision  SNL,  
double precision  CSL  
) 
DLASV2 computes the singular value decomposition of a 2by2 triangular matrix.
Download DLASV2 + dependencies [TGZ] [ZIP] [TXT]DLASV2 computes the singular value decomposition of a 2by2 triangular matrix [ F G ] [ 0 H ]. On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and right singular vectors for abs(SSMAX), giving the decomposition [ CSL SNL ] [ F G ] [ CSR SNR ] = [ SSMAX 0 ] [SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].
[in]  F  F is DOUBLE PRECISION The (1,1) element of the 2by2 matrix. 
[in]  G  G is DOUBLE PRECISION The (1,2) element of the 2by2 matrix. 
[in]  H  H is DOUBLE PRECISION The (2,2) element of the 2by2 matrix. 
[out]  SSMIN  SSMIN is DOUBLE PRECISION abs(SSMIN) is the smaller singular value. 
[out]  SSMAX  SSMAX is DOUBLE PRECISION abs(SSMAX) is the larger singular value. 
[out]  SNL  SNL is DOUBLE PRECISION 
[out]  CSL  CSL is DOUBLE PRECISION The vector (CSL, SNL) is a unit left singular vector for the singular value abs(SSMAX). 
[out]  SNR  SNR is DOUBLE PRECISION 
[out]  CSR  CSR is DOUBLE PRECISION The vector (CSR, SNR) is a unit right singular vector for the singular value abs(SSMAX). 
Any input parameter may be aliased with any output parameter. Barring over/underflow and assuming a guard digit in subtraction, all output quantities are correct to within a few units in the last place (ulps). In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.
Definition at line 139 of file dlasv2.f.
DOUBLE PRECISION function dsecnd  (  ) 
DSECND Using ETIME
DSECND returns nothing
DSECND Using the INTERNAL function ETIME.
DSECND Using INTERNAL function CPU_TIME.
DSECND Using ETIME_
DSECND returns the user time for a process in seconds. This version gets the time from the EXTERNAL system function ETIME.
DSECND returns the user time for a process in seconds. This version gets the time from the system function ETIME_.
DSECND returns the user time for a process in seconds. This version gets the time from the INTERNAL function CPU_TIME.
DSECND returns the user time for a process in seconds. This version gets the time from the INTERNAL function ETIME.
DSECND returns nothing instead of returning the user time for a process in seconds. If you are using that routine, it means that neither EXTERNAL ETIME, EXTERNAL ETIME_, INTERNAL ETIME, INTERNAL CPU_TIME is available on your machine.
Definition at line 36 of file dsecnd_EXT_ETIME.f.
INTEGER function ieeeck  (  integer  ISPEC, 
real  ZERO,  
real  ONE  
) 
IEEECK
Download IEEECK + dependencies [TGZ] [ZIP] [TXT]IEEECK is called from the ILAENV to verify that Infinity and possibly NaN arithmetic is safe (i.e. will not trap).
[in]  ISPEC  ISPEC is INTEGER Specifies whether to test just for inifinity arithmetic or whether to test for infinity and NaN arithmetic. = 0: Verify infinity arithmetic only. = 1: Verify infinity and NaN arithmetic. 
[in]  ZERO  ZERO is REAL Must contain the value 0.0 This is passed to prevent the compiler from optimizing away this code. 
[in]  ONE  ONE is REAL Must contain the value 1.0 This is passed to prevent the compiler from optimizing away this code. RETURN VALUE: INTEGER = 0: Arithmetic failed to produce the correct answers = 1: Arithmetic produced the correct answers 
Definition at line 83 of file ieeeck.f.
INTEGER function iladlc  (  integer  M, 
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA  
) 
ILADLC scans a matrix for its last nonzero column.
Download ILADLC + dependencies [TGZ] [ZIP] [TXT]ILADLC scans A for its last nonzero column.
[in]  M  M is INTEGER The number of rows of the matrix A. 
[in]  N  N is INTEGER The number of columns of the matrix A. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
Definition at line 79 of file iladlc.f.
INTEGER function iladlr  (  integer  M, 
integer  N,  
double precision, dimension( lda, * )  A,  
integer  LDA  
) 
ILADLR scans a matrix for its last nonzero row.
Download ILADLR + dependencies [TGZ] [ZIP] [TXT]ILADLR scans A for its last nonzero row.
[in]  M  M is INTEGER The number of rows of the matrix A. 
[in]  N  N is INTEGER The number of columns of the matrix A. 
[in]  A  A is DOUBLE PRECISION array, dimension (LDA,N) The m by n matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
Definition at line 79 of file iladlr.f.
INTEGER function ilaenv  (  integer  ISPEC, 
character*( * )  NAME,  
character*( * )  OPTS,  
integer  N1,  
integer  N2,  
integer  N3,  
integer  N4  
) 
ILAENV
Download ILAENV + dependencies [TGZ] [ZIP] [TXT]ILAENV is called from the LAPACK routines to choose problemdependent parameters for the local environment. See ISPEC for a description of the parameters. ILAENV returns an INTEGER if ILAENV >= 0: ILAENV returns the value of the parameter specified by ISPEC if ILAENV < 0: if ILAENV = k, the kth argument had an illegal value. This version provides a set of parameters which should give good, but not optimal, performance on many of the currently available computers. Users are encouraged to modify this subroutine to set the tuning parameters for their particular machine using the option and problem size information in the arguments. This routine will not function correctly if it is converted to all lower case. Converting it to all upper case is allowed.
[in]  ISPEC  ISPEC is INTEGER Specifies the parameter to be returned as the value of ILAENV. = 1: the optimal blocksize; if this value is 1, an unblocked algorithm will give the best performance. = 2: the minimum block size for which the block routine should be used; if the usable block size is less than this value, an unblocked routine should be used. = 3: the crossover point (in a block routine, for N less than this value, an unblocked routine should be used) = 4: the number of shifts, used in the nonsymmetric eigenvalue routines (DEPRECATED) = 5: the minimum column dimension for blocking to be used; rectangular blocks must have dimension at least k by m, where k is given by ILAENV(2,...) and m by ILAENV(5,...) = 6: the crossover point for the SVD (when reducing an m by n matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds this value, a QR factorization is used first to reduce the matrix to a triangular form.) = 7: the number of processors = 8: the crossover point for the multishift QR method for nonsymmetric eigenvalue problems (DEPRECATED) = 9: maximum size of the subproblems at the bottom of the computation tree in the divideandconquer algorithm (used by xGELSD and xGESDD) =10: ieee NaN arithmetic can be trusted not to trap =11: infinity arithmetic can be trusted not to trap 12 <= ISPEC <= 16: xHSEQR or one of its subroutines, see IPARMQ for detailed explanation 
[in]  NAME  NAME is CHARACTER*(*) The name of the calling subroutine, in either upper case or lower case. 
[in]  OPTS  OPTS is CHARACTER*(*) The character options to the subroutine NAME, concatenated into a single character string. For example, UPLO = 'U', TRANS = 'T', and DIAG = 'N' for a triangular routine would be specified as OPTS = 'UTN'. 
[in]  N1  N1 is INTEGER 
[in]  N2  N2 is INTEGER 
[in]  N3  N3 is INTEGER 
[in]  N4  N4 is INTEGER Problem dimensions for the subroutine NAME; these may not all be required. 
The following conventions have been used when calling ILAENV from the LAPACK routines: 1) OPTS is a concatenation of all of the character options to subroutine NAME, in the same order that they appear in the argument list for NAME, even if they are not used in determining the value of the parameter specified by ISPEC. 2) The problem dimensions N1, N2, N3, N4 are specified in the order that they appear in the argument list for NAME. N1 is used first, N2 second, and so on, and unused problem dimensions are passed a value of 1. 3) The parameter value returned by ILAENV is checked for validity in the calling subroutine. For example, ILAENV is used to retrieve the optimal blocksize for STRTRI as follows: NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, 1, 1, 1 ) IF( NB.LE.1 ) NB = MAX( 1, N )
Definition at line 163 of file ilaenv.f.
subroutine ilaver  (  integer  VERS_MAJOR, 
integer  VERS_MINOR,  
integer  VERS_PATCH  
) 
ILAVER returns the LAPACK version.
This subroutine returns the LAPACK version.
[out]  VERS_MAJOR  return the lapack major version 
[out]  VERS_MINOR  return the lapack minor version from the major version 
[out]  VERS_PATCH  return the lapack patch version from the minor version 
Definition at line 49 of file ilaver.f.
INTEGER function iparmq  (  integer  ISPEC, 
character, dimension( * )  NAME,  
character, dimension( * )  OPTS,  
integer  N,  
integer  ILO,  
integer  IHI,  
integer  LWORK  
) 
IPARMQ
Download IPARMQ + dependencies [TGZ] [ZIP] [TXT]This program sets problem and machine dependent parameters useful for xHSEQR and its subroutines. It is called whenever ILAENV is called with 12 <= ISPEC <= 16
[in]  ISPEC  ISPEC is integer scalar ISPEC specifies which tunable parameter IPARMQ should return. ISPEC=12: (INMIN) Matrices of order nmin or less are sent directly to xLAHQR, the implicit double shift QR algorithm. NMIN must be at least 11. ISPEC=13: (INWIN) Size of the deflation window. This is best set greater than or equal to the number of simultaneous shifts NS. Larger matrices benefit from larger deflation windows. ISPEC=14: (INIBL) Determines when to stop nibbling and invest in an (expensive) multishift QR sweep. If the aggressive early deflation subroutine finds LD converged eigenvalues from an order NW deflation window and LD.GT.(NW*NIBBLE)/100, then the next QR sweep is skipped and early deflation is applied immediately to the remaining active diagonal block. Setting IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a multishift QR sweep whenever early deflation finds a converged eigenvalue. Setting IPARMQ(ISPEC=14) greater than or equal to 100 prevents TTQRE from skipping a multishift QR sweep. ISPEC=15: (NSHFTS) The number of simultaneous shifts in a multishift QR iteration. ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the following meanings. 0: During the multishift QR sweep, xLAQR5 does not accumulate reflections and does not use matrixmatrix multiply to update the farfromdiagonal matrix entries. 1: During the multishift QR sweep, xLAQR5 and/or xLAQRaccumulates reflections and uses matrixmatrix multiply to update the farfromdiagonal matrix entries. 2: During the multishift QR sweep. xLAQR5 accumulates reflections and takes advantage of 2by2 block structure during matrixmatrix multiplies. (If xTRMM is slower than xGEMM, then IPARMQ(ISPEC=16)=1 may be more efficient than IPARMQ(ISPEC=16)=2 despite the greater level of arithmetic work implied by the latter choice.) 
[in]  NAME  NAME is character string Name of the calling subroutine 
[in]  OPTS  OPTS is character string This is a concatenation of the string arguments to TTQRE. 
[in]  N  N is integer scalar N is the order of the Hessenberg matrix H. 
[in]  ILO  ILO is INTEGER 
[in]  IHI  IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO1 and IHI+1:N. 
[in]  LWORK  LWORK is integer scalar The amount of workspace available. 
Little is known about how best to choose these parameters. It is possible to use different values of the parameters for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR. It is probably best to choose different parameters for different matrices and different parameters at different times during the iteration, but this has not been implemented  yet. The best choices of most of the parameters depend in an illunderstood way on the relative execution rate of xLAQR3 and xLAQR5 and on the nature of each particular eigenvalue problem. Experiment may be the only practical way to determine which choices are most effective. Following is a list of default values supplied by IPARMQ. These defaults may be adjusted in order to attain better performance in any particular computational environment. IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point. Default: 75. (Must be at least 11.) IPARMQ(ISPEC=13) Recommended deflation window size. This depends on ILO, IHI and NS, the number of simultaneous shifts returned by IPARMQ(ISPEC=15). The default for (IHIILO+1).LE.500 is NS. The default for (IHIILO+1).GT.500 is 3*NS/2. IPARMQ(ISPEC=14) Nibble crossover point. Default: 14. IPARMQ(ISPEC=15) Number of simultaneous shifts, NS. a multishift QR iteration. If IHIILO+1 is ... greater than ...but less ... the or equal to ... than default is 0 30 NS = 2+ 30 60 NS = 4+ 60 150 NS = 10 150 590 NS = ** 590 3000 NS = 64 3000 6000 NS = 128 6000 infinity NS = 256 (+) By default matrices of this order are passed to the implicit double shift routine xLAHQR. See IPARMQ(ISPEC=12) above. These values of NS are used only in case of a rare xLAHQR failure. (**) The asterisks (**) indicate an adhoc function increasing from 10 to 64. IPARMQ(ISPEC=16) Select structured matrix multiply. (See ISPEC=16 above for details.) Default: 3.
Definition at line 215 of file iparmq.f.
LOGICAL function lsame  (  character  CA, 
character  CB  
) 
LOGICAL function lsamen  (  integer  N, 
character*( * )  CA,  
character*( * )  CB  
) 
LSAMEN
Download LSAMEN + dependencies [TGZ] [ZIP] [TXT]LSAMEN tests if the first N letters of CA are the same as the first N letters of CB, regardless of case. LSAMEN returns .TRUE. if CA and CB are equivalent except for case and .FALSE. otherwise. LSAMEN also returns .FALSE. if LEN( CA ) or LEN( CB ) is less than N.
[in]  N  N is INTEGER The number of characters in CA and CB to be compared. 
[in]  CA  CA is CHARACTER*(*) 
[in]  CB  CB is CHARACTER*(*) CA and CB specify two character strings of length at least N. Only the first N characters of each string will be accessed. 
Definition at line 75 of file lsamen.f.
REAL function second  (  ) 
SECOND Using ETIME
SECOND returns nothing
SECOND Using the INTERNAL function ETIME.
SECOND Using INTERNAL function CPU_TIME.
SECOND Using ETIME_
SECOND returns the user time for a process in seconds. This version gets the time from the EXTERNAL system function ETIME.
SECOND returns the user time for a process in seconds. This version gets the time from the system function ETIME_.
SECOND returns the user time for a process in seconds. This version gets the time from the INTERNAL function CPU_TIME.
SECOND returns the user time for a process in seconds. This version gets the time from the INTERNAL function ETIME.
SECOND returns nothing instead of returning the user time for a process in seconds. If you are using that routine, it means that neither EXTERNAL ETIME, EXTERNAL ETIME_, INTERNAL ETIME, INTERNAL CPU_TIME is available on your machine.
Definition at line 36 of file second_EXT_ETIME.f.
LOGICAL function sisnan  (  real  SIN  ) 
SISNAN tests input for NaN.
Download SISNAN + dependencies [TGZ] [ZIP] [TXT]SISNAN returns .TRUE. if its argument is NaN, and .FALSE. otherwise. To be replaced by the Fortran 2003 intrinsic in the future.
[in]  SIN  SIN is REAL Input to test for NaN. 
Definition at line 60 of file sisnan.f.
subroutine slabad  (  real  SMALL, 
real  LARGE  
) 
SLABAD
Download SLABAD + dependencies [TGZ] [ZIP] [TXT]SLABAD takes as input the values computed by SLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by SLAMCH. This subroutine is needed because SLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray.
[in,out]  SMALL  SMALL is REAL On entry, the underflow threshold as computed by SLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of SMALL, otherwise unchanged. 
[in,out]  LARGE  LARGE is REAL On entry, the overflow threshold as computed by SLAMCH. On exit, if LOG10(LARGE) is sufficiently large, the square root of LARGE, otherwise unchanged. 
subroutine slacpy  (  character  UPLO, 
integer  M,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
real, dimension( ldb, * )  B,  
integer  LDB  
) 
SLACPY copies all or part of one twodimensional array to another.
Download SLACPY + dependencies [TGZ] [ZIP] [TXT]SLACPY copies all or part of a twodimensional matrix A to another matrix B.
[in]  UPLO  UPLO is CHARACTER*1 Specifies the part of the matrix A to be copied to B. = 'U': Upper triangular part = 'L': Lower triangular part Otherwise: All of the matrix A 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  A  A is REAL array, dimension (LDA,N) The m by n matrix A. If UPLO = 'U', only the upper triangle or trapezoid is accessed; if UPLO = 'L', only the lower triangle or trapezoid is accessed. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[out]  B  B is REAL array, dimension (LDB,N) On exit, B = A in the locations specified by UPLO. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). 
Definition at line 104 of file slacpy.f.
subroutine sladiv  (  real  A, 
real  B,  
real  C,  
real  D,  
real  P,  
real  Q  
) 
SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
Download SLADIV + dependencies [TGZ] [ZIP] [TXT]SLADIV performs complex division in real arithmetic a + i*b p + i*q =  c + i*d The algorithm is due to Robert L. Smith and can be found in D. Knuth, The art of Computer Programming, Vol.2, p.195
[in]  A  A is REAL 
[in]  B  B is REAL 
[in]  C  C is REAL 
[in]  D  D is REAL The scalars a, b, c, and d in the above expression. 
[out]  P  P is REAL 
[out]  Q  Q is REAL The scalars p and q in the above expression. 
Definition at line 91 of file sladiv.f.
subroutine slae2  (  real  A, 
real  B,  
real  C,  
real  RT1,  
real  RT2  
) 
SLAE2 computes the eigenvalues of a 2by2 symmetric matrix.
Download SLAE2 + dependencies [TGZ] [ZIP] [TXT]SLAE2 computes the eigenvalues of a 2by2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, and RT2 is the eigenvalue of smaller absolute value.
[in]  A  A is REAL The (1,1) element of the 2by2 matrix. 
[in]  B  B is REAL The (1,2) and (2,1) elements of the 2by2 matrix. 
[in]  C  C is REAL The (2,2) element of the 2by2 matrix. 
[out]  RT1  RT1 is REAL The eigenvalue of larger absolute value. 
[out]  RT2  RT2 is REAL The eigenvalue of smaller absolute value. 
RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*CB*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.
Definition at line 103 of file slae2.f.
subroutine slaebz  (  integer  IJOB, 
integer  NITMAX,  
integer  N,  
integer  MMAX,  
integer  MINP,  
integer  NBMIN,  
real  ABSTOL,  
real  RELTOL,  
real  PIVMIN,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( * )  E2,  
integer, dimension( * )  NVAL,  
real, dimension( mmax, * )  AB,  
real, dimension( * )  C,  
integer  MOUT,  
integer, dimension( mmax, * )  NAB,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SLAEBZ computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine sstebz.
Download SLAEBZ + dependencies [TGZ] [ZIP] [TXT]SLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: IJOB=1, followed by IJOB=2: It takes as input a list of intervals and returns a list of sufficiently small intervals whose union contains the same eigenvalues as the union of the original intervals. The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP. The output interval (AB(j,1),AB(j,2)] will contain eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT. IJOB=3: It performs a binary search in each input interval (AB(j,1),AB(j,2)] for a point w(j) such that N(w(j))=NVAL(j), and uses C(j) as the starting point of the search. If such a w(j) is found, then on output AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output (AB(j,1),AB(j,2)] will be a small interval containing the point where N(w) jumps through NVAL(j), unless that point lies outside the initial interval. Note that the intervals are in all cases halfopen intervals, i.e., of the form (a,b] , which includes b but not a . To avoid underflow, the matrix should be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value. To assure the most accurate computation of small eigenvalues, the matrix should be scaled to be not much smaller than that, either. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966 Note: the arguments are, in general, *not* checked for unreasonable values.
[in]  IJOB  IJOB is INTEGER Specifies what is to be done: = 1: Compute NAB for the initial intervals. = 2: Perform bisection iteration to find eigenvalues of T. = 3: Perform bisection iteration to invert N(w), i.e., to find a point which has a specified number of eigenvalues of T to its left. Other values will cause SLAEBZ to return with INFO=1. 
[in]  NITMAX  NITMAX is INTEGER The maximum number of "levels" of bisection to be performed, i.e., an interval of width W will not be made smaller than 2^(NITMAX) * W. If not all intervals have converged after NITMAX iterations, then INFO is set to the number of nonconverged intervals. 
[in]  N  N is INTEGER The dimension n of the tridiagonal matrix T. It must be at least 1. 
[in]  MMAX  MMAX is INTEGER The maximum number of intervals. If more than MMAX intervals are generated, then SLAEBZ will quit with INFO=MMAX+1. 
[in]  MINP  MINP is INTEGER The initial number of intervals. It may not be greater than MMAX. 
[in]  NBMIN  NBMIN is INTEGER The smallest number of intervals that should be processed using a vector loop. If zero, then only the scalar loop will be used. 
[in]  ABSTOL  ABSTOL is REAL The minimum (absolute) width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. This must be at least zero. 
[in]  RELTOL  RELTOL is REAL The minimum relative width of an interval. When an interval is narrower than ABSTOL, or than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. 
[in]  PIVMIN  PIVMIN is REAL The minimum absolute value of a "pivot" in the Sturm sequence loop. This must be at least max e(j)**2*safe_min and at least safe_min, where safe_min is at least the smallest number that can divide one without overflow. 
[in]  D  D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T. 
[in]  E  E is REAL array, dimension (N) The offdiagonal elements of the tridiagonal matrix T in positions 1 through N1. E(N) is arbitrary. 
[in]  E2  E2 is REAL array, dimension (N) The squares of the offdiagonal elements of the tridiagonal matrix T. E2(N) is ignored. 
[in,out]  NVAL  NVAL is INTEGER array, dimension (MINP) If IJOB=1 or 2, not referenced. If IJOB=3, the desired values of N(w). The elements of NVAL will be reordered to correspond with the intervals in AB. Thus, NVAL(j) on output will not, in general be the same as NVAL(j) on input, but it will correspond with the interval (AB(j,1),AB(j,2)] on output. 
[in,out]  AB  AB is REAL array, dimension (MMAX,2) The endpoints of the intervals. AB(j,1) is a(j), the left endpoint of the jth interval, and AB(j,2) is b(j), the right endpoint of the jth interval. The input intervals will, in general, be modified, split, and reordered by the calculation. 
[in,out]  C  C is REAL array, dimension (MMAX) If IJOB=1, ignored. If IJOB=2, workspace. If IJOB=3, then on input C(j) should be initialized to the first search point in the binary search. 
[out]  MOUT  MOUT is INTEGER If IJOB=1, the number of eigenvalues in the intervals. If IJOB=2 or 3, the number of intervals output. If IJOB=3, MOUT will equal MINP. 
[in,out]  NAB  NAB is INTEGER array, dimension (MMAX,2) If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)). If IJOB=2, then on input, NAB(i,j) should be set. It must satisfy the condition: N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)), which means that in interval i only eigenvalues NAB(i,1)+1,...,NAB(i,2) will be considered. Usually, NAB(i,j)=N(AB(i,j)), from a previous call to SLAEBZ with IJOB=1. On output, NAB(i,j) will contain max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of the input interval that the output interval (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the the input values of NAB(k,1) and NAB(k,2). If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)), unless N(w) > NVAL(i) for all search points w , in which case NAB(i,1) will not be modified, i.e., the output value will be the same as the input value (modulo reorderings  see NVAL and AB), or unless N(w) < NVAL(i) for all search points w , in which case NAB(i,2) will not be modified. Normally, NAB should be set to some distinctive value(s) before SLAEBZ is called. 
[out]  WORK  WORK is REAL array, dimension (MMAX) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (MMAX) Workspace. 
[out]  INFO  INFO is INTEGER = 0: All intervals converged. = 1MMAX: The last INFO intervals did not converge. = MMAX+1: More than MMAX intervals were generated. 
This routine is intended to be called only by other LAPACK routines, thus the interface is less userfriendly. It is intended for two purposes: (a) finding eigenvalues. In this case, SLAEBZ should have one or more initial intervals set up in AB, and SLAEBZ should be called with IJOB=1. This sets up NAB, and also counts the eigenvalues. Intervals with no eigenvalues would usually be thrown out at this point. Also, if not all the eigenvalues in an interval i are desired, NAB(i,1) can be increased or NAB(i,2) decreased. For example, set NAB(i,1)=NAB(i,2)1 to get the largest eigenvalue. SLAEBZ is then called with IJOB=2 and MMAX no smaller than the value of MOUT returned by the call with IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1 through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the tolerance specified by ABSTOL and RELTOL. (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l). In this case, start with a Gershgorin interval (a,b). Set up AB to contain 2 search intervals, both initially (a,b). One NVAL element should contain f1 and the other should contain l , while C should contain a and b, resp. NAB(i,1) should be 1 and NAB(i,2) should be N+1, to flag an error if the desired interval does not lie in (a,b). SLAEBZ is then called with IJOB=3. On exit, if w(f1) < w(f), then one of the intervals  j  will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f1, while if, to the specified tolerance, w(fk)=...=w(f+r), k > 0 and r >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=fk and N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and w(lr)=...=w(l+k) are handled similarly.
Definition at line 318 of file slaebz.f.
subroutine slaev2  (  real  A, 
real  B,  
real  C,  
real  RT1,  
real  RT2,  
real  CS1,  
real  SN1  
) 
SLAEV2 computes the eigenvalues and eigenvectors of a 2by2 symmetric/Hermitian matrix.
Download SLAEV2 + dependencies [TGZ] [ZIP] [TXT]SLAEV2 computes the eigendecomposition of a 2by2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition [ CS1 SN1 ] [ A B ] [ CS1 SN1 ] = [ RT1 0 ] [SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
[in]  A  A is REAL The (1,1) element of the 2by2 matrix. 
[in]  B  B is REAL The (1,2) element and the conjugate of the (2,1) element of the 2by2 matrix. 
[in]  C  C is REAL The (2,2) element of the 2by2 matrix. 
[out]  RT1  RT1 is REAL The eigenvalue of larger absolute value. 
[out]  RT2  RT2 is REAL The eigenvalue of smaller absolute value. 
[out]  CS1  CS1 is REAL 
[out]  SN1  SN1 is REAL The vector (CS1, SN1) is a unit right eigenvector for RT1. 
RT1 is accurate to a few ulps barring over/underflow. RT2 may be inaccurate if there is massive cancellation in the determinant A*CB*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases. CS1 and SN1 are accurate to a few ulps barring over/underflow. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds underflow_threshold / macheps.
Definition at line 121 of file slaev2.f.
subroutine slag2d  (  integer  M, 
integer  N,  
real, dimension( ldsa, * )  SA,  
integer  LDSA,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
integer  INFO  
) 
SLAG2D converts a single precision matrix to a double precision matrix.
Download SLAG2D + dependencies [TGZ] [ZIP] [TXT]SLAG2D converts a SINGLE PRECISION matrix, SA, to a DOUBLE PRECISION matrix, A. Note that while it is possible to overflow while converting from double to single, it is not possible to overflow when converting from single to double. This is an auxiliary routine so there is no argument checking.
[in]  M  M is INTEGER The number of lines of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  SA  SA is REAL array, dimension (LDSA,N) On entry, the MbyN coefficient matrix SA. 
[in]  LDSA  LDSA is INTEGER The leading dimension of the array SA. LDSA >= max(1,M). 
[out]  A  A is DOUBLE PRECISION array, dimension (LDA,N) On exit, the MbyN coefficient matrix A. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[out]  INFO  INFO is INTEGER = 0: successful exit 
Definition at line 105 of file slag2d.f.
subroutine slagts  (  integer  JOB, 
integer  N,  
real, dimension( * )  A,  
real, dimension( * )  B,  
real, dimension( * )  C,  
real, dimension( * )  D,  
integer, dimension( * )  IN,  
real, dimension( * )  Y,  
real  TOL,  
integer  INFO  
) 
SLAGTS solves the system of equations (TλI)x = y or (TλI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.
Download SLAGTS + dependencies [TGZ] [ZIP] [TXT]SLAGTS may be used to solve one of the systems of equations (T  lambda*I)*x = y or (T  lambda*I)**T*x = y, where T is an n by n tridiagonal matrix, for x, following the factorization of (T  lambda*I) as (T  lambda*I) = P*L*U , by routine SLAGTF. The choice of equation to be solved is controlled by the argument JOB, and in each case there is an option to perturb zero or very small diagonal elements of U, this option being intended for use in applications such as inverse iteration.
[in]  JOB  JOB is INTEGER Specifies the job to be performed by SLAGTS as follows: = 1: The equations (T  lambda*I)x = y are to be solved, but diagonal elements of U are not to be perturbed. = 1: The equations (T  lambda*I)x = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. = 2: The equations (T  lambda*I)**Tx = y are to be solved, but diagonal elements of U are not to be perturbed. = 2: The equations (T  lambda*I)**Tx = y are to be solved and, if overflow would otherwise occur, the diagonal elements of U are to be perturbed. See argument TOL below. 
[in]  N  N is INTEGER The order of the matrix T. 
[in]  A  A is REAL array, dimension (N) On entry, A must contain the diagonal elements of U as returned from SLAGTF. 
[in]  B  B is REAL array, dimension (N1) On entry, B must contain the first superdiagonal elements of U as returned from SLAGTF. 
[in]  C  C is REAL array, dimension (N1) On entry, C must contain the subdiagonal elements of L as returned from SLAGTF. 
[in]  D  D is REAL array, dimension (N2) On entry, D must contain the second superdiagonal elements of U as returned from SLAGTF. 
[in]  IN  IN is INTEGER array, dimension (N) On entry, IN must contain details of the matrix P as returned from SLAGTF. 
[in,out]  Y  Y is REAL array, dimension (N) On entry, the right hand side vector y. On exit, Y is overwritten by the solution vector x. 
[in,out]  TOL  TOL is REAL On entry, with JOB .lt. 0, TOL should be the minimum perturbation to be made to very small diagonal elements of U. TOL should normally be chosen as about eps*norm(U), where eps is the relative machine precision, but if TOL is supplied as nonpositive, then it is reset to eps*max( abs( u(i,j) ) ). If JOB .gt. 0 then TOL is not referenced. On exit, TOL is changed as described above, only if TOL is nonpositive on entry. Otherwise TOL is unchanged. 
[out]  INFO  INFO is INTEGER = 0 : successful exit .lt. 0: if INFO = i, the ith argument had an illegal value .gt. 0: overflow would occur when computing the INFO(th) element of the solution vector x. This can only occur when JOB is supplied as positive and either means that a diagonal element of U is very small, or that the elements of the righthand side vector y are very large. 
Definition at line 162 of file slagts.f.
LOGICAL function slaisnan  (  real  SIN1, 
real  SIN2  
) 
SLAISNAN tests input for NaN by comparing two arguments for inequality.
Download SLAISNAN + dependencies [TGZ] [ZIP] [TXT]This routine is not for general use. It exists solely to avoid overoptimization in SISNAN. SLAISNAN checks for NaNs by comparing its two arguments for inequality. NaN is the only floatingpoint value where NaN != NaN returns .TRUE. To check for NaNs, pass the same variable as both arguments. A compiler must assume that the two arguments are not the same variable, and the test will not be optimized away. Interprocedural or wholeprogram optimization may delete this test. The ISNAN functions will be replaced by the correct Fortran 03 intrinsic once the intrinsic is widely available.
[in]  SIN1  SIN1 is REAL 
[in]  SIN2  SIN2 is REAL Two numbers to compare for inequality. 
Definition at line 75 of file slaisnan.f.
subroutine slamc1  (  integer  BETA, 
integer  T,  
logical  RND,  
logical  IEEE1  
) 
SLAMC1
Purpose:
SLAMC1 determines the machine parameters given by BETA, T, RND, and IEEE1.
[out]  BETA  The base of the machine. 
[out]  T  The number of ( BETA ) digits in the mantissa. 
[out]  RND  Specifies whether proper rounding ( RND = .TRUE. ) or chopping ( RND = .FALSE. ) occurs in addition. This may not be a reliable guide to the way in which the machine performs its arithmetic. 
[out]  IEEE1  Specifies whether rounding appears to be done in the IEEE 'round to nearest' style. 
Further Details
The routine is based on the routine ENVRON by Malcolm and incorporates suggestions by Gentleman and Marovich. See Malcolm M. A. (1972) Algorithms to reveal properties of floatingpoint arithmetic. Comms. of the ACM, 15, 949951. Gentleman W. M. and Marovich S. B. (1974) More on algorithms that reveal properties of floating point arithmetic units. Comms. of the ACM, 17, 276277.
Definition at line 210 of file slamchf77.f.
subroutine slamc2  (  integer  BETA, 
integer  T,  
logical  RND,  
real  EPS,  
integer  EMIN,  
real  RMIN,  
integer  EMAX,  
real  RMAX  
) 
SLAMC2
Purpose:
SLAMC2 determines the machine parameters specified in its argument list.
[out]  BETA  The base of the machine. 
[out]  T  The number of ( BETA ) digits in the mantissa. 
[out]  RND  Specifies whether proper rounding ( RND = .TRUE. ) or chopping ( RND = .FALSE. ) occurs in addition. This may not be a reliable guide to the way in which the machine performs its arithmetic. 
[out]  EPS  The smallest positive number such that fl( 1.0  EPS ) .LT. 1.0, where fl denotes the computed value. 
[out]  EMIN  The minimum exponent before (gradual) underflow occurs. 
[out]  RMIN  The smallest normalized number for the machine, given by BASE**( EMIN  1 ), where BASE is the floating point value of BETA. 
[out]  EMAX  The maximum exponent before overflow occurs. 
[out]  RMAX  The largest positive number for the machine, given by BASE**EMAX * ( 1  EPS ), where BASE is the floating point value of BETA. 
Further Details
The computation of EPS is based on a routine PARANOIA by W. Kahan of the University of California at Berkeley.
Definition at line 423 of file slamchf77.f.
REAL function slamc3  (  real  A, 
real  B  
) 
SLAMC3
Purpose:
SLAMC3 is intended to force A and B to be stored prior to doing the addition of A and B , for use in situations where optimizers might hold one of these in a register.
[in]  A  
[in]  B  The values A and B. 
Definition at line 171 of file slamch.f.
REAL function slamch  (  character  CMACH  ) 
SLAMCH
SLAMCHF77 deprecated
SLAMCH determines single precision machine parameters.
[in]  CMACH  Specifies the value to be returned by SLAMCH: = 'E' or 'e', SLAMCH := eps = 'S' or 's , SLAMCH := sfmin = 'B' or 'b', SLAMCH := base = 'P' or 'p', SLAMCH := eps*base = 'N' or 'n', SLAMCH := t = 'R' or 'r', SLAMCH := rnd = 'M' or 'm', SLAMCH := emin = 'U' or 'u', SLAMCH := rmin = 'L' or 'l', SLAMCH := emax = 'O' or 'o', SLAMCH := rmax where eps = relative machine precision sfmin = safe minimum, such that 1/sfmin does not overflow base = base of the machine prec = eps*base t = number of (base) digits in the mantissa rnd = 1.0 when rounding occurs in addition, 0.0 otherwise emin = minimum exponent before (gradual) underflow rmin = underflow threshold  base**(emin1) emax = largest exponent before overflow rmax = overflow threshold  (base**emax)*(1eps) 
SLAMCH determines single precision machine parameters.
[in]  CMACH  Specifies the value to be returned by SLAMCH: = 'E' or 'e', SLAMCH := eps = 'S' or 's , SLAMCH := sfmin = 'B' or 'b', SLAMCH := base = 'P' or 'p', SLAMCH := eps*base = 'N' or 'n', SLAMCH := t = 'R' or 'r', SLAMCH := rnd = 'M' or 'm', SLAMCH := emin = 'U' or 'u', SLAMCH := rmin = 'L' or 'l', SLAMCH := emax = 'O' or 'o', SLAMCH := rmax where eps = relative machine precision sfmin = safe minimum, such that 1/sfmin does not overflow base = base of the machine prec = eps*base t = number of (base) digits in the mantissa rnd = 1.0 when rounding occurs in addition, 0.0 otherwise emin = minimum exponent before (gradual) underflow rmin = underflow threshold  base**(emin1) emax = largest exponent before overflow rmax = overflow threshold  (base**emax)*(1eps) 
Definition at line 68 of file slamch.f.
INTEGER function slaneg  (  integer  N, 
real, dimension( * )  D,  
real, dimension( * )  LLD,  
real  SIGMA,  
real  PIVMIN,  
integer  R  
) 
SLANEG computes the Sturm count.
Download SLANEG + dependencies [TGZ] [ZIP] [TXT]SLANEG computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T  sigma I = L D L^T. This implementation works directly on the factors without forming the tridiagonal matrix T. The Sturm count is also the number of eigenvalues of T less than sigma. This routine is called from SLARRB. The current routine does not use the PIVMIN parameter but rather requires IEEE754 propagation of Infinities and NaNs. This routine also has no input range restrictions but does require default exception handling such that x/0 produces Inf when x is nonzero, and Inf/Inf produces NaN. For more information, see: Marques, Riedy, and Voemel, "Benefits of IEEE754 Features in Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624 (Tech report version in LAWN 172 with the same title.)
[in]  N  N is INTEGER The order of the matrix. 
[in]  D  D is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D. 
[in]  LLD  LLD is REAL array, dimension (N1) The (N1) elements L(i)*L(i)*D(i). 
[in]  SIGMA  SIGMA is REAL Shift amount in T  sigma I = L D L^T. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot in the Sturm sequence. May be used when zero pivots are encountered on nonIEEE754 architectures. 
[in]  R  R is INTEGER The twist index for the twisted factorization that is used for the negcount. 
Definition at line 119 of file slaneg.f.
REAL function slanst  (  character  NORM, 
integer  N,  
real, dimension( * )  D,  
real, dimension( * )  E  
) 
SLANST returns the value of the 1norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix.
Download SLANST + dependencies [TGZ] [ZIP] [TXT]SLANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A.
SLANST = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
[in]  NORM  NORM is CHARACTER*1 Specifies the value to be returned in SLANST as described above. 
[in]  N  N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANST is set to zero. 
[in]  D  D is REAL array, dimension (N) The diagonal elements of A. 
[in]  E  E is REAL array, dimension (N1) The (n1) subdiagonal or superdiagonal elements of A. 
Definition at line 101 of file slanst.f.
REAL function slapy2  (  real  X, 
real  Y  
) 
SLAPY2 returns sqrt(x2+y2).
Download SLAPY2 + dependencies [TGZ] [ZIP] [TXT]SLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary overflow.
[in]  X  X is REAL 
[in]  Y  Y is REAL X and Y specify the values x and y. 
Definition at line 64 of file slapy2.f.
REAL function slapy3  (  real  X, 
real  Y,  
real  Z  
) 
SLAPY3 returns sqrt(x2+y2+z2).
Download SLAPY3 + dependencies [TGZ] [ZIP] [TXT]SLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause unnecessary overflow.
[in]  X  X is REAL 
[in]  Y  Y is REAL 
[in]  Z  Z is REAL X, Y and Z specify the values x, y and z. 
Definition at line 69 of file slapy3.f.
subroutine slarnv  (  integer  IDIST, 
integer, dimension( 4 )  ISEED,  
integer  N,  
real, dimension( * )  X  
) 
SLARNV returns a vector of random numbers from a uniform or normal distribution.
Download SLARNV + dependencies [TGZ] [ZIP] [TXT]SLARNV returns a vector of n random real numbers from a uniform or normal distribution.
[in]  IDIST  IDIST is INTEGER Specifies the distribution of the random numbers: = 1: uniform (0,1) = 2: uniform (1,1) = 3: normal (0,1) 
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated. 
[in]  N  N is INTEGER The number of random numbers to be generated. 
[out]  X  X is REAL array, dimension (N) The generated random numbers. 
This routine calls the auxiliary routine SLARUV to generate random real numbers from a uniform (0,1) distribution, in batches of up to 128 using vectorisable code. The BoxMuller method is used to transform numbers from a uniform to a normal distribution.
Definition at line 98 of file slarnv.f.
subroutine slarra  (  integer  N, 
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( * )  E2,  
real  SPLTOL,  
real  TNRM,  
integer  NSPLIT,  
integer, dimension( * )  ISPLIT,  
integer  INFO  
) 
SLARRA computes the splitting points with the specified threshold.
Download SLARRA + dependencies [TGZ] [ZIP] [TXT]Compute the splitting points with threshold SPLTOL. SLARRA sets any "small" offdiagonal elements to zero.
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in]  D  D is REAL array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T. 
[in,out]  E  E is REAL array, dimension (N) On entry, the first (N1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, are set to zero, the other entries of E are untouched. 
[in,out]  E2  E2 is REAL array, dimension (N) On entry, the first (N1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zero 
[in]  SPLTOL  SPLTOL is REAL The threshold for splitting. Two criteria can be used: SPLTOL<0 : criterion based on absolute offdiagonal value SPLTOL>0 : criterion that preserves relative accuracy 
[in]  TNRM  TNRM is REAL The norm of the matrix. 
[out]  NSPLIT  NSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N. 
[out]  ISPLIT  ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLITth consists of rows/columns ISPLIT(NSPLIT1)+1 through ISPLIT(NSPLIT)=N. 
[out]  INFO  INFO is INTEGER = 0: successful exit 
Definition at line 136 of file slarra.f.
subroutine slarrb  (  integer  N, 
real, dimension( * )  D,  
real, dimension( * )  LLD,  
integer  IFIRST,  
integer  ILAST,  
real  RTOL1,  
real  RTOL2,  
integer  OFFSET,  
real, dimension( * )  W,  
real, dimension( * )  WGAP,  
real, dimension( * )  WERR,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
real  PIVMIN,  
real  SPDIAM,  
integer  TWIST,  
integer  INFO  
) 
SLARRB provides limited bisection to locate eigenvalues for more accuracy.
Download SLARRB + dependencies [TGZ] [ZIP] [TXT]Given the relatively robust representation(RRR) L D L^T, SLARRB does "limited" bisection to refine the eigenvalues of L D L^T, W( IFIRSTOFFSET ) through W( ILASTOFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses and their gaps are input in WERR and WGAP, respectively. During bisection, intervals [left, right] are maintained by storing their midpoints and semiwidths in the arrays W and WERR respectively.
[in]  N  N is INTEGER The order of the matrix. 
[in]  D  D is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D. 
[in]  LLD  LLD is REAL array, dimension (N1) The (N1) elements L(i)*L(i)*D(i). 
[in]  IFIRST  IFIRST is INTEGER The index of the first eigenvalue to be computed. 
[in]  ILAST  ILAST is INTEGER The index of the last eigenvalue to be computed. 
[in]  RTOL1  RTOL1 is REAL 
[in]  RTOL2  RTOL2 is REAL Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHTLEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(LEFT,RIGHT) ) where GAP is the (estimated) distance to the nearest eigenvalue. 
[in]  OFFSET  OFFSET is INTEGER Offset for the arrays W, WGAP and WERR, i.e., the IFIRSTOFFSET through ILASTOFFSET elements of these arrays are to be used. 
[in,out]  W  W is REAL array, dimension (N) On input, W( IFIRSTOFFSET ) through W( ILASTOFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST throug ILAST. On output, these estimates are refined. 
[in,out]  WGAP  WGAP is REAL array, dimension (N1) On input, the (estimated) gaps between consecutive eigenvalues of L D L^T, i.e., WGAP(IOFFSET) is the gap between eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST then WGAP(IFIRSTOFFSET) must be set to ZERO. On output, these gaps are refined. 
[in,out]  WERR  WERR is REAL array, dimension (N) On input, WERR( IFIRSTOFFSET ) through WERR( ILASTOFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined. 
[out]  WORK  WORK is REAL array, dimension (2*N) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (2*N) Workspace. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot in the Sturm sequence. 
[in]  SPDIAM  SPDIAM is REAL The spectral diameter of the matrix. 
[in]  TWIST  TWIST is INTEGER The twist index for the twisted factorization that is used for the negcount. TWIST = N: Compute negcount from L D L^T  LAMBDA I = L+ D+ L+^T TWIST = 1: Compute negcount from L D L^T  LAMBDA I = U D U^T TWIST = R: Compute negcount from L D L^T  LAMBDA I = N(r) D(r) N(r) 
[out]  INFO  INFO is INTEGER Error flag. 
Definition at line 195 of file slarrb.f.
subroutine slarrc  (  character  JOBT, 
integer  N,  
real  VL,  
real  VU,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real  PIVMIN,  
integer  EIGCNT,  
integer  LCNT,  
integer  RCNT,  
integer  INFO  
) 
SLARRC computes the number of eigenvalues of the symmetric tridiagonal matrix.
Download SLARRC + dependencies [TGZ] [ZIP] [TXT]Find the number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] if JOBT = 'T', and of L D L^T if JOBT = 'L'.
[in]  JOBT  JOBT is CHARACTER*1 = 'T': Compute Sturm count for matrix T. = 'L': Compute Sturm count for matrix L D L^T. 
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in]  VL  VL is DOUBLE PRECISION 
[in]  VU  VU is DOUBLE PRECISION The lower and upper bounds for the eigenvalues. 
[in]  D  D is DOUBLE PRECISION array, dimension (N) JOBT = 'T': The N diagonal elements of the tridiagonal matrix T. JOBT = 'L': The N diagonal elements of the diagonal matrix D. 
[in]  E  E is DOUBLE PRECISION array, dimension (N) JOBT = 'T': The N1 offdiagonal elements of the matrix T. JOBT = 'L': The N1 offdiagonal elements of the matrix L. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot in the Sturm sequence for T. 
[out]  EIGCNT  EIGCNT is INTEGER The number of eigenvalues of the symmetric tridiagonal matrix T that are in the interval (VL,VU] 
[out]  LCNT  LCNT is INTEGER 
[out]  RCNT  RCNT is INTEGER The left and right negcounts of the interval. 
[out]  INFO  INFO is INTEGER 
Definition at line 136 of file slarrc.f.
subroutine slarrd  (  character  RANGE, 
character  ORDER,  
integer  N,  
real  VL,  
real  VU,  
integer  IL,  
integer  IU,  
real, dimension( * )  GERS,  
real  RELTOL,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( * )  E2,  
real  PIVMIN,  
integer  NSPLIT,  
integer, dimension( * )  ISPLIT,  
integer  M,  
real, dimension( * )  W,  
real, dimension( * )  WERR,  
real  WL,  
real  WU,  
integer, dimension( * )  IBLOCK,  
integer, dimension( * )  INDEXW,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
Download SLARRD + dependencies [TGZ] [ZIP] [TXT]SLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. The user may ask for all eigenvalues, all eigenvalues in the halfopen interval (VL, VU], or the ILth through IUth eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
[in]  RANGE  RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the halfopen interval (VL, VU] will be found. = 'I': ("Index") the ILth through IUth eigenvalues (of the entire matrix) will be found. 
[in]  ORDER  ORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by splitoff block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest. 
[in]  N  N is INTEGER The order of the tridiagonal matrix T. N >= 0. 
[in]  VL  VL is REAL 
[in]  VU  VU is REAL If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'. 
[in]  IL  IL is INTEGER 
[in]  IU  IU is INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'. 
[in]  GERS  GERS is REAL array, dimension (2*N) The N Gerschgorin intervals (the ith Gerschgorin interval is (GERS(2*i1), GERS(2*i)). 
[in]  RELTOL  RELTOL is REAL The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. 
[in]  D  D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T. 
[in]  E  E is REAL array, dimension (N1) The (n1) offdiagonal elements of the tridiagonal matrix T. 
[in]  E2  E2 is REAL array, dimension (N1) The (n1) squared offdiagonal elements of the tridiagonal matrix T. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot allowed in the Sturm sequence for T. 
[in]  NSPLIT  NSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N. 
[in]  ISPLIT  ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLITth consists of rows/columns ISPLIT(NSPLIT1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.) 
[out]  M  M is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.) 
[out]  W  W is REAL array, dimension (N) On exit, the first M elements of W will contain the eigenvalue approximations. SLARRD computes an interval I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue approximation is given as the interval midpoint W(j)= ( a_j + b_j)/2. The corresponding error is bounded by WERR(j) = abs( a_j  b_j)/2 
[out]  WERR  WERR is REAL array, dimension (N) The error bound on the corresponding eigenvalue approximation in W. 
[out]  WL  WL is REAL 
[out]  WU  WU is REAL The interval (WL, WU] contains all the wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin bounds on the spectrum. If RANGE='I', then WL and WU are computed by SLAEBZ from the index range specified. 
[out]  IBLOCK  IBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (SLARRD may use the remaining NM elements as workspace.) 
[out]  INDEXW  INDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= j and IBLOCK(i)=k imply that the ith eigenvalue W(i) is the jth eigenvalue in block k. 
[out]  WORK  WORK is REAL array, dimension (4*N) 
[out]  IWORK  IWORK is INTEGER array, dimension (3*N) 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1IL Cause: nonmonotonic arithmetic, causing the Sturm sequence to be nonmonotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floatingpoint arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again. 
FUDGE REAL, default = 2 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution.
Definition at line 319 of file slarrd.f.
subroutine slarre  (  character  RANGE, 
integer  N,  
real  VL,  
real  VU,  
integer  IL,  
integer  IU,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( * )  E2,  
real  RTOL1,  
real  RTOL2,  
real  SPLTOL,  
integer  NSPLIT,  
integer, dimension( * )  ISPLIT,  
integer  M,  
real, dimension( * )  W,  
real, dimension( * )  WERR,  
real, dimension( * )  WGAP,  
integer, dimension( * )  IBLOCK,  
integer, dimension( * )  INDEXW,  
real, dimension( * )  GERS,  
real  PIVMIN,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SLARRE given the tridiagonal matrix T, sets small offdiagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
Download SLARRE + dependencies [TGZ] [ZIP] [TXT]To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, SLARRE sets any "small" offdiagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i  sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T. The representations and eigenvalues found are then used by SSTEMR to compute the eigenvectors of T. The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to conpute all and then discard any unwanted one. As an added benefit, SLARRE also outputs the n Gerschgorin intervals for the matrices L_i D_i L_i^T.
[in]  RANGE  RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the halfopen interval (VL, VU] will be found. = 'I': ("Index") the ILth through IUth eigenvalues (of the entire matrix) will be found. 
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in,out]  VL  VL is REAL 
[in,out]  VU  VU is REAL If RANGE='V', the lower and upper bounds for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', SLARRE computes bounds on the desired part of the spectrum. 
[in]  IL  IL is INTEGER 
[in]  IU  IU is INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N. 
[in,out]  D  D is REAL array, dimension (N) On entry, the N diagonal elements of the tridiagonal matrix T. On exit, the N diagonal elements of the diagonal matrices D_i. 
[in,out]  E  E is REAL array, dimension (N) On entry, the first (N1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, E contains the subdiagonal elements of the unit bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, contain the base points sigma_i on output. 
[in,out]  E2  E2 is REAL array, dimension (N) On entry, the first (N1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zero 
[in]  RTOL1  RTOL1 is REAL 
[in]  RTOL2  RTOL2 is REAL Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHTLEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(LEFT,RIGHT) ) 
[in]  SPLTOL  SPLTOL is REAL The threshold for splitting. 
[out]  NSPLIT  NSPLIT is INTEGER The number of blocks T splits into. 1 <= NSPLIT <= N. 
[out]  ISPLIT  ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLITth consists of rows/columns ISPLIT(NSPLIT1)+1 through ISPLIT(NSPLIT)=N. 
[out]  M  M is INTEGER The total number of eigenvalues (of all L_i D_i L_i^T) found. 
[out]  W  W is REAL array, dimension (N) The first M elements contain the eigenvalues. The eigenvalues of each of the blocks, L_i D_i L_i^T, are sorted in ascending order ( SLARRE may use the remaining NM elements as workspace). 
[out]  WERR  WERR is REAL array, dimension (N) The error bound on the corresponding eigenvalue in W. 
[out]  WGAP  WGAP is REAL array, dimension (N) The separation from the right neighbor eigenvalue in W. The gap is only with respect to the eigenvalues of the same block as each block has its own representation tree. Exception: at the right end of a block we store the left gap 
[out]  IBLOCK  IBLOCK is INTEGER array, dimension (N) The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc. 
[out]  INDEXW  INDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the ith eigenvalue W(i) is the 10th eigenvalue in block 2 
[out]  GERS  GERS is REAL array, dimension (2*N) The N Gerschgorin intervals (the ith Gerschgorin interval is (GERS(2*i1), GERS(2*i)). 
[out]  PIVMIN  PIVMIN is REAL The minimum pivot in the Sturm sequence for T. 
[out]  WORK  WORK is REAL array, dimension (6*N) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (5*N) Workspace. 
[out]  INFO  INFO is INTEGER = 0: successful exit > 0: A problem occured in SLARRE. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =1: Problem in SLARRD. = 2: No base representation could be found in MAXTRY iterations. Increasing MAXTRY and recompilation might be a remedy. =3: Problem in SLARRB when computing the refined root representation for SLASQ2. =4: Problem in SLARRB when preforming bisection on the desired part of the spectrum. =5: Problem in SLASQ2. =6: Problem in SLASQ2. 
The base representations are required to suffer very little element growth and consequently define all their eigenvalues to high relative accuracy.
Definition at line 295 of file slarre.f.
subroutine slarrf  (  integer  N, 
real, dimension( * )  D,  
real, dimension( * )  L,  
real, dimension( * )  LD,  
integer  CLSTRT,  
integer  CLEND,  
real, dimension( * )  W,  
real, dimension( * )  WGAP,  
real, dimension( * )  WERR,  
real  SPDIAM,  
real  CLGAPL,  
real  CLGAPR,  
real  PIVMIN,  
real  SIGMA,  
real, dimension( * )  DPLUS,  
real, dimension( * )  LPLUS,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
Download SLARRF + dependencies [TGZ] [ZIP] [TXT]Given the initial representation L D L^T and its cluster of close eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ... W( CLEND ), SLARRF finds a new relatively robust representation L D L^T  SIGMA I = L(+) D(+) L(+)^T such that at least one of the eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
[in]  N  N is INTEGER The order of the matrix (subblock, if the matrix splitted). 
[in]  D  D is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D. 
[in]  L  L is REAL array, dimension (N1) The (N1) subdiagonal elements of the unit bidiagonal matrix L. 
[in]  LD  LD is REAL array, dimension (N1) The (N1) elements L(i)*D(i). 
[in]  CLSTRT  CLSTRT is INTEGER The index of the first eigenvalue in the cluster. 
[in]  CLEND  CLEND is INTEGER The index of the last eigenvalue in the cluster. 
[in]  W  W is REAL array, dimension dimension is >= (CLENDCLSTRT+1) The eigenvalue APPROXIMATIONS of L D L^T in ascending order. W( CLSTRT ) through W( CLEND ) form the cluster of relatively close eigenalues. 
[in,out]  WGAP  WGAP is REAL array, dimension dimension is >= (CLENDCLSTRT+1) The separation from the right neighbor eigenvalue in W. 
[in]  WERR  WERR is REAL array, dimension dimension is >= (CLENDCLSTRT+1) WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in W 
[in]  SPDIAM  SPDIAM is REAL estimate of the spectral diameter obtained from the Gerschgorin intervals 
[in]  CLGAPL  CLGAPL is REAL 
[in]  CLGAPR  CLGAPR is REAL absolute gap on each end of the cluster. Set by the calling routine to protect against shifts too close to eigenvalues outside the cluster. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot allowed in the Sturm sequence. 
[out]  SIGMA  SIGMA is REAL The shift used to form L(+) D(+) L(+)^T. 
[out]  DPLUS  DPLUS is REAL array, dimension (N) The N diagonal elements of the diagonal matrix D(+). 
[out]  LPLUS  LPLUS is REAL array, dimension (N1) The first (N1) elements of LPLUS contain the subdiagonal elements of the unit bidiagonal matrix L(+). 
[out]  WORK  WORK is REAL array, dimension (2*N) Workspace. 
[out]  INFO  INFO is INTEGER Signals processing OK (=0) or failure (=1) 
Definition at line 191 of file slarrf.f.
subroutine slarrj  (  integer  N, 
real, dimension( * )  D,  
real, dimension( * )  E2,  
integer  IFIRST,  
integer  ILAST,  
real  RTOL,  
integer  OFFSET,  
real, dimension( * )  W,  
real, dimension( * )  WERR,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
real  PIVMIN,  
real  SPDIAM,  
integer  INFO  
) 
SLARRJ performs refinement of the initial estimates of the eigenvalues of the matrix T.
Download SLARRJ + dependencies [TGZ] [ZIP] [TXT]Given the initial eigenvalue approximations of T, SLARRJ does bisection to refine the eigenvalues of T, W( IFIRSTOFFSET ) through W( ILASTOFFSET ), to more accuracy. Initial guesses for these eigenvalues are input in W, the corresponding estimate of the error in these guesses in WERR. During bisection, intervals [left, right] are maintained by storing their midpoints and semiwidths in the arrays W and WERR respectively.
[in]  N  N is INTEGER The order of the matrix. 
[in]  D  D is REAL array, dimension (N) The N diagonal elements of T. 
[in]  E2  E2 is REAL array, dimension (N1) The Squares of the (N1) subdiagonal elements of T. 
[in]  IFIRST  IFIRST is INTEGER The index of the first eigenvalue to be computed. 
[in]  ILAST  ILAST is INTEGER The index of the last eigenvalue to be computed. 
[in]  RTOL  RTOL is REAL Tolerance for the convergence of the bisection intervals. An interval [LEFT,RIGHT] has converged if RIGHTLEFT.LT.RTOL*MAX(LEFT,RIGHT). 
[in]  OFFSET  OFFSET is INTEGER Offset for the arrays W and WERR, i.e., the IFIRSTOFFSET through ILASTOFFSET elements of these arrays are to be used. 
[in,out]  W  W is REAL array, dimension (N) On input, W( IFIRSTOFFSET ) through W( ILASTOFFSET ) are estimates of the eigenvalues of L D L^T indexed IFIRST through ILAST. On output, these estimates are refined. 
[in,out]  WERR  WERR is REAL array, dimension (N) On input, WERR( IFIRSTOFFSET ) through WERR( ILASTOFFSET ) are the errors in the estimates of the corresponding elements in W. On output, these errors are refined. 
[out]  WORK  WORK is REAL array, dimension (2*N) Workspace. 
[out]  IWORK  IWORK is INTEGER array, dimension (2*N) Workspace. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot in the Sturm sequence for T. 
[in]  SPDIAM  SPDIAM is REAL The spectral diameter of T. 
[out]  INFO  INFO is INTEGER Error flag. 
Definition at line 167 of file slarrj.f.
subroutine slarrk  (  integer  N, 
integer  IW,  
real  GL,  
real  GU,  
real, dimension( * )  D,  
real, dimension( * )  E2,  
real  PIVMIN,  
real  RELTOL,  
real  W,  
real  WERR,  
integer  INFO  
) 
SLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.
Download SLARRK + dependencies [TGZ] [ZIP] [TXT]SLARRK computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from SSTEMR. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
[in]  N  N is INTEGER The order of the tridiagonal matrix T. N >= 0. 
[in]  IW  IW is INTEGER The index of the eigenvalues to be returned. 
[in]  GL  GL is REAL 
[in]  GU  GU is REAL An upper and a lower bound on the eigenvalue. 
[in]  D  D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T. 
[in]  E2  E2 is REAL array, dimension (N1) The (n1) squared offdiagonal elements of the tridiagonal matrix T. 
[in]  PIVMIN  PIVMIN is REAL The minimum pivot allowed in the Sturm sequence for T. 
[in]  RELTOL  RELTOL is REAL The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon. 
[out]  W  W is REAL 
[out]  WERR  WERR is REAL The error bound on the corresponding eigenvalue approximation in W. 
[out]  INFO  INFO is INTEGER = 0: Eigenvalue converged = 1: Eigenvalue did NOT converge 
FUDGE REAL , default = 2 A "fudge factor" to widen the Gershgorin intervals.
Definition at line 145 of file slarrk.f.
subroutine slarrr  (  integer  N, 
real, dimension( * )  D,  
real, dimension( * )  E,  
integer  INFO  
) 
SLARRR performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
Download SLARRR + dependencies [TGZ] [ZIP] [TXT]Perform tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
[in]  N  N is INTEGER The order of the matrix. N > 0. 
[in]  D  D is REAL array, dimension (N) The N diagonal elements of the tridiagonal matrix T. 
[in,out]  E  E is REAL array, dimension (N) On entry, the first (N1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) is set to ZERO. 
[out]  INFO  INFO is INTEGER INFO = 0(default) : the matrix warrants computations preserving relative accuracy. INFO = 1 : the matrix warrants computations guaranteeing only absolute accuracy. 
Definition at line 95 of file slarrr.f.
subroutine slartg  (  real  F, 
real  G,  
real  CS,  
real  SN,  
real  R  
) 
SLARTG generates a plane rotation with real cosine and real sine.
Download SLARTG + dependencies [TGZ] [ZIP] [TXT]SLARTG generate a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the BLAS1 routine SROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=1 without doing any floating point operations (saves work in SBDSQR when there are zeros on the diagonal). If F exceeds G in magnitude, CS will be positive.
[in]  F  F is REAL The first component of vector to be rotated. 
[in]  G  G is REAL The second component of vector to be rotated. 
[out]  CS  CS is REAL The cosine of the rotation. 
[out]  SN  SN is REAL The sine of the rotation. 
[out]  R  R is REAL The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH. 
Definition at line 98 of file slartg.f.
subroutine slartgp  (  real  F, 
real  G,  
real  CS,  
real  SN,  
real  R  
) 
SLARTGP generates a plane rotation so that the diagonal is nonnegative.
Download SLARTGP + dependencies [TGZ] [ZIP] [TXT]SLARTGP generates a plane rotation so that [ CS SN ] . [ F ] = [ R ] where CS**2 + SN**2 = 1. [ SN CS ] [ G ] [ 0 ] This is a slower, more accurate version of the Level 1 BLAS routine SROTG, with the following other differences: F and G are unchanged on return. If G=0, then CS=(+/)1 and SN=0. If F=0 and (G .ne. 0), then CS=0 and SN=(+/)1. The sign is chosen so that R >= 0.
[in]  F  F is REAL The first component of vector to be rotated. 
[in]  G  G is REAL The second component of vector to be rotated. 
[out]  CS  CS is REAL The cosine of the rotation. 
[out]  SN  SN is REAL The sine of the rotation. 
[out]  R  R is REAL The nonzero component of the rotated vector. This version has a few statements commented out for thread safety (machine parameters are computed on each entry). 10 feb 03, SJH. 
Definition at line 96 of file slartgp.f.
subroutine slaruv  (  integer, dimension( 4 )  ISEED, 
integer  N,  
real, dimension( n )  X  
) 
SLARUV returns a vector of n random real numbers from a uniform distribution.
Download SLARUV + dependencies [TGZ] [ZIP] [TXT]SLARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). This is an auxiliary routine called by SLARNV and CLARNV.
[in,out]  ISEED  ISEED is INTEGER array, dimension (4) On entry, the seed of the random number generator; the array elements must be between 0 and 4095, and ISEED(4) must be odd. On exit, the seed is updated. 
[in]  N  N is INTEGER The number of random numbers to be generated. N <= 128. 
[out]  X  X is REAL array, dimension (N) The generated random numbers. 
This routine uses a multiplicative congruential method with modulus 2**48 and multiplier 33952834046453 (see G.S.Fishman, 'Multiplicative congruential random number generators with modulus 2**b: an exhaustive analysis for b = 32 and a partial analysis for b = 48', Math. Comp. 189, pp 331344, 1990). 48bit integers are stored in 4 integer array elements with 12 bits per element. Hence the routine is portable across machines with integers of 32 bits or more.
Definition at line 96 of file slaruv.f.
subroutine slas2  (  real  F, 
real  G,  
real  H,  
real  SSMIN,  
real  SSMAX  
) 
SLAS2 computes singular values of a 2by2 triangular matrix.
Download SLAS2 + dependencies [TGZ] [ZIP] [TXT]SLAS2 computes the singular values of the 2by2 matrix [ F G ] [ 0 H ]. On return, SSMIN is the smaller singular value and SSMAX is the larger singular value.
[in]  F  F is REAL The (1,1) element of the 2by2 matrix. 
[in]  G  G is REAL The (1,2) element of the 2by2 matrix. 
[in]  H  H is REAL The (2,2) element of the 2by2 matrix. 
[out]  SSMIN  SSMIN is REAL The smaller singular value. 
[out]  SSMAX  SSMAX is REAL The larger singular value. 
Barring over/underflow, all output quantities are correct to within a few units in the last place (ulps), even in the absence of a guard digit in addition/subtraction. In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows, or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.
Definition at line 108 of file slas2.f.
subroutine slascl  (  character  TYPE, 
integer  KL,  
integer  KU,  
real  CFROM,  
real  CTO,  
integer  M,  
integer  N,  
real, dimension( lda, * )  A,  
integer  LDA,  
integer  INFO  
) 
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Download SLASCL + dependencies [TGZ] [ZIP] [TXT]SLASCL multiplies the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.
[in]  TYPE  TYPE is CHARACTER*1 TYPE indices the storage type of the input matrix. = 'G': A is a full matrix. = 'L': A is a lower triangular matrix. = 'U': A is an upper triangular matrix. = 'H': A is an upper Hessenberg matrix. = 'B': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the lower half stored. = 'Q': A is a symmetric band matrix with lower bandwidth KL and upper bandwidth KU and with the only the upper half stored. = 'Z': A is a band matrix with lower bandwidth KL and upper bandwidth KU. See SGBTRF for storage details. 
[in]  KL  KL is INTEGER The lower bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'. 
[in]  KU  KU is INTEGER The upper bandwidth of A. Referenced only if TYPE = 'B', 'Q' or 'Z'. 
[in]  CFROM  CFROM is REAL 
[in]  CTO  CTO is REAL The matrix A is multiplied by CTO/CFROM. A(I,J) is computed without over/underflow if the final result CTO*A(I,J)/CFROM can be represented without over/underflow. CFROM must be nonzero. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in,out]  A  A is REAL array, dimension (LDA,N) The matrix to be multiplied by CTO/CFROM. See TYPE for the storage type. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[out]  INFO  INFO is INTEGER 0  successful exit <0  if INFO = i, the ith argument had an illegal value. 
Definition at line 140 of file slascl.f.
subroutine slasd0  (  integer  N, 
integer  SQRE,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( ldu, * )  U,  
integer  LDU,  
real, dimension( ldvt, * )  VT,  
integer  LDVT,  
integer  SMLSIZ,  
integer, dimension( * )  IWORK,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SLASD0 computes the singular values of a real upper bidiagonal nbym matrix B with diagonal d and offdiagonal e. Used by sbdsdc.
Download SLASD0 + dependencies [TGZ] [ZIP] [TXT]Using a divide and conquer approach, SLASD0 computes the singular value decomposition (SVD) of a real upper bidiagonal NbyM matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes orthogonal matrices U and VT such that B = U * S * VT. The singular values S are overwritten on D. A related subroutine, SLASDA, computes only the singular values, and optionally, the singular vectors in compact form.
[in]  N  N is INTEGER On entry, the row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D. 
[in]  SQRE  SQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N+1; 
[in,out]  D  D is REAL array, dimension (N) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values. 
[in]  E  E is REAL array, dimension (M1) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed. 
[out]  U  U is REAL array, dimension at least (LDQ, N) On exit, U contains the left singular vectors. 
[in]  LDU  LDU is INTEGER On entry, leading dimension of U. 
[out]  VT  VT is REAL array, dimension at least (LDVT, M) On exit, VT**T contains the right singular vectors. 
[in]  LDVT  LDVT is INTEGER On entry, leading dimension of VT. 
[in]  SMLSIZ  SMLSIZ is INTEGER On entry, maximum size of the subproblems at the bottom of the computation tree. 
[out]  IWORK  IWORK is INTEGER array, dimension (8*N) 
[out]  WORK  WORK is REAL array, dimension (3*M**2+2*M) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 150 of file slasd0.f.
subroutine slasd1  (  integer  NL, 
integer  NR,  
integer  SQRE,  
real, dimension( * )  D,  
real  ALPHA,  
real  BETA,  
real, dimension( ldu, * )  U,  
integer  LDU,  
real, dimension( ldvt, * )  VT,  
integer  LDVT,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  IWORK,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
Download SLASD1 + dependencies [TGZ] [ZIP] [TXT]SLASD1 computes the SVD of an upper bidiagonal NbyM matrix B, where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0. A related subroutine SLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired. SLASD1 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. For each such occurence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLASD2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine SLASD4 (as called by SLASD3). This routine also calculates the singular vectors of the current problem. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE. 
[in,out]  D  D is REAL array, dimension (NL+NR+1). N = NL+NR+1 On entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix. 
[in,out]  ALPHA  ALPHA is REAL Contains the diagonal element associated with the added row. 
[in,out]  BETA  BETA is REAL Contains the offdiagonal element associated with the added row. 
[in,out]  U  U is REAL array, dimension (LDU,N) On entry U(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. On exit U contains the left singular vectors of the bidiagonal matrix. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max( 1, N ). 
[in,out]  VT  VT is REAL array, dimension (LDVT,M) where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. On exit VT**T contains the right singular vectors of the bidiagonal matrix. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= max( 1, M ). 
[out]  IDXQ  IDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order. 
[out]  IWORK  IWORK is INTEGER array, dimension (4*N) 
[out]  WORK  WORK is REAL array, dimension (3*M**2+2*M) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 204 of file slasd1.f.
subroutine slasd2  (  integer  NL, 
integer  NR,  
integer  SQRE,  
integer  K,  
real, dimension( * )  D,  
real, dimension( * )  Z,  
real  ALPHA,  
real  BETA,  
real, dimension( ldu, * )  U,  
integer  LDU,  
real, dimension( ldvt, * )  VT,  
integer  LDVT,  
real, dimension( * )  DSIGMA,  
real, dimension( ldu2, * )  U2,  
integer  LDU2,  
real, dimension( ldvt2, * )  VT2,  
integer  LDVT2,  
integer, dimension( * )  IDXP,  
integer, dimension( * )  IDX,  
integer, dimension( * )  IDXC,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  COLTYP,  
integer  INFO  
) 
SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
Download SLASD2 + dependencies [TGZ] [ZIP] [TXT]SLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. SLASD2 is called from SLASD1.
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[out]  K  K is INTEGER Contains the dimension of the nondeflated matrix, This is the order of the related secular equation. 1 <= K <=N. 
[in,out]  D  D is REAL array, dimension (N) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (NK) updated singular values (those which were deflated) sorted into increasing order. 
[out]  Z  Z is REAL array, dimension (N) On exit Z contains the updating row vector in the secular equation. 
[in]  ALPHA  ALPHA is REAL Contains the diagonal element associated with the added row. 
[in]  BETA  BETA is REAL Contains the offdiagonal element associated with the added row. 
[in,out]  U  U is REAL array, dimension (LDU,N) On entry U contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (NK) updated left singular vectors (those which were deflated) in its last NK columns. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= N. 
[in,out]  VT  VT is REAL array, dimension (LDVT,M) On entry VT**T contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT**T contains the trailing (NK) updated right singular vectors (those which were deflated) in its last NK columns. In case SQRE =1, the last row of VT spans the right null space. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= M. 
[out]  DSIGMA  DSIGMA is REAL array, dimension (N) Contains a copy of the diagonal elements (K1 singular values and one zero) in the secular equation. 
[out]  U2  U2 is REAL array, dimension (LDU2,N) Contains a copy of the first K1 left singular vectors which will be used by SLASD3 in a matrix multiply (SGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains nonzero entries only at and above NL; the third contains nonzero entries only below NL+1; and the fourth is dense. 
[in]  LDU2  LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N. 
[out]  VT2  VT2 is REAL array, dimension (LDVT2,N) VT2**T contains a copy of the first K right singular vectors which will be used by SLASD3 in a matrix multiply (SGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains nonzeros only at and before NL +1; the third block contains nonzeros only at and after NL +2. 
[in]  LDVT2  LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= M. 
[out]  IDXP  IDXP is INTEGER array, dimension (N) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated Dvalues and IDXP(K+1:N) points to the deflated singular values. 
[out]  IDX  IDX is INTEGER array, dimension (N) This will contain the permutation used to sort the contents of D into ascending order. 
[out]  IDXC  IDXC is INTEGER array, dimension (N) This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains nonzero entries only at and above NL, the second contains nonzero entries only below NL+2, and the third is dense. 
[in,out]  IDXQ  IDXQ is INTEGER array, dimension (N) This contains the permutation which separately sorts the two subproblems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values. 
[out]  COLTYP  COLTYP is INTEGER array, dimension (N) As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is: 1 : nonzero in the upper half only 2 : nonzero in the lower half only 3 : dense 4 : deflated On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the Ith type columns. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
Definition at line 268 of file slasd2.f.
subroutine slasd3  (  integer  NL, 
integer  NR,  
integer  SQRE,  
integer  K,  
real, dimension( * )  D,  
real, dimension( ldq, * )  Q,  
integer  LDQ,  
real, dimension( * )  DSIGMA,  
real, dimension( ldu, * )  U,  
integer  LDU,  
real, dimension( ldu2, * )  U2,  
integer  LDU2,  
real, dimension( ldvt, * )  VT,  
integer  LDVT,  
real, dimension( ldvt2, * )  VT2,  
integer  LDVT2,  
integer, dimension( * )  IDXC,  
integer, dimension( * )  CTOT,  
real, dimension( * )  Z,  
integer  INFO  
) 
SLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
Download SLASD3 + dependencies [TGZ] [ZIP] [TXT]SLASD3 finds all the square roots of the roots of the secular equation, as defined by the values in D and Z. It makes the appropriate calls to SLASD4 and then updates the singular vectors by matrix multiplication. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. SLASD3 is called from SLASD1.
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[in]  K  K is INTEGER The size of the secular equation, 1 =< K = < N. 
[out]  D  D is REAL array, dimension(K) On exit the square roots of the roots of the secular equation, in ascending order. 
[out]  Q  Q is REAL array, dimension at least (LDQ,K). 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= K. 
[in,out]  DSIGMA  DSIGMA is REAL array, dimension(K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. 
[out]  U  U is REAL array, dimension (LDU, N) The last N  K columns of this matrix contain the deflated left singular vectors. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= N. 
[in]  U2  U2 is REAL array, dimension (LDU2, N) The first K columns of this matrix contain the nondeflated left singular vectors for the split problem. 
[in]  LDU2  LDU2 is INTEGER The leading dimension of the array U2. LDU2 >= N. 
[out]  VT  VT is REAL array, dimension (LDVT, M) The last M  K columns of VT**T contain the deflated right singular vectors. 
[in]  LDVT  LDVT is INTEGER The leading dimension of the array VT. LDVT >= N. 
[in,out]  VT2  VT2 is REAL array, dimension (LDVT2, N) The first K columns of VT2**T contain the nondeflated right singular vectors for the split problem. 
[in]  LDVT2  LDVT2 is INTEGER The leading dimension of the array VT2. LDVT2 >= N. 
[in]  IDXC  IDXC is INTEGER array, dimension (N) The permutation used to arrange the columns of U (and rows of VT) into three groups: the first group contains nonzero entries only at and above (or before) NL +1; the second contains nonzero entries only at and below (or after) NL+2; and the third is dense. The first column of U and the row of VT are treated separately, however. The rows of the singular vectors found by SLASD4 must be likewise permuted before the matrix multiplies can take place. 
[in]  CTOT  CTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in U (or rows in VT), as described in IDXC. The fourth column type is any column which has been deflated. 
[in,out]  Z  Z is REAL array, dimension (K) The first K elements of this array contain the components of the deflationadjusted updating row vector. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 224 of file slasd3.f.
subroutine slasd4  (  integer  N, 
integer  I,  
real, dimension( * )  D,  
real, dimension( * )  Z,  
real, dimension( * )  DELTA,  
real  RHO,  
real  SIGMA,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SLASD4 computes the square root of the ith updated eigenvalue of a positive symmetric rankone modification to a positive diagonal matrix. Used by sbdsdc.
Download SLASD4 + dependencies [TGZ] [ZIP] [TXT]This subroutine computes the square root of the Ith updated eigenvalue of a positive symmetric rankone modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rankone modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.
[in]  N  N is INTEGER The length of all arrays. 
[in]  I  I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N. 
[in]  D  D is REAL array, dimension ( N ) The original eigenvalues. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J. 
[in]  Z  Z is REAL array, dimension (N) The components of the updating vector. 
[out]  DELTA  DELTA is REAL array, dimension (N) If N .ne. 1, DELTA contains (D(j)  sigma_I) in its jth component. If N = 1, then DELTA(1) = 1. The vector DELTA contains the information necessary to construct the (singular) eigenvectors. 
[in]  RHO  RHO is REAL The scalar in the symmetric updating formula. 
[out]  SIGMA  SIGMA is REAL The computed sigma_I, the Ith updated eigenvalue. 
[out]  WORK  WORK is REAL array, dimension (N) If N .ne. 1, WORK contains (D(j) + sigma_I) in its jth component. If N = 1, then WORK( 1 ) = 1. 
[out]  INFO  INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed. 
Logical variable ORGATI (originati?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switchfor3poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.
Definition at line 154 of file slasd4.f.
subroutine slasd5  (  integer  I, 
real, dimension( 2 )  D,  
real, dimension( 2 )  Z,  
real, dimension( 2 )  DELTA,  
real  RHO,  
real  DSIGMA,  
real, dimension( 2 )  WORK  
) 
SLASD5 computes the square root of the ith eigenvalue of a positive symmetric rankone modification of a 2by2 diagonal matrix. Used by sbdsdc.
Download SLASD5 + dependencies [TGZ] [ZIP] [TXT]This subroutine computes the square root of the Ith eigenvalue of a positive symmetric rankone modification of a 2by2 diagonal matrix diag( D ) * diag( D ) + RHO * Z * transpose(Z) . The diagonal entries in the array D are assumed to satisfy 0 <= D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.
[in]  I  I is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2. 
[in]  D  D is REAL array, dimension (2) The original eigenvalues. We assume 0 <= D(1) < D(2). 
[in]  Z  Z is REAL array, dimension (2) The components of the updating vector. 
[out]  DELTA  DELTA is REAL array, dimension (2) Contains (D(j)  sigma_I) in its jth component. The vector DELTA contains the information necessary to construct the eigenvectors. 
[in]  RHO  RHO is REAL The scalar in the symmetric updating formula. 
[out]  DSIGMA  DSIGMA is REAL The computed sigma_I, the Ith updated eigenvalue. 
[out]  WORK  WORK is REAL array, dimension (2) WORK contains (D(j) + sigma_I) in its jth component. 
Definition at line 117 of file slasd5.f.
subroutine slasd6  (  integer  ICOMPQ, 
integer  NL,  
integer  NR,  
integer  SQRE,  
real, dimension( * )  D,  
real, dimension( * )  VF,  
real, dimension( * )  VL,  
real  ALPHA,  
real  BETA,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  PERM,  
integer  GIVPTR,  
integer, dimension( ldgcol, * )  GIVCOL,  
integer  LDGCOL,  
real, dimension( ldgnum, * )  GIVNUM,  
integer  LDGNUM,  
real, dimension( ldgnum, * )  POLES,  
real, dimension( * )  DIFL,  
real, dimension( * )  DIFR,  
real, dimension( * )  Z,  
integer  K,  
real  C,  
real  S,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
Download SLASD6 + dependencies [TGZ] [ZIP] [TXT]SLASD6 computes the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row. This routine is used only for the problem which requires all singular values and optionally singular vector matrices in factored form. B is an NbyM matrix with N = NL + NR + 1 and M = N + SQRE. A related subroutine, SLASD1, handles the case in which all singular values and singular vectors of the bidiagonal matrix are desired. SLASD6 computes the SVD as follows: ( D1(in) 0 0 0 ) B = U(in) * ( Z1**T a Z2**T b ) * VT(in) ( 0 0 D2(in) 0 ) = U(out) * ( D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The singular values of B can be computed using D1, D2, the first components of all the right singular vectors of the lower block, and the last components of all the right singular vectors of the upper block. These components are stored and updated in VF and VL, respectively, in SLASD6. Hence U and VT are not explicitly referenced. The singular values are stored in D. The algorithm consists of two stages: The first stage consists of deflating the size of the problem when there are multiple singular values or if there is a zero in the Z vector. For each such occurence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLASD7. The second stage consists of calculating the updated singular values. This is done by finding the roots of the secular equation via the routine SLASD4 (as called by SLASD8). This routine also updates VF and VL and computes the distances between the updated singular values and the old singular values. SLASD6 is called from SLASDA.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well. 
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE. 
[in,out]  D  D is REAL array, dimension (NL+NR+1). On entry D(1:NL,1:NL) contains the singular values of the upper block, and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix. 
[in,out]  VF  VF is REAL array, dimension (M) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix. 
[in,out]  VL  VL is REAL array, dimension (M) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix. 
[in,out]  ALPHA  ALPHA is REAL Contains the diagonal element associated with the added row. 
[in,out]  BETA  BETA is REAL Contains the offdiagonal element associated with the added row. 
[out]  IDXQ  IDXQ is INTEGER array, dimension (N) This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D( IDXQ( I = 1, N ) ) will be in ascending order. 
[out]  PERM  PERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0. 
[out]  GIVPTR  GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0. 
[out]  GIVCOL  GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGCOL  LDGCOL is INTEGER leading dimension of GIVCOL, must be at least N. 
[out]  GIVNUM  GIVNUM is REAL array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGNUM  LDGNUM is INTEGER The leading dimension of GIVNUM and POLES, must be at least N. 
[out]  POLES  POLES is REAL array, dimension ( LDGNUM, 2 ) On exit, POLES(1,*) is an array containing the new singular values obtained from solving the secular equation, and POLES(2,*) is an array containing the poles in the secular equation. Not referenced if ICOMPQ = 0. 
[out]  DIFL  DIFL is REAL array, dimension ( N ) On exit, DIFL(I) is the distance between Ith updated (undeflated) singular value and the Ith (undeflated) old singular value. 
[out]  DIFR  DIFR is REAL array, dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. On exit, DIFR(I, 1) is the distance between Ith updated (undeflated) singular value and the I+1th (undeflated) old singular value. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. See SLASD8 for details on DIFL and DIFR. 
[out]  Z  Z is REAL array, dimension ( M ) The first elements of this array contain the components of the deflationadjusted updating row vector. 
[out]  K  K is INTEGER Contains the dimension of the nondeflated matrix, This is the order of the related secular equation. 1 <= K <=N. 
[out]  C  C is REAL C contains garbage if SQRE =0 and the Cvalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  S  S is REAL S contains garbage if SQRE =0 and the Svalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  WORK  WORK is REAL array, dimension ( 4 * M ) 
[out]  IWORK  IWORK is INTEGER array, dimension ( 3 * N ) 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 312 of file slasd6.f.
subroutine slasd7  (  integer  ICOMPQ, 
integer  NL,  
integer  NR,  
integer  SQRE,  
integer  K,  
real, dimension( * )  D,  
real, dimension( * )  Z,  
real, dimension( * )  ZW,  
real, dimension( * )  VF,  
real, dimension( * )  VFW,  
real, dimension( * )  VL,  
real, dimension( * )  VLW,  
real  ALPHA,  
real  BETA,  
real, dimension( * )  DSIGMA,  
integer, dimension( * )  IDX,  
integer, dimension( * )  IDXP,  
integer, dimension( * )  IDXQ,  
integer, dimension( * )  PERM,  
integer  GIVPTR,  
integer, dimension( ldgcol, * )  GIVCOL,  
integer  LDGCOL,  
real, dimension( ldgnum, * )  GIVNUM,  
integer  LDGNUM,  
real  C,  
real  S,  
integer  INFO  
) 
SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc.
Download SLASD7 + dependencies [TGZ] [ZIP] [TXT]SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one. SLASD7 is called from SLASD6.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows: = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form. 
[in]  NL  NL is INTEGER The row dimension of the upper block. NL >= 1. 
[in]  NR  NR is INTEGER The row dimension of the lower block. NR >= 1. 
[in]  SQRE  SQRE is INTEGER = 0: the lower block is an NRbyNR square matrix. = 1: the lower block is an NRby(NR+1) rectangular matrix. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[out]  K  K is INTEGER Contains the dimension of the nondeflated matrix, this is the order of the related secular equation. 1 <= K <=N. 
[in,out]  D  D is REAL array, dimension ( N ) On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (NK) updated singular values (those which were deflated) sorted into increasing order. 
[out]  Z  Z is REAL array, dimension ( M ) On exit Z contains the updating row vector in the secular equation. 
[out]  ZW  ZW is REAL array, dimension ( M ) Workspace for Z. 
[in,out]  VF  VF is REAL array, dimension ( M ) On entry, VF(1:NL+1) contains the first components of all right singular vectors of the upper block; and VF(NL+2:M) contains the first components of all right singular vectors of the lower block. On exit, VF contains the first components of all right singular vectors of the bidiagonal matrix. 
[out]  VFW  VFW is REAL array, dimension ( M ) Workspace for VF. 
[in,out]  VL  VL is REAL array, dimension ( M ) On entry, VL(1:NL+1) contains the last components of all right singular vectors of the upper block; and VL(NL+2:M) contains the last components of all right singular vectors of the lower block. On exit, VL contains the last components of all right singular vectors of the bidiagonal matrix. 
[out]  VLW  VLW is REAL array, dimension ( M ) Workspace for VL. 
[in]  ALPHA  ALPHA is REAL Contains the diagonal element associated with the added row. 
[in]  BETA  BETA is REAL Contains the offdiagonal element associated with the added row. 
[out]  DSIGMA  DSIGMA is REAL array, dimension ( N ) Contains a copy of the diagonal elements (K1 singular values and one zero) in the secular equation. 
[out]  IDX  IDX is INTEGER array, dimension ( N ) This will contain the permutation used to sort the contents of D into ascending order. 
[out]  IDXP  IDXP is INTEGER array, dimension ( N ) This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K) points to the nondeflated Dvalues and IDXP(K+1:N) points to the deflated singular values. 
[in]  IDXQ  IDXQ is INTEGER array, dimension ( N ) This contains the permutation which separately sorts the two subproblems in D into ascending order. Note that entries in the first half of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values. 
[out]  PERM  PERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) to be applied to each singular block. Not referenced if ICOMPQ = 0. 
[out]  GIVPTR  GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0. 
[out]  GIVCOL  GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGCOL  LDGCOL is INTEGER The leading dimension of GIVCOL, must be at least N. 
[out]  GIVNUM  GIVNUM is REAL array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value to be used in the corresponding Givens rotation. Not referenced if ICOMPQ = 0. 
[in]  LDGNUM  LDGNUM is INTEGER The leading dimension of GIVNUM, must be at least N. 
[out]  C  C is REAL C contains garbage if SQRE =0 and the Cvalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  S  S is REAL S contains garbage if SQRE =0 and the Svalue of a Givens rotation related to the right null space if SQRE = 1. 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. 
Definition at line 278 of file slasd7.f.
subroutine slasd8  (  integer  ICOMPQ, 
integer  K,  
real, dimension( * )  D,  
real, dimension( * )  Z,  
real, dimension( * )  VF,  
real, dimension( * )  VL,  
real, dimension( * )  DIFL,  
real, dimension( lddifr, * )  DIFR,  
integer  LDDIFR,  
real, dimension( * )  DSIGMA,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SLASD8 finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles. Used by sbdsdc.
Download SLASD8 + dependencies [TGZ] [ZIP] [TXT]SLASD8 finds the square roots of the roots of the secular equation, as defined by the values in DSIGMA and Z. It makes the appropriate calls to SLASD4, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA). It also updates the arrays VF and VL, the first and last components of all the right singular vectors of the original bidiagonal matrix. SLASD8 is called from SLASD6.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form in the calling routine: = 0: Compute singular values only. = 1: Compute singular vectors in factored form as well. 
[in]  K  K is INTEGER The number of terms in the rational function to be solved by SLASD4. K >= 1. 
[out]  D  D is REAL array, dimension ( K ) On output, D contains the updated singular values. 
[in,out]  Z  Z is REAL array, dimension ( K ) On entry, the first K elements of this array contain the components of the deflationadjusted updating row vector. On exit, Z is updated. 
[in,out]  VF  VF is REAL array, dimension ( K ) On entry, VF contains information passed through DBEDE8. On exit, VF contains the first K components of the first components of all right singular vectors of the bidiagonal matrix. 
[in,out]  VL  VL is REAL array, dimension ( K ) On entry, VL contains information passed through DBEDE8. On exit, VL contains the first K components of the last components of all right singular vectors of the bidiagonal matrix. 
[out]  DIFL  DIFL is REAL array, dimension ( K ) On exit, DIFL(I) = D(I)  DSIGMA(I). 
[out]  DIFR  DIFR is REAL array, dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and dimension ( K ) if ICOMPQ = 0. On exit, DIFR(I,1) = D(I)  DSIGMA(I+1), DIFR(K,1) is not defined and will not be referenced. If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing factors for the right singular vector matrix. 
[in]  LDDIFR  LDDIFR is INTEGER The leading dimension of DIFR, must be at least K. 
[in,out]  DSIGMA  DSIGMA is REAL array, dimension ( K ) On entry, the first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. On exit, the elements of DSIGMA may be very slightly altered in value. 
[out]  WORK  WORK is REAL array, dimension at least 3 * K 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 166 of file slasd8.f.
subroutine slasda  (  integer  ICOMPQ, 
integer  SMLSIZ,  
integer  N,  
integer  SQRE,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( ldu, * )  U,  
integer  LDU,  
real, dimension( ldu, * )  VT,  
integer, dimension( * )  K,  
real, dimension( ldu, * )  DIFL,  
real, dimension( ldu, * )  DIFR,  
real, dimension( ldu, * )  Z,  
real, dimension( ldu, * )  POLES,  
integer, dimension( * )  GIVPTR,  
integer, dimension( ldgcol, * )  GIVCOL,  
integer  LDGCOL,  
integer, dimension( ldgcol, * )  PERM,  
real, dimension( ldu, * )  GIVNUM,  
real, dimension( * )  C,  
real, dimension( * )  S,  
real, dimension( * )  WORK,  
integer, dimension( * )  IWORK,  
integer  INFO  
) 
SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.
Download SLASDA + dependencies [TGZ] [ZIP] [TXT]Using a divide and conquer approach, SLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal NbyM matrix B with diagonal D and offdiagonal E, where M = N + SQRE. The algorithm computes the singular values in the SVD B = U * S * VT. The orthogonal matrices U and VT are optionally computed in compact form. A related subroutine, SLASD0, computes the singular values and the singular vectors in explicit form.
[in]  ICOMPQ  ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in compact form, as follows = 0: Compute singular values only. = 1: Compute singular vectors of upper bidiagonal matrix in compact form. 
[in]  SMLSIZ  SMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree. 
[in]  N  N is INTEGER The row dimension of the upper bidiagonal matrix. This is also the dimension of the main diagonal array D. 
[in]  SQRE  SQRE is INTEGER Specifies the column dimension of the bidiagonal matrix. = 0: The bidiagonal matrix has column dimension M = N; = 1: The bidiagonal matrix has column dimension M = N + 1. 
[in,out]  D  D is REAL array, dimension ( N ) On entry D contains the main diagonal of the bidiagonal matrix. On exit D, if INFO = 0, contains its singular values. 
[in]  E  E is REAL array, dimension ( M1 ) Contains the subdiagonal entries of the bidiagonal matrix. On exit, E has been destroyed. 
[out]  U  U is REAL array, dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left singular vector matrices of all subproblems at the bottom level. 
[in]  LDU  LDU is INTEGER, LDU = > N. The leading dimension of arrays U, VT, DIFL, DIFR, POLES, GIVNUM, and Z. 
[out]  VT  VT is REAL array, dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right singular vector matrices of all subproblems at the bottom level. 
[out]  K  K is INTEGER array, dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1, on exit, K(I) is the dimension of the Ith secular equation on the computation tree. 
[out]  DIFL  DIFL is REAL array, dimension ( LDU, NLVL ), where NLVL = floor(log_2 (N/SMLSIZ))). 
[out]  DIFR  DIFR is REAL array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I  1) record distances between singular values on the Ith level and singular values on the (I 1)th level, and DIFR(1:N, 2 * I ) contains the normalizing factors for the right singular vector matrix. See SLASD8 for details. 
[out]  Z  Z is REAL array, dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension ( N ) if ICOMPQ = 0. The first K elements of Z(1, I) contain the components of the deflationadjusted updating row vector for subproblems on the Ith level. 
[out]  POLES  POLES is REAL array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I  1) and POLES(1, 2*I) contain the new and old singular values involved in the secular equations on the Ith level. 
[out]  GIVPTR  GIVPTR is INTEGER array, dimension ( N ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records the number of Givens rotations performed on the Ith problem on the computation tree. 
[out]  GIVCOL  GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVCOL(1, 2 *I  1) and GIVCOL(1, 2 *I) record the locations of Givens rotations performed on the Ith level on the computation tree. 
[in]  LDGCOL  LDGCOL is INTEGER, LDGCOL = > N. The leading dimension of arrays GIVCOL and PERM. 
[out]  PERM  PERM is INTEGER array, dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records permutations done on the Ith level of the computation tree. 
[out]  GIVNUM  GIVNUM is REAL array, dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, GIVNUM(1, 2 *I  1) and GIVNUM(1, 2 *I) record the C and S values of Givens rotations performed on the Ith level on the computation tree. 
[out]  C  C is REAL array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the Ith subproblem is not square, on exit, C( I ) contains the Cvalue of a Givens rotation related to the right null space of the Ith subproblem. 
[out]  S  S is REAL array, dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the Ith subproblem is not square, on exit, S( I ) contains the Svalue of a Givens rotation related to the right null space of the Ith subproblem. 
[out]  WORK  WORK is REAL array, dimension (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). 
[out]  IWORK  IWORK is INTEGER array, dimension (7*N). 
[out]  INFO  INFO is INTEGER = 0: successful exit. < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, a singular value did not converge 
Definition at line 272 of file slasda.f.
subroutine slasdq  (  character  UPLO, 
integer  SQRE,  
integer  N,  
integer  NCVT,  
integer  NRU,  
integer  NCC,  
real, dimension( * )  D,  
real, dimension( * )  E,  
real, dimension( ldvt, * )  VT,  
integer  LDVT,  
real, dimension( ldu, * )  U,  
integer  LDU,  
real, dimension( ldc, * )  C,  
integer  LDC,  
real, dimension( * )  WORK,  
integer  INFO  
) 
SLASDQ computes the SVD of a real bidiagonal matrix with diagonal d and offdiagonal e. Used by sbdsdc.
Download SLASDQ + dependencies [TGZ] [ZIP] [TXT]SLASDQ computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired. Letting B denote the input bidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q * S * P**T (P**T denotes the transpose of P). The singular values S are overwritten on D. The input matrix U is changed to U * Q if desired. The input matrix VT is changed to P**T * VT if desired. The input matrix C is changed to Q**T * C if desired. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3, for a detailed description of the algorithm.
[in]  UPLO  UPLO is CHARACTER*1 On entry, UPLO specifies whether the input bidiagonal matrix is upper or lower bidiagonal, and wether it is square are not. UPLO = 'U' or 'u' B is upper bidiagonal. UPLO = 'L' or 'l' B is lower bidiagonal. 
[in]  SQRE  SQRE is INTEGER = 0: then the input matrix is NbyN. = 1: then the input matrix is Nby(N+1) if UPLU = 'U' and (N+1)byN if UPLU = 'L'. The bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE >= N columns. 
[in]  N  N is INTEGER On entry, N specifies the number of rows and columns in the matrix. N must be at least 0. 
[in]  NCVT  NCVT is INTEGER On entry, NCVT specifies the number of columns of the matrix VT. NCVT must be at least 0. 
[in]  NRU  NRU is INTEGER On entry, NRU specifies the number of rows of the matrix U. NRU must be at least 0. 
[in]  NCC  NCC is INTEGER On entry, NCC specifies the number of columns of the matrix C. NCC must be at least 0. 
[in,out]  D  D is REAL array, dimension (N) On entry, D contains the diagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in ascending order. 
[in,out]  E  E is REAL array. dimension is (N1) if SQRE = 0 and N if SQRE = 1. On entry, the entries of E contain the offdiagonal entries of the bidiagonal matrix whose SVD is desired. On normal exit, E will contain 0. If the algorithm does not converge, D and E will contain the diagonal and superdiagonal entries of a bidiagonal matrix orthogonally equivalent to the one given as input. 
[in,out]  VT  VT is REAL array, dimension (LDVT, NCVT) On entry, contains a matrix which on exit has been premultiplied by P**T, dimension NbyNCVT if SQRE = 0 and (N+1)byNCVT if SQRE = 1 (not referenced if NCVT=0). 
[in]  LDVT  LDVT is INTEGER On entry, LDVT specifies the leading dimension of VT as declared in the calling (sub) program. LDVT must be at least 1. If NCVT is nonzero LDVT must also be at least N. 
[in,out]  U  U is REAL array, dimension (LDU, N) On entry, contains a matrix which on exit has been postmultiplied by Q, dimension NRUbyN if SQRE = 0 and NRUby(N+1) if SQRE = 1 (not referenced if NRU=0). 
[in]  LDU  LDU is INTEGER On entry, LDU specifies the leading dimension of U as declared in the calling (sub) program. LDU must be at least max( 1, NRU ) . 
[in,out]  C  C is REAL array, dimension (LDC, NCC) On entry, contains an NbyNCC matrix which on exit has been premultiplied by Q**T dimension NbyNCC if SQRE = 0 and (N+1)byNCC if SQRE = 1 (not referenced if NCC=0). 
[in]  LDC  LDC is INTEGER On entry, LDC specifies the leading dimension of C as declared in the calling (sub) program. LDC must be at least 1. If NCC is nonzero, LDC must also be at least N. 
[out]  WORK  WORK is REAL array, dimension (4*N) Workspace. Only referenced if one of NCVT, NRU, or NCC is nonzero, and if N is at least 2. 
[out]  INFO  INFO is INTEGER On exit, a value of 0 indicates a successful exit. If INFO < 0, argument number INFO is illegal. If INFO > 0, the algorithm did not converge, and INFO specifies how many superdiagonals did not converge. 
Definition at line 211 of file slasdq.f.
subroutine slasdt  (  integer  N, 
integer  LVL,  
integer  ND,  
integer, dimension( * )  INODE,  
integer, dimension( * )  NDIML,  
integer, dimension( * )  NDIMR,  
integer  MSUB  
) 
SLASDT creates a tree of subproblems for bidiagonal divide and conquer. Used by sbdsdc.
Download SLASDT + dependencies [TGZ] [ZIP] [TXT]SLASDT creates a tree of subproblems for bidiagonal divide and conquer.
[in]  N  N is INTEGER On entry, the number of diagonal elements of the bidiagonal matrix. 
[out]  LVL  LVL is INTEGER On exit, the number of levels on the computation tree. 
[out]  ND  ND is INTEGER On exit, the number of nodes on the tree. 
[out]  INODE  INODE is INTEGER array, dimension ( N ) On exit, centers of subproblems. 
[out]  NDIML  NDIML is INTEGER array, dimension ( N ) On exit, row dimensions of left children. 
[out]  NDIMR  NDIMR is INTEGER array, dimension ( N ) On exit, row dimensions of right children. 
[in]  MSUB  MSUB is INTEGER On entry, the maximum row dimension each subproblem at the bottom of the tree can be of. 
Definition at line 106 of file slasdt.f.
subroutine slaset  (  character  UPLO, 
integer  M,  
integer  N,  
real  ALPHA,  
real  BETA,  
real, dimension( lda, * )  A,  
integer  LDA  
) 
SLASET initializes the offdiagonal elements and the diagonal elements of a matrix to given values.
Download SLASET + dependencies [TGZ] [ZIP] [TXT]SLASET initializes an mbyn matrix A to BETA on the diagonal and ALPHA on the offdiagonals.
[in]  UPLO  UPLO is CHARACTER*1 Specifies the part of the matrix A to be set. = 'U': Upper triangular part is set; the strictly lower triangular part of A is not changed. = 'L': Lower triangular part is set; the strictly upper triangular part of A is not changed. Otherwise: All of the matrix A is set. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  ALPHA  ALPHA is REAL The constant to which the offdiagonal elements are to be set. 
[in]  BETA  BETA is REAL The constant to which the diagonal elements are to be set. 
[in,out]  A  A is REAL array, dimension (LDA,N) On exit, the leading mbyn submatrix of A is set as follows: if UPLO = 'U', A(i,j) = ALPHA, 1<=i<=j1, 1<=j<=n, if UPLO = 'L', A(i,j) = ALPHA, j+1<=i<=m, 1<=j<=n, otherwise, A(i,j) = ALPHA, 1<=i<=m, 1<=j<=n, i.ne.j, and, for all UPLO, A(i,i) = BETA, 1<=i<=min(m,n). 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
Definition at line 111 of file slaset.f.
subroutine slasr  (  character  SIDE, 
character  PIVOT,  
character  DIRECT,  
integer  M,  
integer  N,  
real, dimension( * )  C,  
real, dimension( * )  S,  
real, dimension( lda, * )  A,  
integer  LDA  
) 
SLASR applies a sequence of plane rotations to a general rectangular matrix.
Download SLASR + dependencies [TGZ] [ZIP] [TXT]SLASR applies a sequence of plane rotations to a real matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form A := P*A and when SIDE = 'R', the transformation takes the form A := A*P**T where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and P**T is the transpose of P. When DIRECT = 'F' (Forward sequence), then P = P(z1) * ... * P(2) * P(1) and when DIRECT = 'B' (Backward sequence), then P = P(1) * P(2) * ... * P(z1) where P(k) is a plane rotation matrix defined by the 2by2 rotation R(k) = ( c(k) s(k) ) = ( s(k) c(k) ). When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears as a rank2 modification to the identity matrix in rows and columns k and k+1. When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) where R(k) appears in rows and columns 1 and k+1. Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( s(k) c(k) ) where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.
[in]  SIDE  SIDE is CHARACTER*1 Specifies whether the plane rotation matrix P is applied to A on the left or the right. = 'L': Left, compute A := P*A = 'R': Right, compute A:= A*P**T 
[in]  PIVOT  PIVOT is CHARACTER*1 Specifies the plane for which P(k) is a plane rotation matrix. = 'V': Variable pivot, the plane (k,k+1) = 'T': Top pivot, the plane (1,k+1) = 'B': Bottom pivot, the plane (k,z) 
[in]  DIRECT  DIRECT is CHARACTER*1 Specifies whether P is a forward or backward sequence of plane rotations. = 'F': Forward, P = P(z1)*...*P(2)*P(1) = 'B': Backward, P = P(1)*P(2)*...*P(z1) 
[in]  M  M is INTEGER The number of rows of the matrix A. If m <= 1, an immediate return is effected. 
[in]  N  N is INTEGER The number of columns of the matrix A. If n <= 1, an immediate return is effected. 
[in]  C  C is REAL array, dimension (M1) if SIDE = 'L' (N1) if SIDE = 'R' The cosines c(k) of the plane rotations. 
[in]  S  S is REAL array, dimension (M1) if SIDE = 'L' (N1) if SIDE = 'R' The sines s(k) of the plane rotations. The 2by2 plane rotation part of the matrix P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( s(k) c(k) ). 
[in,out]  A  A is REAL array, dimension (LDA,N) The MbyN matrix A. On exit, A is overwritten by P*A if SIDE = 'R' or by A*P**T if SIDE = 'L'. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
Definition at line 200 of file slasr.f.
subroutine slassq  (  integer  N, 
real, dimension( * )  X,  
integer  INCX,  
real  SCALE,  
real  SUMSQ  
) 
SLASSQ updates a sum of squares represented in scaled form.
Download SLASSQ + dependencies [TGZ] [ZIP] [TXT]SLASSQ returns the values scl and smsq such that ( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq, where x( i ) = X( 1 + ( i  1 )*INCX ). The value of sumsq is assumed to be nonnegative and scl returns the value scl = max( scale, abs( x( i ) ) ). scale and sumsq must be supplied in SCALE and SUMSQ and scl and smsq are overwritten on SCALE and SUMSQ respectively. The routine makes only one pass through the vector x.
[in]  N  N is INTEGER The number of elements to be used from the vector X. 
[in]  X  X is REAL array, dimension (N) The vector for which a scaled sum of squares is computed. x( i ) = X( 1 + ( i  1 )*INCX ), 1 <= i <= n. 
[in]  INCX  INCX is INTEGER The increment between successive values of the vector X. INCX > 0. 
[in,out]  SCALE  SCALE is REAL On entry, the value scale in the equation above. On exit, SCALE is overwritten with scl , the scaling factor for the sum of squares. 
[in,out]  SUMSQ  SUMSQ is REAL On entry, the value sumsq in the equation above. On exit, SUMSQ is overwritten with smsq , the basic sum of squares from which scl has been factored out. 
Definition at line 104 of file slassq.f.
subroutine slasv2  (  real  F, 
real  G,  
real  H,  
real  SSMIN,  
real  SSMAX,  
real  SNR,  
real  CSR,  
real  SNL,  
real  CSL  
) 
SLASV2 computes the singular value decomposition of a 2by2 triangular matrix.
Download SLASV2 + dependencies [TGZ] [ZIP] [TXT]SLASV2 computes the singular value decomposition of a 2by2 triangular matrix [ F G ] [ 0 H ]. On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and right singular vectors for abs(SSMAX), giving the decomposition [ CSL SNL ] [ F G ] [ CSR SNR ] = [ SSMAX 0 ] [SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].
[in]  F  F is REAL The (1,1) element of the 2by2 matrix. 
[in]  G  G is REAL The (1,2) element of the 2by2 matrix. 
[in]  H  H is REAL The (2,2) element of the 2by2 matrix. 
[out]  SSMIN  SSMIN is REAL abs(SSMIN) is the smaller singular value. 
[out]  SSMAX  SSMAX is REAL abs(SSMAX) is the larger singular value. 
[out]  SNL  SNL is REAL 
[out]  CSL  CSL is REAL The vector (CSL, SNL) is a unit left singular vector for the singular value abs(SSMAX). 
[out]  SNR  SNR is REAL 
[out]  CSR  CSR is REAL The vector (CSR, SNR) is a unit right singular vector for the singular value abs(SSMAX). 
Any input parameter may be aliased with any output parameter. Barring over/underflow and assuming a guard digit in subtraction, all output quantities are correct to within a few units in the last place (ulps). In IEEE arithmetic, the code works correctly if one matrix element is infinite. Overflow will not occur unless the largest singular value itself overflows or is within a few ulps of overflow. (On machines with partial overflow, like the Cray, overflow may occur if the largest singular value is within a factor of 2 of overflow.) Underflow is harmless if underflow is gradual. Otherwise, results may correspond to a matrix modified by perturbations of size near the underflow threshold.
Definition at line 139 of file slasv2.f.
program tstiee  (  ) 
subroutine xerbla  (  character*(*)  SRNAME, 
integer  INFO  
) 
XERBLA
Download XERBLA + dependencies [TGZ] [ZIP] [TXT]XERBLA is an error handler for the LAPACK routines. It is called by an LAPACK routine if an input parameter has an invalid value. A message is printed and execution stops. Installers may consider modifying the STOP statement in order to call systemspecific exceptionhandling facilities.
[in]  SRNAME  SRNAME is CHARACTER*(*) The name of the routine which called XERBLA. 
[in]  INFO  INFO is INTEGER The position of the invalid parameter in the parameter list of the calling routine. 
subroutine xerbla_array  (  character(1), dimension(srname_len)  SRNAME_ARRAY, 
integer  SRNAME_LEN,  
integer  INFO  
) 
XERBLA_ARRAY
Download XERBLA_ARRAY + dependencies [TGZ] [ZIP] [TXT]XERBLA_ARRAY assists other languages in calling XERBLA, the LAPACK and BLAS error handler. Rather than taking a Fortran string argument as the function's name, XERBLA_ARRAY takes an array of single characters along with the array's length. XERBLA_ARRAY then copies up to 32 characters of that array into a Fortran string and passes that to XERBLA. If called with a nonpositive SRNAME_LEN, XERBLA_ARRAY will call XERBLA with a string of all blank characters. Say some macro or other device makes XERBLA_ARRAY available to C99 by a name lapack_xerbla and with a common Fortran calling convention. Then a C99 program could invoke XERBLA via: { int flen = strlen(__func__); lapack_xerbla(__func__, &flen, &info); } Providing XERBLA_ARRAY is not necessary for intercepting LAPACK errors. XERBLA_ARRAY calls XERBLA.
[in]  SRNAME_ARRAY  SRNAME_ARRAY is CHARACTER(1) array, dimension (SRNAME_LEN) The name of the routine which called XERBLA_ARRAY. 
[in]  SRNAME_LEN  SRNAME_LEN is INTEGER The length of the name in SRNAME_ARRAY. 
[in]  INFO  INFO is INTEGER The position of the invalid parameter in the parameter list of the calling routine. 
Definition at line 91 of file xerbla_array.f.