LAPACK  3.4.2
LAPACK: Linear Algebra PACKage
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double
Computational routines
Collaboration diagram for double:

Functions/Subroutines

subroutine dgbbrd (VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, LDQ, PT, LDPT, C, LDC, WORK, INFO)
 DGBBRD
subroutine dgbcon (NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
 DGBCON
subroutine dgbequ (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
 DGBEQU
subroutine dgbequb (M, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, AMAX, INFO)
 DGBEQUB
subroutine dgbrfs (TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
 DGBRFS
subroutine dgbrfsx (TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
 DGBRFSX
subroutine dgbtf2 (M, N, KL, KU, AB, LDAB, IPIV, INFO)
 DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
subroutine dgbtrf (M, N, KL, KU, AB, LDAB, IPIV, INFO)
 DGBTRF
subroutine dgbtrs (TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
 DGBTRS
subroutine dggbak (JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO)
 DGGBAK
subroutine dggbal (JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
 DGGBAL
subroutine dla_gbamv (TRANS, M, N, KL, KU, ALPHA, AB, LDAB, X, INCX, BETA, Y, INCY)
 DLA_GBAMV performs a matrix-vector operation to calculate error bounds.
DOUBLE PRECISION function dla_gbrcond (TRANS, N, KL, KU, AB, LDAB, AFB, LDAFB, IPIV, CMODE, C, INFO, WORK, IWORK)
 DLA_GBRCOND estimates the Skeel condition number for a general banded matrix.
subroutine dla_gbrfsx_extended (PREC_TYPE, TRANS_TYPE, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
 DLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
DOUBLE PRECISION function dla_gbrpvgrw (N, KL, KU, NCOLS, AB, LDAB, AFB, LDAFB)
 DLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
subroutine dorgbr (VECT, M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
 DORGBR

Detailed Description

This is the group of double computational functions for GB matrices

Function/Subroutine Documentation

subroutine dgbbrd ( character  VECT,
integer  M,
integer  N,
integer  NCC,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( ldq, * )  Q,
integer  LDQ,
double precision, dimension( ldpt, * )  PT,
integer  LDPT,
double precision, dimension( ldc, * )  C,
integer  LDC,
double precision, dimension( * )  WORK,
integer  INFO 
)

DGBBRD

Download DGBBRD + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBBRD reduces a real general m-by-n band matrix A to upper
 bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.

 The routine computes B, and optionally forms Q or P**T, or computes
 Q**T*C for a given matrix C.
Parameters:
[in]VECT
          VECT is CHARACTER*1
          Specifies whether or not the matrices Q and P**T are to be
          formed.
          = 'N': do not form Q or P**T;
          = 'Q': form Q only;
          = 'P': form P**T only;
          = 'B': form both.
[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]NCC
          NCC is INTEGER
          The number of columns of the matrix C.  NCC >= 0.
[in]KL
          KL is INTEGER
          The number of subdiagonals of the matrix A. KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals of the matrix A. KU >= 0.
[in,out]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the m-by-n band matrix A, stored in rows 1 to
          KL+KU+1. The j-th column of A is stored in the j-th column of
          the array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
          On exit, A is overwritten by values generated during the
          reduction.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array A. LDAB >= KL+KU+1.
[out]D
          D is DOUBLE PRECISION array, dimension (min(M,N))
          The diagonal elements of the bidiagonal matrix B.
[out]E
          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
          The superdiagonal elements of the bidiagonal matrix B.
[out]Q
          Q is DOUBLE PRECISION array, dimension (LDQ,M)
          If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
          If VECT = 'N' or 'P', the array Q is not referenced.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
[out]PT
          PT is DOUBLE PRECISION array, dimension (LDPT,N)
          If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
          If VECT = 'N' or 'Q', the array PT is not referenced.
[in]LDPT
          LDPT is INTEGER
          The leading dimension of the array PT.
          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
[in,out]C
          C is DOUBLE PRECISION array, dimension (LDC,NCC)
          On entry, an m-by-ncc matrix C.
          On exit, C is overwritten by Q**T*C.
          C is not referenced if NCC = 0.
[in]LDC
          LDC is INTEGER
          The leading dimension of the array C.
          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (2*max(M,N))
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011

Definition at line 187 of file dgbbrd.f.

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subroutine dgbcon ( character  NORM,
integer  N,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
integer, dimension( * )  IPIV,
double precision  ANORM,
double precision  RCOND,
double precision, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

DGBCON

Download DGBCON + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBCON estimates the reciprocal of the condition number of a real
 general band matrix A, in either the 1-norm or the infinity-norm,
 using the LU factorization computed by DGBTRF.

 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as
    RCOND = 1 / ( norm(A) * norm(inv(A)) ).
Parameters:
[in]NORM
          NORM is CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          Details of the LU factorization of the band matrix A, as
          computed by DGBTRF.  U is stored as an upper triangular band
          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
[in]IPIV
          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= N, row i of the matrix was
          interchanged with row IPIV(i).
[in]ANORM
          ANORM is DOUBLE PRECISION
          If NORM = '1' or 'O', the 1-norm of the original matrix A.
          If NORM = 'I', the infinity-norm of the original matrix A.
[out]RCOND
          RCOND is DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (3*N)
[out]IWORK
          IWORK is INTEGER array, dimension (N)
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011

Definition at line 146 of file dgbcon.f.

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subroutine dgbequ ( integer  M,
integer  N,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
double precision  ROWCND,
double precision  COLCND,
double precision  AMAX,
integer  INFO 
)

DGBEQU

Download DGBEQU + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBEQU computes row and column scalings intended to equilibrate an
 M-by-N band matrix A and reduce its condition number.  R returns the
 row scale factors and C the column scale factors, chosen to try to
 make the largest element in each row and column of the matrix B with
 elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

 R(i) and C(j) are restricted to be between SMLNUM = smallest safe
 number and BIGNUM = largest safe number.  Use of these scaling
 factors is not guaranteed to reduce the condition number of A but
 works well in practice.
Parameters:
[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]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
          column of A is stored in the j-th column of the array AB as
          follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.
[out]R
          R is DOUBLE PRECISION array, dimension (M)
          If INFO = 0, or INFO > M, R contains the row scale factors
          for A.
[out]C
          C is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, C contains the column scale factors for A.
[out]ROWCND
          ROWCND is DOUBLE PRECISION
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
          AMAX is neither too large nor too small, it is not worth
          scaling by R.
[out]COLCND
          COLCND is DOUBLE PRECISION
          If INFO = 0, COLCND contains the ratio of the smallest
          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
          worth scaling by C.
[out]AMAX
          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= M:  the i-th row of A is exactly zero
                >  M:  the (i-M)-th column of A is exactly zero
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011

Definition at line 153 of file dgbequ.f.

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subroutine dgbequb ( integer  M,
integer  N,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
double precision  ROWCND,
double precision  COLCND,
double precision  AMAX,
integer  INFO 
)

DGBEQUB

Download DGBEQUB + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBEQUB computes row and column scalings intended to equilibrate an
 M-by-N matrix A and reduce its condition number.  R returns the row
 scale factors and C the column scale factors, chosen to try to make
 the largest element in each row and column of the matrix B with
 elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
 the radix.

 R(i) and C(j) are restricted to be a power of the radix between
 SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
 of these scaling factors is not guaranteed to reduce the condition
 number of A but works well in practice.

 This routine differs from DGEEQU by restricting the scaling factors
 to a power of the radix.  Baring over- and underflow, scaling by
 these factors introduces no additional rounding errors.  However, the
 scaled entries' magnitured are no longer approximately 1 but lie
 between sqrt(radix) and 1/sqrt(radix).
Parameters:
[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]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array A.  LDAB >= max(1,M).
[out]R
          R is DOUBLE PRECISION array, dimension (M)
          If INFO = 0 or INFO > M, R contains the row scale factors
          for A.
[out]C
          C is DOUBLE PRECISION array, dimension (N)
          If INFO = 0,  C contains the column scale factors for A.
[out]ROWCND
          ROWCND is DOUBLE PRECISION
          If INFO = 0 or INFO > M, ROWCND contains the ratio of the
          smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
          AMAX is neither too large nor too small, it is not worth
          scaling by R.
[out]COLCND
          COLCND is DOUBLE PRECISION
          If INFO = 0, COLCND contains the ratio of the smallest
          C(i) to the largest C(i).  If COLCND >= 0.1, it is not
          worth scaling by C.
[out]AMAX
          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i,  and i is
                <= M:  the i-th row of A is exactly zero
                >  M:  the (i-M)-th column of A is exactly zero
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011

Definition at line 160 of file dgbequb.f.

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subroutine dgbrfs ( character  TRANS,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( ldx, * )  X,
integer  LDX,
double precision, dimension( * )  FERR,
double precision, dimension( * )  BERR,
double precision, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

DGBRFS

Download DGBRFS + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is banded, and provides
 error bounds and backward error estimates for the solution.
Parameters:
[in]TRANS
          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in]NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The original band matrix A, stored in rows 1 to KL+KU+1.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KL+KU+1.
[in]AFB
          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
          Details of the LU factorization of the band matrix A, as
          computed by DGBTRF.  U is stored as an upper triangular band
          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.
[in]LDAFB
          LDAFB is INTEGER
          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
[in]IPIV
          IPIV is INTEGER array, dimension (N)
          The pivot indices from DGBTRF; for 1<=i<=N, row i of the
          matrix was interchanged with row IPIV(i).
[in]B
          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
[in,out]X
          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DGBTRS.
          On exit, the improved solution matrix X.
[in]LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
[out]FERR
          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.
[out]BERR
          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (3*N)
[out]IWORK
          IWORK is INTEGER array, dimension (N)
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Internal Parameters: 
  ITMAX is the maximum number of steps of iterative refinement.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011

Definition at line 204 of file dgbrfs.f.

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subroutine dgbrfsx ( character  TRANS,
character  EQUED,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
double precision, dimension( * )  R,
double precision, dimension( * )  C,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( ldx , * )  X,
integer  LDX,
double precision  RCOND,
double precision, dimension( * )  BERR,
integer  N_ERR_BNDS,
double precision, dimension( nrhs, * )  ERR_BNDS_NORM,
double precision, dimension( nrhs, * )  ERR_BNDS_COMP,
integer  NPARAMS,
double precision, dimension( * )  PARAMS,
double precision, dimension( * )  WORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

DGBRFSX

Download DGBRFSX + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
    DGBRFSX improves the computed solution to a system of linear
    equations and provides error bounds and backward error estimates
    for the solution.  In addition to normwise error bound, the code
    provides maximum componentwise error bound if possible.  See
    comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
    error bounds.

    The original system of linear equations may have been equilibrated
    before calling this routine, as described by arguments EQUED, R
    and C below. In this case, the solution and error bounds returned
    are for the original unequilibrated system.
     Some optional parameters are bundled in the PARAMS array.  These
     settings determine how refinement is performed, but often the
     defaults are acceptable.  If the defaults are acceptable, users
     can pass NPARAMS = 0 which prevents the source code from accessing
     the PARAMS argument.
Parameters:
[in]TRANS
          TRANS is CHARACTER*1
     Specifies the form of the system of equations:
       = 'N':  A * X = B     (No transpose)
       = 'T':  A**T * X = B  (Transpose)
       = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
[in]EQUED
          EQUED is CHARACTER*1
     Specifies the form of equilibration that was done to A
     before calling this routine. This is needed to compute
     the solution and error bounds correctly.
       = 'N':  No equilibration
       = 'R':  Row equilibration, i.e., A has been premultiplied by
               diag(R).
       = 'C':  Column equilibration, i.e., A has been postmultiplied
               by diag(C).
       = 'B':  Both row and column equilibration, i.e., A has been
               replaced by diag(R) * A * diag(C).
               The right hand side B has been changed accordingly.
[in]N
          N is INTEGER
     The order of the matrix A.  N >= 0.
[in]KL
          KL is INTEGER
     The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
     The number of superdiagonals within the band of A.  KU >= 0.
[in]NRHS
          NRHS is INTEGER
     The number of right hand sides, i.e., the number of columns
     of the matrices B and X.  NRHS >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
     The original band matrix A, stored in rows 1 to KL+KU+1.
     The j-th column of A is stored in the j-th column of the
     array AB as follows:
     AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
[in]LDAB
          LDAB is INTEGER
     The leading dimension of the array AB.  LDAB >= KL+KU+1.
[in]AFB
          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
     Details of the LU factorization of the band matrix A, as
     computed by DGBTRF.  U is stored as an upper triangular band
     matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
     the multipliers used during the factorization are stored in
     rows KL+KU+2 to 2*KL+KU+1.
[in]LDAFB
          LDAFB is INTEGER
     The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
[in]IPIV
          IPIV is INTEGER array, dimension (N)
     The pivot indices from DGETRF; for 1<=i<=N, row i of the
     matrix was interchanged with row IPIV(i).
[in,out]R
          R is DOUBLE PRECISION array, dimension (N)
     The row scale factors for A.  If EQUED = 'R' or 'B', A is
     multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
     is not accessed.  R is an input argument if FACT = 'F';
     otherwise, R is an output argument.  If FACT = 'F' and
     EQUED = 'R' or 'B', each element of R must be positive.
     If R is output, each element of R is a power of the radix.
     If R is input, each element of R should be a power of the radix
     to ensure a reliable solution and error estimates. Scaling by
     powers of the radix does not cause rounding errors unless the
     result underflows or overflows. Rounding errors during scaling
     lead to refining with a matrix that is not equivalent to the
     input matrix, producing error estimates that may not be
     reliable.
[in,out]C
          C is DOUBLE PRECISION array, dimension (N)
     The column scale factors for A.  If EQUED = 'C' or 'B', A is
     multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
     is not accessed.  C is an input argument if FACT = 'F';
     otherwise, C is an output argument.  If FACT = 'F' and
     EQUED = 'C' or 'B', each element of C must be positive.
     If C is output, each element of C is a power of the radix.
     If C is input, each element of C should be a power of the radix
     to ensure a reliable solution and error estimates. Scaling by
     powers of the radix does not cause rounding errors unless the
     result underflows or overflows. Rounding errors during scaling
     lead to refining with a matrix that is not equivalent to the
     input matrix, producing error estimates that may not be
     reliable.
[in]B
          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
     The right hand side matrix B.
[in]LDB
          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).
[in,out]X
          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
     On entry, the solution matrix X, as computed by DGETRS.
     On exit, the improved solution matrix X.
[in]LDX
          LDX is INTEGER
     The leading dimension of the array X.  LDX >= max(1,N).
[out]RCOND
          RCOND is DOUBLE PRECISION
     Reciprocal scaled condition number.  This is an estimate of the
     reciprocal Skeel condition number of the matrix A after
     equilibration (if done).  If this is less than the machine
     precision (in particular, if it is zero), the matrix is singular
     to working precision.  Note that the error may still be small even
     if this number is very small and the matrix appears ill-
     conditioned.
[out]BERR
          BERR is DOUBLE PRECISION array, dimension (NRHS)
     Componentwise relative backward error.  This is the
     componentwise relative backward error of each solution vector X(j)
     (i.e., the smallest relative change in any element of A or B that
     makes X(j) an exact solution).
[in]N_ERR_BNDS
          N_ERR_BNDS is INTEGER
     Number of error bounds to return for each right hand side
     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
     ERR_BNDS_COMP below.
[out]ERR_BNDS_NORM
          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     normwise relative error, which is defined as follows:

     Normwise relative error in the ith solution vector:
             max_j (abs(XTRUE(j,i) - X(j,i)))
            ------------------------------
                  max_j abs(X(j,i))

     The array is indexed by the type of error information as described
     below. There currently are up to three pieces of information
     returned.

     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
     right-hand side.

     The second index in ERR_BNDS_NORM(:,err) contains the following
     three fields:
     err = 1 "Trust/don't trust" boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * dlamch('Epsilon').

     err = 2 "Guaranteed" error bound: The estimated forward error,
              almost certainly within a factor of 10 of the true error
              so long as the next entry is greater than the threshold
              sqrt(n) * dlamch('Epsilon'). This error bound should only
              be trusted if the previous boolean is true.

     err = 3  Reciprocal condition number: Estimated normwise
              reciprocal condition number.  Compared with the threshold
              sqrt(n) * dlamch('Epsilon') to determine if the error
              estimate is "guaranteed". These reciprocal condition
              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
              appropriately scaled matrix Z.
              Let Z = S*A, where S scales each row by a power of the
              radix so all absolute row sums of Z are approximately 1.

     See Lapack Working Note 165 for further details and extra
     cautions.
[out]ERR_BNDS_COMP
          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     componentwise relative error, which is defined as follows:

     Componentwise relative error in the ith solution vector:
                    abs(XTRUE(j,i) - X(j,i))
             max_j ----------------------
                         abs(X(j,i))

     The array is indexed by the right-hand side i (on which the
     componentwise relative error depends), and the type of error
     information as described below. There currently are up to three
     pieces of information returned for each right-hand side. If
     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
     the first (:,N_ERR_BNDS) entries are returned.

     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
     right-hand side.

     The second index in ERR_BNDS_COMP(:,err) contains the following
     three fields:
     err = 1 "Trust/don't trust" boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * dlamch('Epsilon').

     err = 2 "Guaranteed" error bound: The estimated forward error,
              almost certainly within a factor of 10 of the true error
              so long as the next entry is greater than the threshold
              sqrt(n) * dlamch('Epsilon'). This error bound should only
              be trusted if the previous boolean is true.

     err = 3  Reciprocal condition number: Estimated componentwise
              reciprocal condition number.  Compared with the threshold
              sqrt(n) * dlamch('Epsilon') to determine if the error
              estimate is "guaranteed". These reciprocal condition
              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
              appropriately scaled matrix Z.
              Let Z = S*(A*diag(x)), where x is the solution for the
              current right-hand side and S scales each row of
              A*diag(x) by a power of the radix so all absolute row
              sums of Z are approximately 1.

     See Lapack Working Note 165 for further details and extra
     cautions.
[in]NPARAMS
          NPARAMS is INTEGER
     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
     PARAMS array is never referenced and default values are used.
[in,out]PARAMS
          PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
     that entry will be filled with default value used for that
     parameter.  Only positions up to NPARAMS are accessed; defaults
     are used for higher-numbered parameters.

       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
            refinement or not.
         Default: 1.0D+0
            = 0.0 : No refinement is performed, and no error bounds are
                    computed.
            = 1.0 : Use the double-precision refinement algorithm,
                    possibly with doubled-single computations if the
                    compilation environment does not support DOUBLE
                    PRECISION.
              (other values are reserved for future use)

       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
            computations allowed for refinement.
         Default: 10
         Aggressive: Set to 100 to permit convergence using approximate
                     factorizations or factorizations other than LU. If
                     the factorization uses a technique other than
                     Gaussian elimination, the guarantees in
                     err_bnds_norm and err_bnds_comp may no longer be
                     trustworthy.

       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
            will attempt to find a solution with small componentwise
            relative error in the double-precision algorithm.  Positive
            is true, 0.0 is false.
         Default: 1.0 (attempt componentwise convergence)
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (4*N)
[out]IWORK
          IWORK is INTEGER array, dimension (N)
[out]INFO
          INFO is INTEGER
       = 0:  Successful exit. The solution to every right-hand side is
         guaranteed.
       < 0:  If INFO = -i, the i-th argument had an illegal value
       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
         has been completed, but the factor U is exactly singular, so
         the solution and error bounds could not be computed. RCOND = 0
         is returned.
       = N+J: The solution corresponding to the Jth right-hand side is
         not guaranteed. The solutions corresponding to other right-
         hand sides K with K > J may not be guaranteed as well, but
         only the first such right-hand side is reported. If a small
         componentwise error is not requested (PARAMS(3) = 0.0) then
         the Jth right-hand side is the first with a normwise error
         bound that is not guaranteed (the smallest J such
         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
         the Jth right-hand side is the first with either a normwise or
         componentwise error bound that is not guaranteed (the smallest
         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
         about all of the right-hand sides check ERR_BNDS_NORM or
         ERR_BNDS_COMP.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
April 2012

Definition at line 437 of file dgbrfsx.f.

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subroutine dgbtf2 ( integer  M,
integer  N,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
integer, dimension( * )  IPIV,
integer  INFO 
)

DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Download DGBTF2 + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBTF2 computes an LU factorization of a real m-by-n band matrix A
 using partial pivoting with row interchanges.

 This is the unblocked version of the algorithm, calling Level 2 BLAS.
Parameters:
[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]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in,out]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
[out]IPIV
          IPIV is INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).
[out]INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used
               to solve a system of equations.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
Further Details: 
  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

      *    *    *    +    +    +       *    *    *   u14  u25  u36
      *    *    +    +    +    +       *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked
  + need not be set on entry, but are required by the routine to store
  elements of U, because of fill-in resulting from the row
  interchanges.

Definition at line 146 of file dgbtf2.f.

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subroutine dgbtrf ( integer  M,
integer  N,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
integer, dimension( * )  IPIV,
integer  INFO 
)

DGBTRF

Download DGBTRF + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBTRF computes an LU factorization of a real m-by-n band matrix A
 using partial pivoting with row interchanges.

 This is the blocked version of the algorithm, calling Level 3 BLAS.
Parameters:
[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]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in,out]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the matrix A in band storage, in rows KL+1 to
          2*KL+KU+1; rows 1 to KL of the array need not be set.
          The j-th column of A is stored in the j-th column of the
          array AB as follows:
          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

          On exit, details of the factorization: U is stored as an
          upper triangular band matrix with KL+KU superdiagonals in
          rows 1 to KL+KU+1, and the multipliers used during the
          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
          See below for further details.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
[out]IPIV
          IPIV is INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).
[out]INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
               has been completed, but the factor U is exactly
               singular, and division by zero will occur if it is used
               to solve a system of equations.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details: 
  The band storage scheme is illustrated by the following example, when
  M = N = 6, KL = 2, KU = 1:

  On entry:                       On exit:

      *    *    *    +    +    +       *    *    *   u14  u25  u36
      *    *    +    +    +    +       *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

  Array elements marked * are not used by the routine; elements marked
  + need not be set on entry, but are required by the routine to store
  elements of U because of fill-in resulting from the row interchanges.

Definition at line 145 of file dgbtrf.f.

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subroutine dgbtrs ( character  TRANS,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
integer, dimension( * )  IPIV,
double precision, dimension( ldb, * )  B,
integer  LDB,
integer  INFO 
)

DGBTRS

Download DGBTRS + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGBTRS solves a system of linear equations
    A * X = B  or  A**T * X = B
 with a general band matrix A using the LU factorization computed
 by DGBTRF.
Parameters:
[in]TRANS
          TRANS is CHARACTER*1
          Specifies the form of the system of equations.
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T* X = B  (Transpose)
          = 'C':  A**T* X = B  (Conjugate transpose = Transpose)
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]KL
          KL is INTEGER
          The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
          The number of superdiagonals within the band of A.  KU >= 0.
[in]NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          Details of the LU factorization of the band matrix A, as
          computed by DGBTRF.  U is stored as an upper triangular band
          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
          the multipliers used during the factorization are stored in
          rows KL+KU+2 to 2*KL+KU+1.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
[in]IPIV
          IPIV is INTEGER array, dimension (N)
          The pivot indices; for 1 <= i <= N, row i of the matrix was
          interchanged with row IPIV(i).
[in,out]B
          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011

Definition at line 138 of file dgbtrs.f.

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subroutine dggbak ( character  JOB,
character  SIDE,
integer  N,
integer  ILO,
integer  IHI,
double precision, dimension( * )  LSCALE,
double precision, dimension( * )  RSCALE,
integer  M,
double precision, dimension( ldv, * )  V,
integer  LDV,
integer  INFO 
)

DGGBAK

Download DGGBAK + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGGBAK forms the right or left eigenvectors of a real generalized
 eigenvalue problem A*x = lambda*B*x, by backward transformation on
 the computed eigenvectors of the balanced pair of matrices output by
 DGGBAL.
Parameters:
[in]JOB
          JOB is CHARACTER*1
          Specifies the type of backward transformation required:
          = 'N':  do nothing, return immediately;
          = 'P':  do backward transformation for permutation only;
          = 'S':  do backward transformation for scaling only;
          = 'B':  do backward transformations for both permutation and
                  scaling.
          JOB must be the same as the argument JOB supplied to DGGBAL.
[in]SIDE
          SIDE is CHARACTER*1
          = 'R':  V contains right eigenvectors;
          = 'L':  V contains left eigenvectors.
[in]N
          N is INTEGER
          The number of rows of the matrix V.  N >= 0.
[in]ILO
          ILO is INTEGER
[in]IHI
          IHI is INTEGER
          The integers ILO and IHI determined by DGGBAL.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
[in]LSCALE
          LSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and/or scaling factors applied
          to the left side of A and B, as returned by DGGBAL.
[in]RSCALE
          RSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and/or scaling factors applied
          to the right side of A and B, as returned by DGGBAL.
[in]M
          M is INTEGER
          The number of columns of the matrix V.  M >= 0.
[in,out]V
          V is DOUBLE PRECISION array, dimension (LDV,M)
          On entry, the matrix of right or left eigenvectors to be
          transformed, as returned by DTGEVC.
          On exit, V is overwritten by the transformed eigenvectors.
[in]LDV
          LDV is INTEGER
          The leading dimension of the matrix V. LDV >= max(1,N).
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details: 
  See R.C. Ward, Balancing the generalized eigenvalue problem,
                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 147 of file dggbak.f.

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subroutine dggbal ( character  JOB,
integer  N,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldb, * )  B,
integer  LDB,
integer  ILO,
integer  IHI,
double precision, dimension( * )  LSCALE,
double precision, dimension( * )  RSCALE,
double precision, dimension( * )  WORK,
integer  INFO 
)

DGGBAL

Download DGGBAL + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DGGBAL balances a pair of general real matrices (A,B).  This
 involves, first, permuting A and B by similarity transformations to
 isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
 elements on the diagonal; and second, applying a diagonal similarity
 transformation to rows and columns ILO to IHI to make the rows
 and columns as close in norm as possible. Both steps are optional.

 Balancing may reduce the 1-norm of the matrices, and improve the
 accuracy of the computed eigenvalues and/or eigenvectors in the
 generalized eigenvalue problem A*x = lambda*B*x.
Parameters:
[in]JOB
          JOB is CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
                  and RSCALE(I) = 1.0 for i = 1,...,N.
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
[in]N
          N is INTEGER
          The order of the matrices A and B.  N >= 0.
[in,out]A
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the input matrix A.
          On exit,  A is overwritten by the balanced matrix.
          If JOB = 'N', A is not referenced.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).
[in,out]B
          B is DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the input matrix B.
          On exit,  B is overwritten by the balanced matrix.
          If JOB = 'N', B is not referenced.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).
[out]ILO
          ILO is INTEGER
[out]IHI
          IHI is INTEGER
          ILO and IHI are set to integers such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
[out]LSCALE
          LSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If P(j) is the index of the
          row interchanged with row j, and D(j)
          is the scaling factor applied to row j, then
            LSCALE(j) = P(j)    for J = 1,...,ILO-1
                      = D(j)    for J = ILO,...,IHI
                      = P(j)    for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.
[out]RSCALE
          RSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If P(j) is the index of the
          column interchanged with column j, and D(j)
          is the scaling factor applied to column j, then
            LSCALE(j) = P(j)    for J = 1,...,ILO-1
                      = D(j)    for J = ILO,...,IHI
                      = P(j)    for J = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
          at least 1 when JOB = 'N' or 'P'.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details: 
  See R.C. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 177 of file dggbal.f.

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subroutine dla_gbamv ( integer  TRANS,
integer  M,
integer  N,
integer  KL,
integer  KU,
double precision  ALPHA,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( * )  X,
integer  INCX,
double precision  BETA,
double precision, dimension( * )  Y,
integer  INCY 
)

DLA_GBAMV performs a matrix-vector operation to calculate error bounds.

Download DLA_GBAMV + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DLA_GBAMV  performs one of the matrix-vector operations

         y := alpha*abs(A)*abs(x) + beta*abs(y),
    or   y := alpha*abs(A)**T*abs(x) + beta*abs(y),

 where alpha and beta are scalars, x and y are vectors and A is an
 m by n matrix.

 This function is primarily used in calculating error bounds.
 To protect against underflow during evaluation, components in
 the resulting vector are perturbed away from zero by (N+1)
 times the underflow threshold.  To prevent unnecessarily large
 errors for block-structure embedded in general matrices,
 "symbolically" zero components are not perturbed.  A zero
 entry is considered "symbolic" if all multiplications involved
 in computing that entry have at least one zero multiplicand.
Parameters:
[in]TRANS
          TRANS is INTEGER
           On entry, TRANS specifies the operation to be performed as
           follows:

             BLAS_NO_TRANS      y := alpha*abs(A)*abs(x) + beta*abs(y)
             BLAS_TRANS         y := alpha*abs(A**T)*abs(x) + beta*abs(y)
             BLAS_CONJ_TRANS    y := alpha*abs(A**T)*abs(x) + beta*abs(y)

           Unchanged on exit.
[in]M
          M is INTEGER
           On entry, M specifies the number of rows of the matrix A.
           M must be at least zero.
           Unchanged on exit.
[in]N
          N is INTEGER
           On entry, N specifies the number of columns of the matrix A.
           N must be at least zero.
           Unchanged on exit.
[in]KL
          KL is INTEGER
           The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
           The number of superdiagonals within the band of A.  KU >= 0.
[in]ALPHA
          ALPHA is DOUBLE PRECISION
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.
[in]AB
          AB is DOUBLE PRECISION array of DIMENSION ( LDAB, n )
           Before entry, the leading m by n part of the array AB must
           contain the matrix of coefficients.
           Unchanged on exit.
[in]LDAB
          LDAB is INTEGER
           On entry, LDA specifies the first dimension of AB as declared
           in the calling (sub) program. LDAB must be at least
           max( 1, m ).
           Unchanged on exit.
[in]X
          X is DOUBLE PRECISION array, dimension
           ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
           and at least
           ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
           Before entry, the incremented array X must contain the
           vector x.
           Unchanged on exit.
[in]INCX
          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
           Unchanged on exit.
[in]BETA
          BETA is DOUBLE PRECISION
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
           Unchanged on exit.
[in,out]Y
          Y is DOUBLE PRECISION array, dimension
           ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
           and at least
           ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
           Before entry with BETA non-zero, the incremented array Y
           must contain the vector y. On exit, Y is overwritten by the
           updated vector y.
[in]INCY
          INCY is INTEGER
           On entry, INCY specifies the increment for the elements of
           Y. INCY must not be zero.
           Unchanged on exit.

  Level 2 Blas routine.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012

Definition at line 185 of file dla_gbamv.f.

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DOUBLE PRECISION function dla_gbrcond ( character  TRANS,
integer  N,
integer  KL,
integer  KU,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
integer  CMODE,
double precision, dimension( * )  C,
integer  INFO,
double precision, dimension( * )  WORK,
integer, dimension( * )  IWORK 
)

DLA_GBRCOND estimates the Skeel condition number for a general banded matrix.

Download DLA_GBRCOND + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
    DLA_GBRCOND Estimates the Skeel condition number of  op(A) * op2(C)
    where op2 is determined by CMODE as follows
    CMODE =  1    op2(C) = C
    CMODE =  0    op2(C) = I
    CMODE = -1    op2(C) = inv(C)
    The Skeel condition number  cond(A) = norminf( |inv(A)||A| )
    is computed by computing scaling factors R such that
    diag(R)*A*op2(C) is row equilibrated and computing the standard
    infinity-norm condition number.
Parameters:
[in]TRANS
          TRANS is CHARACTER*1
     Specifies the form of the system of equations:
       = 'N':  A * X = B     (No transpose)
       = 'T':  A**T * X = B  (Transpose)
       = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)
[in]N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
[in]KL
          KL is INTEGER
     The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
     The number of superdiagonals within the band of A.  KU >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
     The j-th column of A is stored in the j-th column of the
     array AB as follows:
     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
[in]LDAB
          LDAB is INTEGER
     The leading dimension of the array AB.  LDAB >= KL+KU+1.
[in]AFB
          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
     Details of the LU factorization of the band matrix A, as
     computed by DGBTRF.  U is stored as an upper triangular
     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
     and the multipliers used during the factorization are stored
     in rows KL+KU+2 to 2*KL+KU+1.
[in]LDAFB
          LDAFB is INTEGER
     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
[in]IPIV
          IPIV is INTEGER array, dimension (N)
     The pivot indices from the factorization A = P*L*U
     as computed by DGBTRF; row i of the matrix was interchanged
     with row IPIV(i).
[in]CMODE
          CMODE is INTEGER
     Determines op2(C) in the formula op(A) * op2(C) as follows:
     CMODE =  1    op2(C) = C
     CMODE =  0    op2(C) = I
     CMODE = -1    op2(C) = inv(C)
[in]C
          C is DOUBLE PRECISION array, dimension (N)
     The vector C in the formula op(A) * op2(C).
[out]INFO
          INFO is INTEGER
       = 0:  Successful exit.
     i > 0:  The ith argument is invalid.
[in]WORK
          WORK is DOUBLE PRECISION array, dimension (5*N).
     Workspace.
[in]IWORK
          IWORK is INTEGER array, dimension (N).
     Workspace.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012

Definition at line 169 of file dla_gbrcond.f.

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subroutine dla_gbrfsx_extended ( integer  PREC_TYPE,
integer  TRANS_TYPE,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
logical  COLEQU,
double precision, dimension( * )  C,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( ldy, * )  Y,
integer  LDY,
double precision, dimension(*)  BERR_OUT,
integer  N_NORMS,
double precision, dimension( nrhs, * )  ERR_BNDS_NORM,
double precision, dimension( nrhs, * )  ERR_BNDS_COMP,
double precision, dimension(*)  RES,
double precision, dimension(*)  AYB,
double precision, dimension(*)  DY,
double precision, dimension(*)  Y_TAIL,
double precision  RCOND,
integer  ITHRESH,
double precision  RTHRESH,
double precision  DZ_UB,
logical  IGNORE_CWISE,
integer  INFO 
)

DLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Download DLA_GBRFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DLA_GBRFSX_EXTENDED improves the computed solution to a system of
 linear equations by performing extra-precise iterative refinement
 and provides error bounds and backward error estimates for the solution.
 This subroutine is called by DGBRFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERR_BNDS_NORM
 and ERR_BNDS_COMP for details of the error bounds. Note that this
 subroutine is only resonsible for setting the second fields of
 ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters:
[in]PREC_TYPE
          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and
     P    = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X', 'E':  Extra
[in]TRANS_TYPE
          TRANS_TYPE is INTEGER
     Specifies the transposition operation on A.
     The value is defined by ILATRANS(T) where T is a CHARACTER and
     T    = 'N':  No transpose
          = 'T':  Transpose
          = 'C':  Conjugate transpose
[in]N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
[in]KL
          KL is INTEGER
     The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
     The number of superdiagonals within the band of A.  KU >= 0
[in]NRHS
          NRHS is INTEGER
     The number of right-hand-sides, i.e., the number of columns of the
     matrix B.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the N-by-N matrix AB.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of the array AB.  LDBA >= max(1,N).
[in]AFB
          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
     The factors L and U from the factorization
     A = P*L*U as computed by DGBTRF.
[in]LDAFB
          LDAFB is INTEGER
     The leading dimension of the array AF.  LDAFB >= max(1,N).
[in]IPIV
          IPIV is INTEGER array, dimension (N)
     The pivot indices from the factorization A = P*L*U
     as computed by DGBTRF; row i of the matrix was interchanged
     with row IPIV(i).
[in]COLEQU
          COLEQU is LOGICAL
     If .TRUE. then column equilibration was done to A before calling
     this routine. This is needed to compute the solution and error
     bounds correctly.
[in]C
          C is DOUBLE PRECISION array, dimension (N)
     The column scale factors for A. If COLEQU = .FALSE., C
     is not accessed. If C is input, each element of C should be a power
     of the radix to ensure a reliable solution and error estimates.
     Scaling by powers of the radix does not cause rounding errors unless
     the result underflows or overflows. Rounding errors during scaling
     lead to refining with a matrix that is not equivalent to the
     input matrix, producing error estimates that may not be
     reliable.
[in]B
          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
     The right-hand-side matrix B.
[in]LDB
          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).
[in,out]Y
          Y is DOUBLE PRECISION array, dimension
                    (LDY,NRHS)
     On entry, the solution matrix X, as computed by DGBTRS.
     On exit, the improved solution matrix Y.
[in]LDY
          LDY is INTEGER
     The leading dimension of the array Y.  LDY >= max(1,N).
[out]BERR_OUT
          BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
     On exit, BERR_OUT(j) contains the componentwise relative backward
     error for right-hand-side j from the formula
         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
     where abs(Z) is the componentwise absolute value of the matrix
     or vector Z. This is computed by DLA_LIN_BERR.
[in]N_NORMS
          N_NORMS is INTEGER
     Determines which error bounds to return (see ERR_BNDS_NORM
     and ERR_BNDS_COMP).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.
[in,out]ERR_BNDS_NORM
          ERR_BNDS_NORM is DOUBLE PRECISION array, dimension
                    (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     normwise relative error, which is defined as follows:

     Normwise relative error in the ith solution vector:
             max_j (abs(XTRUE(j,i) - X(j,i)))
            ------------------------------
                  max_j abs(X(j,i))

     The array is indexed by the type of error information as described
     below. There currently are up to three pieces of information
     returned.

     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
     right-hand side.

     The second index in ERR_BNDS_NORM(:,err) contains the following
     three fields:
     err = 1 "Trust/don't trust" boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * slamch('Epsilon').

     err = 2 "Guaranteed" error bound: The estimated forward error,
              almost certainly within a factor of 10 of the true error
              so long as the next entry is greater than the threshold
              sqrt(n) * slamch('Epsilon'). This error bound should only
              be trusted if the previous boolean is true.

     err = 3  Reciprocal condition number: Estimated normwise
              reciprocal condition number.  Compared with the threshold
              sqrt(n) * slamch('Epsilon') to determine if the error
              estimate is "guaranteed". These reciprocal condition
              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
              appropriately scaled matrix Z.
              Let Z = S*A, where S scales each row by a power of the
              radix so all absolute row sums of Z are approximately 1.

     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.
[in,out]ERR_BNDS_COMP
          ERR_BNDS_COMP is DOUBLE PRECISION array, dimension
                    (NRHS, N_ERR_BNDS)
     For each right-hand side, this array contains information about
     various error bounds and condition numbers corresponding to the
     componentwise relative error, which is defined as follows:

     Componentwise relative error in the ith solution vector:
                    abs(XTRUE(j,i) - X(j,i))
             max_j ----------------------
                         abs(X(j,i))

     The array is indexed by the right-hand side i (on which the
     componentwise relative error depends), and the type of error
     information as described below. There currently are up to three
     pieces of information returned for each right-hand side. If
     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
     the first (:,N_ERR_BNDS) entries are returned.

     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
     right-hand side.

     The second index in ERR_BNDS_COMP(:,err) contains the following
     three fields:
     err = 1 "Trust/don't trust" boolean. Trust the answer if the
              reciprocal condition number is less than the threshold
              sqrt(n) * slamch('Epsilon').

     err = 2 "Guaranteed" error bound: The estimated forward error,
              almost certainly within a factor of 10 of the true error
              so long as the next entry is greater than the threshold
              sqrt(n) * slamch('Epsilon'). This error bound should only
              be trusted if the previous boolean is true.

     err = 3  Reciprocal condition number: Estimated componentwise
              reciprocal condition number.  Compared with the threshold
              sqrt(n) * slamch('Epsilon') to determine if the error
              estimate is "guaranteed". These reciprocal condition
              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
              appropriately scaled matrix Z.
              Let Z = S*(A*diag(x)), where x is the solution for the
              current right-hand side and S scales each row of
              A*diag(x) by a power of the radix so all absolute row
              sums of Z are approximately 1.

     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.
[in]RES
          RES is DOUBLE PRECISION array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is DOUBLE PRECISION array, dimension (N)
     Workspace. This can be the same workspace passed for Y_TAIL.
[in]DY
          DY is DOUBLE PRECISION array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is DOUBLE PRECISION array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.
[in]RCOND
          RCOND is DOUBLE PRECISION
     Reciprocal scaled condition number.  This is an estimate of the
     reciprocal Skeel condition number of the matrix A after
     equilibration (if done).  If this is less than the machine
     precision (in particular, if it is zero), the matrix is singular
     to working precision.  Note that the error may still be small even
     if this number is very small and the matrix appears ill-
     conditioned.
[in]ITHRESH
          ITHRESH is INTEGER
     The maximum number of residual computations allowed for
     refinement. The default is 10. For 'aggressive' set to 100 to
     permit convergence using approximate factorizations or
     factorizations other than LU. If the factorization uses a
     technique other than Gaussian elimination, the guarantees in
     ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
[in]RTHRESH
          RTHRESH is DOUBLE PRECISION
     Determines when to stop refinement if the error estimate stops
     decreasing. Refinement will stop when the next solution no longer
     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
     default value is 0.5. For 'aggressive' set to 0.9 to permit
     convergence on extremely ill-conditioned matrices. See LAWN 165
     for more details.
[in]DZ_UB
          DZ_UB is DOUBLE PRECISION
     Determines when to start considering componentwise convergence.
     Componentwise convergence is only considered after each component
     of the solution Y is stable, which we definte as the relative
     change in each component being less than DZ_UB. The default value
     is 0.25, requiring the first bit to be stable. See LAWN 165 for
     more details.
[in]IGNORE_CWISE
          IGNORE_CWISE is LOGICAL
     If .TRUE. then ignore componentwise convergence. Default value
     is .FALSE..
[out]INFO
          INFO is INTEGER
       = 0:  Successful exit.
       < 0:  if INFO = -i, the ith argument to DGBTRS had an illegal
             value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012

Definition at line 409 of file dla_gbrfsx_extended.f.

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DOUBLE PRECISION function dla_gbrpvgrw ( integer  N,
integer  KL,
integer  KU,
integer  NCOLS,
double precision, dimension( ldab, * )  AB,
integer  LDAB,
double precision, dimension( ldafb, * )  AFB,
integer  LDAFB 
)

DLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.

Download DLA_GBRPVGRW + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DLA_GBRPVGRW computes the reciprocal pivot growth factor
 norm(A)/norm(U). The "max absolute element" norm is used. If this is
 much less than 1, the stability of the LU factorization of the
 (equilibrated) matrix A could be poor. This also means that the
 solution X, estimated condition numbers, and error bounds could be
 unreliable.
Parameters:
[in]N
          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.
[in]KL
          KL is INTEGER
     The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
     The number of superdiagonals within the band of A.  KU >= 0.
[in]NCOLS
          NCOLS is INTEGER
     The number of columns of the matrix A.  NCOLS >= 0.
[in]AB
          AB is DOUBLE PRECISION array, dimension (LDAB,N)
     On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
     The j-th column of A is stored in the j-th column of the
     array AB as follows:
     AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
[in]LDAB
          LDAB is INTEGER
     The leading dimension of the array AB.  LDAB >= KL+KU+1.
[in]AFB
          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
     Details of the LU factorization of the band matrix A, as
     computed by DGBTRF.  U is stored as an upper triangular
     band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
     and the multipliers used during the factorization are stored
     in rows KL+KU+2 to 2*KL+KU+1.
[in]LDAFB
          LDAFB is INTEGER
     The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012

Definition at line 117 of file dla_gbrpvgrw.f.

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subroutine dorgbr ( character  VECT,
integer  M,
integer  N,
integer  K,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( * )  TAU,
double precision, dimension( * )  WORK,
integer  LWORK,
integer  INFO 
)

DORGBR

Download DORGBR + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 DORGBR generates one of the real orthogonal matrices Q or P**T
 determined by DGEBRD when reducing a real matrix A to bidiagonal
 form: A = Q * B * P**T.  Q and P**T are defined as products of
 elementary reflectors H(i) or G(i) respectively.

 If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
 is of order M:
 if m >= k, Q = H(1) H(2) . . . H(k) and DORGBR returns the first n
 columns of Q, where m >= n >= k;
 if m < k, Q = H(1) H(2) . . . H(m-1) and DORGBR returns Q as an
 M-by-M matrix.

 If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**T
 is of order N:
 if k < n, P**T = G(k) . . . G(2) G(1) and DORGBR returns the first m
 rows of P**T, where n >= m >= k;
 if k >= n, P**T = G(n-1) . . . G(2) G(1) and DORGBR returns P**T as
 an N-by-N matrix.
Parameters:
[in]VECT
          VECT is CHARACTER*1
          Specifies whether the matrix Q or the matrix P**T is
          required, as defined in the transformation applied by DGEBRD:
          = 'Q':  generate Q;
          = 'P':  generate P**T.
[in]M
          M is INTEGER
          The number of rows of the matrix Q or P**T to be returned.
          M >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrix Q or P**T to be returned.
          N >= 0.
          If VECT = 'Q', M >= N >= min(M,K);
          if VECT = 'P', N >= M >= min(N,K).
[in]K
          K is INTEGER
          If VECT = 'Q', the number of columns in the original M-by-K
          matrix reduced by DGEBRD.
          If VECT = 'P', the number of rows in the original K-by-N
          matrix reduced by DGEBRD.
          K >= 0.
[in,out]A
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by DGEBRD.
          On exit, the M-by-N matrix Q or P**T.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
[in]TAU
          TAU is DOUBLE PRECISION array, dimension
                                (min(M,K)) if VECT = 'Q'
                                (min(N,K)) if VECT = 'P'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i), which determines Q or P**T, as
          returned by DGEBRD in its array argument TAUQ or TAUP.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,min(M,N)).
          For optimum performance LWORK >= min(M,N)*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
April 2012

Definition at line 158 of file dorgbr.f.

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