 |
LAPACK
3.4.2
LAPACK: Linear Algebra PACKage
|
Functions/Subroutines | |
| subroutine | sgebak (JOB, SIDE, N, ILO, IHI, SCALE, M, V, LDV, INFO) |
| SGEBAK | |
| subroutine | sgebal (JOB, N, A, LDA, ILO, IHI, SCALE, INFO) |
| SGEBAL | |
| subroutine | sgebd2 (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO) |
| SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. | |
| subroutine | sgebrd (M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) |
| SGEBRD | |
| subroutine | sgecon (NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO) |
| SGECON | |
| subroutine | sgeequ (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO) |
| SGEEQU | |
| subroutine | sgeequb (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO) |
| SGEEQUB | |
| subroutine | sgehd2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO) |
| SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. | |
| subroutine | sgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO) |
| SGEHRD | |
| subroutine | sgelq2 (M, N, A, LDA, TAU, WORK, INFO) |
| SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm. | |
| subroutine | sgelqf (M, N, A, LDA, TAU, WORK, LWORK, INFO) |
| SGELQF | |
| subroutine | sgemqrt (SIDE, TRANS, M, N, K, NB, V, LDV, T, LDT, C, LDC, WORK, INFO) |
| SGEMQRT | |
| subroutine | sgeql2 (M, N, A, LDA, TAU, WORK, INFO) |
| SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm. | |
| subroutine | sgeqlf (M, N, A, LDA, TAU, WORK, LWORK, INFO) |
| SGEQLF | |
| subroutine | sgeqp3 (M, N, A, LDA, JPVT, TAU, WORK, LWORK, INFO) |
| SGEQP3 | |
| subroutine | sgeqpf (M, N, A, LDA, JPVT, TAU, WORK, INFO) |
| SGEQPF | |
| subroutine | sgeqr2 (M, N, A, LDA, TAU, WORK, INFO) |
| SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm. | |
| subroutine | sgeqr2p (M, N, A, LDA, TAU, WORK, INFO) |
| SGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm. | |
| subroutine | sgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO) |
| SGEQRF | |
| subroutine | sgeqrfp (M, N, A, LDA, TAU, WORK, LWORK, INFO) |
| SGEQRFP | |
| subroutine | sgeqrt (M, N, NB, A, LDA, T, LDT, WORK, INFO) |
| SGEQRT | |
| subroutine | sgeqrt2 (M, N, A, LDA, T, LDT, INFO) |
| SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. | |
| recursive subroutine | sgeqrt3 (M, N, A, LDA, T, LDT, INFO) |
| SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. | |
| subroutine | sgerfs (TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO) |
| SGERFS | |
| subroutine | sgerfsx (TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) |
| SGERFSX | |
| subroutine | sgerq2 (M, N, A, LDA, TAU, WORK, INFO) |
| SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm. | |
| subroutine | sgerqf (M, N, A, LDA, TAU, WORK, LWORK, INFO) |
| SGERQF | |
| subroutine | sgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, WORK, LWORK, INFO) |
| SGESVJ | |
| subroutine | sgetf2 (M, N, A, LDA, IPIV, INFO) |
| SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm). | |
| subroutine | sgetrf (M, N, A, LDA, IPIV, INFO) |
| SGETRF | |
| subroutine | sgetri (N, A, LDA, IPIV, WORK, LWORK, INFO) |
| SGETRI | |
| subroutine | sgetrs (TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO) |
| SGETRS | |
| subroutine | shgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, INFO) |
| SHGEQZ | |
| subroutine | sla_geamv (TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) |
| SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds. | |
| REAL function | sla_gercond (TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK) |
| SLA_GERCOND estimates the Skeel condition number for a general matrix. | |
| subroutine | sla_gerfsx_extended (PREC_TYPE, TRANS_TYPE, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERRS_N, ERRS_C, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO) |
| SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. | |
| REAL function | sla_gerpvgrw (N, NCOLS, A, LDA, AF, LDAF) |
| SLA_GERPVGRW | |
| subroutine | stgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO) |
| STGEVC | |
| subroutine | stgexc (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, IFST, ILST, WORK, LWORK, INFO) |
| STGEXC | |
This is the group of real computational functions for GE matrices
| subroutine sgebak | ( | character | JOB, |
| character | SIDE, | ||
| integer | N, | ||
| integer | ILO, | ||
| integer | IHI, | ||
| real, dimension( * ) | SCALE, | ||
| integer | M, | ||
| real, dimension( ldv, * ) | V, | ||
| integer | LDV, | ||
| integer | INFO | ||
| ) |
SGEBAK
Download SGEBAK + dependencies [TGZ] [ZIP] [TXT]SGEBAK forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by SGEBAL.
| [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 SGEBAL. |
| [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 SGEBAL.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. |
| [in] | SCALE | SCALE is REAL array, dimension (N)
Details of the permutation and scaling factors, as returned
by SGEBAL. |
| [in] | M | M is INTEGER
The number of columns of the matrix V. M >= 0. |
| [in,out] | V | V is REAL array, dimension (LDV,M)
On entry, the matrix of right or left eigenvectors to be
transformed, as returned by SHSEIN or STREVC.
On exit, V is overwritten by the transformed eigenvectors. |
| [in] | LDV | LDV is INTEGER
The leading dimension of the array 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. |
Definition at line 130 of file sgebak.f.
| subroutine sgebal | ( | character | JOB, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer | ILO, | ||
| integer | IHI, | ||
| real, dimension( * ) | SCALE, | ||
| integer | INFO | ||
| ) |
SGEBAL
Download SGEBAL + dependencies [TGZ] [ZIP] [TXT]SGEBAL balances a general real matrix A. This involves, first, permuting A by a similarity transformation 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 matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.
| [in] | JOB | JOB is CHARACTER*1
Specifies the operations to be performed on A:
= 'N': none: simply set ILO = 1, IHI = N, SCALE(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 matrix A. N >= 0. |
| [in,out] | A | A is REAL 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.
See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= 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 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N. |
| [out] | SCALE | SCALE is REAL array, dimension (N)
Details of the permutations and scaling factors applied to
A. If P(j) is the index of the row and column interchanged
with row and column j and D(j) is the scaling factor
applied to row and column j, then
SCALE(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] | INFO | INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
The permutations consist of row and column interchanges which put
the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ILO and IHI mark the starting
and ending columns of the submatrix B. Balancing consists of applying
a diagonal similarity transformation inv(D) * B * D to make the
1-norms of each row of B and its corresponding column nearly equal.
The output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix D is
returned in the vector SCALE.
This subroutine is based on the EISPACK routine BALANC.
Modified by Tzu-Yi Chen, Computer Science Division, University of
California at Berkeley, USA Definition at line 161 of file sgebal.f.
| subroutine sgebd2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | D, | ||
| real, dimension( * ) | E, | ||
| real, dimension( * ) | TAUQ, | ||
| real, dimension( * ) | TAUP, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Download SGEBD2 + dependencies [TGZ] [ZIP] [TXT]SGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
| [in] | M | M is INTEGER
The number of rows in the matrix A. M >= 0. |
| [in] | N | N is INTEGER
The number of columns in the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the orthogonal matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the orthogonal matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors.
See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | D | D is REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i). |
| [out] | E | E is REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
| [out] | TAUQ | TAUQ is REAL array dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details. |
| [out] | TAUP | TAUP is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details. |
| [out] | WORK | WORK is REAL array, dimension (max(M,N)) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i). Definition at line 190 of file sgebd2.f.
| subroutine sgebrd | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | D, | ||
| real, dimension( * ) | E, | ||
| real, dimension( * ) | TAUQ, | ||
| real, dimension( * ) | TAUP, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGEBRD
Download SGEBRD + dependencies [TGZ] [ZIP] [TXT]SGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
| [in] | M | M is INTEGER
The number of rows in the matrix A. M >= 0. |
| [in] | N | N is INTEGER
The number of columns in the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the orthogonal matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the orthogonal matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the orthogonal matrix P as
a product of elementary reflectors.
See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | D | D is REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i). |
| [out] | E | E is REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
| [out] | TAUQ | TAUQ is REAL array dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details. |
| [out] | TAUP | TAUP is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details. |
| [out] | WORK | WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
| [in] | LWORK | LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,M,N).
For optimum performance LWORK >= (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. |
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i). Definition at line 205 of file sgebrd.f.
| subroutine sgecon | ( | character | NORM, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real | ANORM, | ||
| real | RCOND, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
SGECON
Download SGECON + dependencies [TGZ] [ZIP] [TXT] SGECON estimates the reciprocal of the condition number of a general
real matrix A, in either the 1-norm or the infinity-norm, using
the LU factorization computed by SGETRF.
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)) ). | [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] | A | A is REAL array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | ANORM | ANORM is REAL
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 REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(norm(A) * norm(inv(A))). |
| [out] | WORK | WORK is REAL array, dimension (4*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 |
Definition at line 124 of file sgecon.f.
| subroutine sgeequ | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | R, | ||
| real, dimension( * ) | C, | ||
| real | ROWCND, | ||
| real | COLCND, | ||
| real | AMAX, | ||
| integer | INFO | ||
| ) |
SGEEQU
Download SGEEQU + dependencies [TGZ] [ZIP] [TXT]SGEEQU 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 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.
| [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 whose equilibration factors are
to be computed. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | R | R is REAL array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A. |
| [out] | C | C is REAL array, dimension (N)
If INFO = 0, C contains the column scale factors for A. |
| [out] | ROWCND | ROWCND is REAL
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 REAL
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 REAL
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 |
Definition at line 139 of file sgeequ.f.
| subroutine sgeequb | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | R, | ||
| real, dimension( * ) | C, | ||
| real | ROWCND, | ||
| real | COLCND, | ||
| real | AMAX, | ||
| integer | INFO | ||
| ) |
SGEEQUB
Download SGEEQUB + dependencies [TGZ] [ZIP] [TXT]SGEEQUB 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 SGEEQU 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).
| [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 whose equilibration factors are
to be computed. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | R | R is REAL array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A. |
| [out] | C | C is REAL array, dimension (N)
If INFO = 0, C contains the column scale factors for A. |
| [out] | ROWCND | ROWCND is REAL
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 REAL
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 REAL
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 |
Definition at line 146 of file sgeequb.f.
| subroutine sgehd2 | ( | integer | N, |
| integer | ILO, | ||
| integer | IHI, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
Download SGEHD2 + dependencies [TGZ] [ZIP] [TXT]SGEHD2 reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q**T * A * Q = H .
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | ILO | ILO is INTEGER |
| [in] | IHI | IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N). |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | TAU | TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i). Definition at line 150 of file sgehd2.f.
| subroutine sgehrd | ( | integer | N, |
| integer | ILO, | ||
| integer | IHI, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGEHRD
Download SGEHRD + dependencies [TGZ] [ZIP] [TXT]SGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q**T * A * Q = H .
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in] | ILO | ILO is INTEGER |
| [in] | IHI | IHI is INTEGER
It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [out] | TAU | TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero. |
| [out] | WORK | WORK is REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
| [in] | LWORK | LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= 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. |
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
This file is a slight modification of LAPACK-3.0's DGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See DLAHR2.) Definition at line 169 of file sgehrd.f.
| subroutine sgelq2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
Download SGELQ2 + dependencies [TGZ] [ZIP] [TXT]SGELQ2 computes an LQ factorization of a real m by n matrix A: A = L * Q.
| [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)
On entry, the m by n matrix A.
On exit, the elements on and below the diagonal of the array
contain the m by min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (M) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i). Definition at line 122 of file sgelq2.f.
| subroutine sgelqf | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGELQF
Download SGELQF + dependencies [TGZ] [ZIP] [TXT]SGELQF computes an LQ factorization of a real M-by-N matrix A: A = L * Q.
| [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)
On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the m-by-min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL 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,M).
For optimum performance LWORK >= M*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 |
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in TAU(i). Definition at line 136 of file sgelqf.f.
| subroutine sgemqrt | ( | character | SIDE, |
| character | TRANS, | ||
| integer | M, | ||
| integer | N, | ||
| integer | K, | ||
| integer | NB, | ||
| real, dimension( ldv, * ) | V, | ||
| integer | LDV, | ||
| real, dimension( ldt, * ) | T, | ||
| integer | LDT, | ||
| real, dimension( ldc, * ) | C, | ||
| integer | LDC, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEMQRT
Download SGEMQRT + dependencies [TGZ] [ZIP] [TXT] SGEMQRT overwrites the general real M-by-N matrix C with
SIDE = 'L' SIDE = 'R'
TRANS = 'N': Q C C Q
TRANS = 'T': Q**T C C Q**T
where Q is a real orthogonal matrix defined as the product of K
elementary reflectors:
Q = H(1) H(2) . . . H(K) = I - V T V**T
generated using the compact WY representation as returned by SGEQRT.
Q is of order M if SIDE = 'L' and of order N if SIDE = 'R'. | [in] | SIDE | SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right. |
| [in] | TRANS | TRANS is CHARACTER*1
= 'N': No transpose, apply Q;
= 'T': Transpose, apply Q**T. |
| [in] | M | M is INTEGER
The number of rows of the matrix C. M >= 0. |
| [in] | N | N is INTEGER
The number of columns of the matrix C. N >= 0. |
| [in] | K | K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0. |
| [in] | NB | NB is INTEGER
The block size used for the storage of T. K >= NB >= 1.
This must be the same value of NB used to generate T
in CGEQRT. |
| [in] | V | V is REAL array, dimension (LDV,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
CGEQRT in the first K columns of its array argument A. |
| [in] | LDV | LDV is INTEGER
The leading dimension of the array V.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N). |
| [in] | T | T is REAL array, dimension (LDT,K)
The upper triangular factors of the block reflectors
as returned by CGEQRT, stored as a NB-by-N matrix. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= NB. |
| [in,out] | C | C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q C, Q**T C, C Q**T or C Q. |
| [in] | LDC | LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M). |
| [out] | WORK | WORK is REAL array. The dimension of WORK is
N*NB if SIDE = 'L', or M*NB if SIDE = 'R'. |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
Definition at line 168 of file sgemqrt.f.
| subroutine sgeql2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
Download SGEQL2 + dependencies [TGZ] [ZIP] [TXT]SGEQL2 computes a QL factorization of a real m by n matrix A: A = Q * L.
| [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)
On entry, the m by n matrix A.
On exit, if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the m by n lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of elementary reflectors
(see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i). Definition at line 124 of file sgeql2.f.
| subroutine sgeqlf | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGEQLF
Download SGEQLF + dependencies [TGZ] [ZIP] [TXT]SGEQLF computes a QL factorization of a real M-by-N matrix A: A = Q * L.
| [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)
On entry, the M-by-N matrix A.
On exit,
if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the M-by-N lower trapezoidal matrix L;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of elementary reflectors
(see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL 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,N).
For optimum performance LWORK >= 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 |
The matrix Q is represented as a product of elementary reflectors
Q = H(k) . . . H(2) H(1), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i). Definition at line 139 of file sgeqlf.f.
| subroutine sgeqp3 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | JPVT, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGEQP3
Download SGEQP3 + dependencies [TGZ] [ZIP] [TXT]SGEQP3 computes a QR factorization with column pivoting of a matrix A: A*P = Q*R using Level 3 BLAS.
| [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)
On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix R; the elements below
the diagonal, together with the array TAU, represent the
orthogonal matrix Q as a product of min(M,N) elementary
reflectors. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [in,out] | JPVT | JPVT is INTEGER array, dimension (N)
On entry, if JPVT(J).ne.0, the J-th column of A is permuted
to the front of A*P (a leading column); if JPVT(J)=0,
the J-th column of A is a free column.
On exit, if JPVT(J)=K, then the J-th column of A*P was the
the K-th column of A. |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors. |
| [out] | WORK | WORK is REAL 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 >= 3*N+1.
For optimal performance LWORK >= 2*N+( N+1 )*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. |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real/complex vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in TAU(i). Definition at line 152 of file sgeqp3.f.
| subroutine sgeqpf | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | JPVT, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEQPF
Download SGEQPF + dependencies [TGZ] [ZIP] [TXT]This routine is deprecated and has been replaced by routine SGEQP3. SGEQPF computes a QR factorization with column pivoting of a real M-by-N matrix A: A*P = Q*R.
| [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)
On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper triangular matrix R; the elements
below the diagonal, together with the array TAU,
represent the orthogonal matrix Q as a product of
min(m,n) elementary reflectors. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [in,out] | JPVT | JPVT is INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
to the front of A*P (a leading column); if JPVT(i) = 0,
the i-th column of A is a free column.
On exit, if JPVT(i) = k, then the i-th column of A*P
was the k-th column of A. |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors. |
| [out] | WORK | WORK is REAL array, dimension (3*N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(n)
Each H(i) has the form
H = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth column of P is the ith canonical unit vector.
Partial column norm updating strategy modified by
Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
University of Zagreb, Croatia.
-- April 2011 --
For more details see LAPACK Working Note 176. Definition at line 143 of file sgeqpf.f.
| subroutine sgeqr2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Download SGEQR2 + dependencies [TGZ] [ZIP] [TXT]SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.
| [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)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i). Definition at line 122 of file sgeqr2.f.
| subroutine sgeqr2p | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEQR2P computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
Download SGEQR2P + dependencies [TGZ] [ZIP] [TXT]SGEQR2P computes a QR factorization of a real m by n matrix A: A = Q * R.
| [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)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of elementary reflectors (see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i). Definition at line 122 of file sgeqr2p.f.
| subroutine sgeqrf | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGEQRF
Download SGEQRF + dependencies [TGZ] [ZIP] [TXT]SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.
| [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)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL 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,N).
For optimum performance LWORK >= 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 |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i). Definition at line 137 of file sgeqrf.f.
| subroutine sgeqrfp | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGEQRFP
Download SGEQRFP + dependencies [TGZ] [ZIP] [TXT]SGEQRFP computes a QR factorization of a real M-by-N matrix A: A = Q * R.
| [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)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors (see Further
Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL 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,N).
For optimum performance LWORK >= 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 |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i). Definition at line 137 of file sgeqrfp.f.
| subroutine sgeqrt | ( | integer | M, |
| integer | N, | ||
| integer | NB, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldt, * ) | T, | ||
| integer | LDT, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGEQRT
Download SGEQRT + dependencies [TGZ] [ZIP] [TXT]SGEQRT computes a blocked QR factorization of a real M-by-N matrix A using the compact WY representation of Q.
| [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] | NB | NB is INTEGER
The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if M >= N); the elements below the diagonal
are the columns of V. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | T | T is REAL array, dimension (LDT,MIN(M,N))
The upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks. See below
for further details. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= NB. |
| [out] | WORK | WORK is REAL array, dimension (NB*N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A.
Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each
block is of order NB except for the last block, which is of order
IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
for the last block) T's are stored in the NB-by-N matrix T as
T = (T1 T2 ... TB). Definition at line 142 of file sgeqrt.f.
| subroutine sgeqrt2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldt, * ) | T, | ||
| integer | LDT, | ||
| integer | INFO | ||
| ) |
SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
Download SGEQRT2 + dependencies [TGZ] [ZIP] [TXT]SGEQRT2 computes a QR factorization of a real M-by-N matrix A, using the compact WY representation of Q.
| [in] | M | M is INTEGER
The number of rows of the matrix A. M >= N. |
| [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)
On entry, the real M-by-N matrix A. On exit, the elements on and
above the diagonal contain the N-by-N upper triangular matrix R; the
elements below the diagonal are the columns of V. See below for
further details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | T | T is REAL array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N). |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A. The
block reflector H is then given by
H = I - V * T * V**T
where V**T is the transpose of V. Definition at line 128 of file sgeqrt2.f.
| recursive subroutine sgeqrt3 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldt, * ) | T, | ||
| integer | LDT, | ||
| integer | INFO | ||
| ) |
SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
Download SGEQRT3 + dependencies [TGZ] [ZIP] [TXT]SGEQRT3 recursively computes a QR factorization of a real M-by-N matrix A, using the compact WY representation of Q. Based on the algorithm of Elmroth and Gustavson, IBM J. Res. Develop. Vol 44 No. 4 July 2000.
| [in] | M | M is INTEGER
The number of rows of the matrix A. M >= N. |
| [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)
On entry, the real M-by-N matrix A. On exit, the elements on and
above the diagonal contain the N-by-N upper triangular matrix R; the
elements below the diagonal are the columns of V. See below for
further details. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | T | T is REAL array, dimension (LDT,N)
The N-by-N upper triangular factor of the block reflector.
The elements on and above the diagonal contain the block
reflector T; the elements below the diagonal are not used.
See below for further details. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N). |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is
V = ( 1 )
( v1 1 )
( v1 v2 1 )
( v1 v2 v3 )
( v1 v2 v3 )
where the vi's represent the vectors which define H(i), which are returned
in the matrix A. The 1's along the diagonal of V are not stored in A. The
block reflector H is then given by
H = I - V * T * V**T
where V**T is the transpose of V.
For details of the algorithm, see Elmroth and Gustavson (cited above). Definition at line 133 of file sgeqrt3.f.
| subroutine sgerfs | ( | character | TRANS, |
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldx, * ) | X, | ||
| integer | LDX, | ||
| real, dimension( * ) | FERR, | ||
| real, dimension( * ) | BERR, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
SGERFS
Download SGERFS + dependencies [TGZ] [ZIP] [TXT]SGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution.
| [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] | 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] | A | A is REAL array, dimension (LDA,N)
The original N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i). |
| [in] | B | B is REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
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 REAL 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 REAL 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 REAL 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 |
ITMAX is the maximum number of steps of iterative refinement.
Definition at line 185 of file sgerfs.f.
| subroutine sgerfsx | ( | character | TRANS, |
| character | EQUED, | ||
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | R, | ||
| real, dimension( * ) | C, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldx , * ) | X, | ||
| integer | LDX, | ||
| real | RCOND, | ||
| real, dimension( * ) | BERR, | ||
| integer | N_ERR_BNDS, | ||
| real, dimension( nrhs, * ) | ERR_BNDS_NORM, | ||
| real, dimension( nrhs, * ) | ERR_BNDS_COMP, | ||
| integer | NPARAMS, | ||
| real, dimension( * ) | PARAMS, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK, | ||
| integer | INFO | ||
| ) |
SGERFSX
Download SGERFSX + dependencies [TGZ] [ZIP] [TXT] SGERFSX 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.| [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] | 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] | A | A is REAL array, dimension (LDA,N)
The original N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i). |
| [in] | R | R is REAL 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.
If R is accessed, 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] | C | C is REAL 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.
If C is accessed, 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 REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
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 REAL
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 REAL 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 REAL 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.
See Lapack Working Note 165 for further details and extra
cautions. |
| [out] | ERR_BNDS_COMP | ERR_BNDS_COMP is REAL 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.
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 REAL 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: 1.0
= 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 REAL 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. |
Definition at line 412 of file sgerfsx.f.
| subroutine sgerq2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
Download SGERQ2 + dependencies [TGZ] [ZIP] [TXT]SGERQ2 computes an RQ factorization of a real m by n matrix A: A = R * Q.
| [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)
On entry, the m by n matrix A.
On exit, if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the m by n upper trapezoidal matrix R; the remaining
elements, with the array TAU, represent the orthogonal matrix
Q as a product of elementary reflectors (see Further
Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL array, dimension (M) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i). Definition at line 124 of file sgerq2.f.
| subroutine sgerqf | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | TAU, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGERQF
Download SGERQF + dependencies [TGZ] [ZIP] [TXT]SGERQF computes an RQ factorization of a real M-by-N matrix A: A = R * Q.
| [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)
On entry, the M-by-N matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see Further Details). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | TAU | TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| [out] | WORK | WORK is REAL 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,M).
For optimum performance LWORK >= M*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 |
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i). Definition at line 139 of file sgerqf.f.
| subroutine sgesvj | ( | character*1 | JOBA, |
| character*1 | JOBU, | ||
| character*1 | JOBV, | ||
| integer | M, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( n ) | SVA, | ||
| integer | MV, | ||
| real, dimension( ldv, * ) | V, | ||
| integer | LDV, | ||
| real, dimension( lwork ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGESVJ
Download SGESVJ + dependencies [TGZ] [ZIP] [TXT] SGESVJ computes the singular value decomposition (SVD) of a real
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively. | [in] | JOBA | JOBA is CHARACTER* 1
Specifies the structure of A.
= 'L': The input matrix A is lower triangular;
= 'U': The input matrix A is upper triangular;
= 'G': The input matrix A is general M-by-N matrix, M >= N. |
| [in] | JOBU | JOBU is CHARACTER*1
Specifies whether to compute the left singular vectors
(columns of U):
= 'U': The left singular vectors corresponding to the nonzero
singular values are computed and returned in the leading
columns of A. See more details in the description of A.
The default numerical orthogonality threshold is set to
approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
= 'C': Analogous to JOBU='U', except that user can control the
level of numerical orthogonality of the computed left
singular vectors. TOL can be set to TOL = CTOL*EPS, where
CTOL is given on input in the array WORK.
No CTOL smaller than ONE is allowed. CTOL greater
than 1 / EPS is meaningless. The option 'C'
can be used if M*EPS is satisfactory orthogonality
of the computed left singular vectors, so CTOL=M could
save few sweeps of Jacobi rotations.
See the descriptions of A and WORK(1).
= 'N': The matrix U is not computed. However, see the
description of A. |
| [in] | JOBV | JOBV is CHARACTER*1
Specifies whether to compute the right singular vectors, that
is, the matrix V:
= 'V' : the matrix V is computed and returned in the array V
= 'A' : the Jacobi rotations are applied to the MV-by-N
array V. In other words, the right singular vector
matrix V is not computed explicitly; instead it is
applied to an MV-by-N matrix initially stored in the
first MV rows of V.
= 'N' : the matrix V is not computed and the array V is not
referenced |
| [in] | M | M is INTEGER
The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0. |
| [in] | N | N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C':
If INFO .EQ. 0 :
RANKA orthonormal columns of U are returned in the
leading RANKA columns of the array A. Here RANKA <= N
is the number of computed singular values of A that are
above the underflow threshold SLAMCH('S'). The singular
vectors corresponding to underflowed or zero singular
values are not computed. The value of RANKA is returned
in the array WORK as RANKA=NINT(WORK(2)). Also see the
descriptions of SVA and WORK. The computed columns of U
are mutually numerically orthogonal up to approximately
TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
see the description of JOBU.
If INFO .GT. 0,
the procedure SGESVJ did not converge in the given number
of iterations (sweeps). In that case, the computed
columns of U may not be orthogonal up to TOL. The output
U (stored in A), SIGMA (given by the computed singular
values in SVA(1:N)) and V is still a decomposition of the
input matrix A in the sense that the residual
||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
If JOBU .EQ. 'N':
If INFO .EQ. 0 :
Note that the left singular vectors are 'for free' in the
one-sided Jacobi SVD algorithm. However, if only the
singular values are needed, the level of numerical
orthogonality of U is not an issue and iterations are
stopped when the columns of the iterated matrix are
numerically orthogonal up to approximately M*EPS. Thus,
on exit, A contains the columns of U scaled with the
corresponding singular values.
If INFO .GT. 0 :
the procedure SGESVJ did not converge in the given number
of iterations (sweeps). |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [out] | SVA | SVA is REAL array, dimension (N)
On exit,
If INFO .EQ. 0 :
depending on the value SCALE = WORK(1), we have:
If SCALE .EQ. ONE:
SVA(1:N) contains the computed singular values of A.
During the computation SVA contains the Euclidean column
norms of the iterated matrices in the array A.
If SCALE .NE. ONE:
The singular values of A are SCALE*SVA(1:N), and this
factored representation is due to the fact that some of the
singular values of A might underflow or overflow.
If INFO .GT. 0 :
the procedure SGESVJ did not converge in the given number of
iterations (sweeps) and SCALE*SVA(1:N) may not be accurate. |
| [in] | MV | MV is INTEGER
If JOBV .EQ. 'A', then the product of Jacobi rotations in SGESVJ
is applied to the first MV rows of V. See the description of JOBV. |
| [in,out] | V | V is REAL array, dimension (LDV,N)
If JOBV = 'V', then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = 'A', then V contains the product of the computed right
singular vector matrix and the initial matrix in
the array V.
If JOBV = 'N', then V is not referenced. |
| [in] | LDV | LDV is INTEGER
The leading dimension of the array V, LDV .GE. 1.
If JOBV .EQ. 'V', then LDV .GE. max(1,N).
If JOBV .EQ. 'A', then LDV .GE. max(1,MV) . |
| [in,out] | WORK | WORK is REAL array, dimension max(4,M+N).
On entry,
If JOBU .EQ. 'C' :
WORK(1) = CTOL, where CTOL defines the threshold for convergence.
The process stops if all columns of A are mutually
orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
It is required that CTOL >= ONE, i.e. it is not
allowed to force the routine to obtain orthogonality
below EPSILON.
On exit,
WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
are the computed singular vcalues of A.
(See description of SVA().)
WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
singular values.
WORK(3) = NINT(WORK(3)) is the number of the computed singular
values that are larger than the underflow threshold.
WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
rotations needed for numerical convergence.
WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
This is useful information in cases when SGESVJ did
not converge, as it can be used to estimate whether
the output is stil useful and for post festum analysis.
WORK(6) = the largest absolute value over all sines of the
Jacobi rotation angles in the last sweep. It can be
useful for a post festum analysis. |
| [in] | LWORK | LWORK is INTEGER
length of WORK, WORK >= MAX(6,M+N) |
| [out] | INFO | INFO is INTEGER
= 0 : successful exit.
< 0 : if INFO = -i, then the i-th argument had an illegal value
> 0 : SGESVJ did not converge in the maximal allowed number (30)
of sweeps. The output may still be useful. See the
description of WORK. |
Definition at line 321 of file sgesvj.f.
| subroutine sgetf2 | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| integer | INFO | ||
| ) |
SGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
Download SGETF2 + dependencies [TGZ] [ZIP] [TXT] SGETF2 computes an LU factorization of a general m-by-n matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the right-looking Level 2 BLAS version of the algorithm. | [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)
On entry, the m by n matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [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 = -k, the k-th argument had an illegal value
> 0: if INFO = k, U(k,k) 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. |
Definition at line 109 of file sgetf2.f.
| subroutine sgetrf | ( | integer | M, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| integer | INFO | ||
| ) |
SGETRF
Download SGETRF + dependencies [TGZ] [ZIP] [TXT] SGETRF computes an LU factorization of a general M-by-N matrix A
using partial pivoting with row interchanges.
The factorization has the form
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
This is the right-looking Level 3 BLAS version of the algorithm. | [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)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M). |
| [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. |
Definition at line 109 of file sgetrf.f.
| subroutine sgetri | ( | integer | N, |
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SGETRI
Download SGETRI + dependencies [TGZ] [ZIP] [TXT]SGETRI computes the inverse of a matrix using the LU factorization computed by SGETRF. This method inverts U and then computes inv(A) by solving the system inv(A)*L = inv(U) for inv(A).
| [in] | N | N is INTEGER
The order of the matrix A. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the factors L and U from the factorization
A = P*L*U as computed by SGETRF.
On exit, if INFO = 0, the inverse of the original matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i). |
| [out] | WORK | WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO=0, then WORK(1) returns the optimal LWORK. |
| [in] | LWORK | LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
For optimal performance LWORK >= N*NB, where NB is
the optimal blocksize returned by ILAENV.
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
> 0: if INFO = i, U(i,i) is exactly zero; the matrix is
singular and its inverse could not be computed. |
Definition at line 115 of file sgetri.f.
| subroutine sgetrs | ( | character | TRANS, |
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| integer, dimension( * ) | IPIV, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| integer | INFO | ||
| ) |
SGETRS
Download SGETRS + dependencies [TGZ] [ZIP] [TXT] SGETRS solves a system of linear equations
A * X = B or A**T * X = B
with a general N-by-N matrix A using the LU factorization computed
by SGETRF. | [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] | NRHS | NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
The factors L and U from the factorization A = P*L*U
as computed by SGETRF. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
The pivot indices from SGETRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i). |
| [in,out] | B | B is REAL 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 |
Definition at line 122 of file sgetrs.f.
| subroutine shgeqz | ( | character | JOB, |
| character | COMPQ, | ||
| character | COMPZ, | ||
| integer | N, | ||
| integer | ILO, | ||
| integer | IHI, | ||
| real, dimension( ldh, * ) | H, | ||
| integer | LDH, | ||
| real, dimension( ldt, * ) | T, | ||
| integer | LDT, | ||
| real, dimension( * ) | ALPHAR, | ||
| real, dimension( * ) | ALPHAI, | ||
| real, dimension( * ) | BETA, | ||
| real, dimension( ldq, * ) | Q, | ||
| integer | LDQ, | ||
| real, dimension( ldz, * ) | Z, | ||
| integer | LDZ, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
SHGEQZ
Download SHGEQZ + dependencies [TGZ] [ZIP] [TXT] SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):
A = Q1*H*Z1**T, B = Q1*T*Z1**T,
as computed by SGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also reduced to generalized Schur form,
H = Q*S*Z**T, T = Q*P*Z**T,
where Q and Z are orthogonal matrices, P is an upper triangular
matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
diagonal blocks.
The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
eigenvalues.
Additionally, the 2-by-2 upper triangular diagonal blocks of P
corresponding to 2-by-2 blocks of S are reduced to positive diagonal
form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
P(j,j) > 0, and P(j+1,j+1) > 0.
Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
generalized nonsymmetric eigenvalue problem (GNEP)
A*x = lambda*B*x
and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
alternate form of the GNEP
mu*A*y = B*y.
Real eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256. | [in] | JOB | JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form. |
| [in] | COMPQ | COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry and
the product Q1*Q is returned. |
| [in] | COMPZ | COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned. |
| [in] | N | N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0. |
| [in] | ILO | ILO is INTEGER |
| [in] | IHI | IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0. |
| [in,out] | H | H is REAL array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal blocks of H match those of S, but
the rest of H is unspecified. |
| [in] | LDH | LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ). |
| [in,out] | T | T is REAL array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if H(j+1,j) is
non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
T(j+1,j+1) > 0.
If JOB = 'E', the diagonal blocks of T match those of P, but
the rest of T is unspecified. |
| [in] | LDT | LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ). |
| [out] | ALPHAR | ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP. |
| [out] | ALPHAI | ALPHAI is REAL array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). |
| [out] | BETA | BETA is REAL array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha. Since either lambda or mu may overflow,
they should not, in general, be computed. |
| [in,out] | Q | Q is REAL array, dimension (LDQ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPZ = 'N'. |
| [in] | LDQ | LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N. |
| [in,out] | Z | Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
orthogonal matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'. |
| [in] | LDZ | LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N. |
| [out] | WORK | WORK is REAL 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,N).
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
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct. |
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom. Definition at line 303 of file shgeqz.f.
| subroutine sla_geamv | ( | integer | TRANS, |
| integer | M, | ||
| integer | N, | ||
| real | ALPHA, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( * ) | X, | ||
| integer | INCX, | ||
| real | BETA, | ||
| real, dimension( * ) | Y, | ||
| integer | INCY | ||
| ) |
SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
Download SLA_GEAMV + dependencies [TGZ] [ZIP] [TXT] SLA_GEAMV 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. | [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] | ALPHA | ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit. |
| [in] | A | A is REAL array of DIMENSION ( LDA, n )
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit. |
| [in] | LDA | LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, m ).
Unchanged on exit. |
| [in] | X | X is REAL 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 REAL
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 REAL
Array of DIMENSION at least
( 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. |
Definition at line 174 of file sla_geamv.f.
| REAL function sla_gercond | ( | character | TRANS, |
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| integer | CMODE, | ||
| real, dimension( * ) | C, | ||
| integer | INFO, | ||
| real, dimension( * ) | WORK, | ||
| integer, dimension( * ) | IWORK | ||
| ) |
SLA_GERCOND estimates the Skeel condition number for a general matrix.
Download SLA_GERCOND + dependencies [TGZ] [ZIP] [TXT] SLA_GERCOND 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. | [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] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by SGETRF; 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 REAL 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 REAL array, dimension (3*N).
Workspace. |
| [in] | IWORK | IWORK is INTEGER array, dimension (N).
Workspace.2 |
Definition at line 150 of file sla_gercond.f.
| subroutine sla_gerfsx_extended | ( | integer | PREC_TYPE, |
| integer | TRANS_TYPE, | ||
| integer | N, | ||
| integer | NRHS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF, | ||
| integer, dimension( * ) | IPIV, | ||
| logical | COLEQU, | ||
| real, dimension( * ) | C, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldy, * ) | Y, | ||
| integer | LDY, | ||
| real, dimension( * ) | BERR_OUT, | ||
| integer | N_NORMS, | ||
| real, dimension( nrhs, * ) | ERRS_N, | ||
| real, dimension( nrhs, * ) | ERRS_C, | ||
| real, dimension( * ) | RES, | ||
| real, dimension( * ) | AYB, | ||
| real, dimension( * ) | DY, | ||
| real, dimension( * ) | Y_TAIL, | ||
| real | RCOND, | ||
| integer | ITHRESH, | ||
| real | RTHRESH, | ||
| real | DZ_UB, | ||
| logical | IGNORE_CWISE, | ||
| integer | INFO | ||
| ) |
SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
Download SLA_GERFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]SLA_GERFSX_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 SGERFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERRS_N and ERRS_C for details of the error bounds. Note that this subroutine is only resonsible for setting the second fields of ERRS_N and ERRS_C.
| [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] | NRHS | NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B. |
| [in] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
| [in] | IPIV | IPIV is INTEGER array, dimension (N)
The pivot indices from the factorization A = P*L*U
as computed by SGETRF; 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 REAL 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 REAL 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 REAL array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by SGETRS.
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 REAL 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 SLA_LIN_BERR. |
| [in] | N_NORMS | N_NORMS is INTEGER
Determines which error bounds to return (see ERRS_N
and ERRS_C).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds. |
| [in,out] | ERRS_N | ERRS_N is REAL 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 ERRS_N(i,:) corresponds to the ith
right-hand side.
The second index in ERRS_N(:,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] | ERRS_C | ERRS_C is REAL 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
ERRS_C is not accessed. If N_ERR_BNDS .LT. 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERRS_C(i,:) corresponds to the ith
right-hand side.
The second index in ERRS_C(:,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 REAL array, dimension (N)
Workspace to hold the intermediate residual. |
| [in] | AYB | AYB is REAL array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL. |
| [in] | DY | DY is REAL array, dimension (N)
Workspace to hold the intermediate solution. |
| [in] | Y_TAIL | Y_TAIL is REAL array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution. |
| [in] | RCOND | RCOND is REAL
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
ERRS_N and ERRS_C may no longer be trustworthy. |
| [in] | RTHRESH | RTHRESH is REAL
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 REAL
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 SGETRS had an illegal
value |
Definition at line 393 of file sla_gerfsx_extended.f.
| REAL function sla_gerpvgrw | ( | integer | N, |
| integer | NCOLS, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldaf, * ) | AF, | ||
| integer | LDAF | ||
| ) |
SLA_GERPVGRW
Download SLA_GERPVGRW + dependencies [TGZ] [ZIP] [TXT]SLA_GERPVGRW 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.
| [in] | N | N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0. |
| [in] | NCOLS | NCOLS is INTEGER
The number of columns of the matrix A. NCOLS >= 0. |
| [in] | A | A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in] | AF | AF is REAL array, dimension (LDAF,N)
The factors L and U from the factorization
A = P*L*U as computed by SGETRF. |
| [in] | LDAF | LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N). |
Definition at line 98 of file sla_gerpvgrw.f.
| subroutine stgevc | ( | character | SIDE, |
| character | HOWMNY, | ||
| logical, dimension( * ) | SELECT, | ||
| integer | N, | ||
| real, dimension( lds, * ) | S, | ||
| integer | LDS, | ||
| real, dimension( ldp, * ) | P, | ||
| integer | LDP, | ||
| real, dimension( ldvl, * ) | VL, | ||
| integer | LDVL, | ||
| real, dimension( ldvr, * ) | VR, | ||
| integer | LDVR, | ||
| integer | MM, | ||
| integer | M, | ||
| real, dimension( * ) | WORK, | ||
| integer | INFO | ||
| ) |
STGEVC
Download STGEVC + dependencies [TGZ] [ZIP] [TXT] STGEVC computes some or all of the right and/or left eigenvectors of
a pair of real matrices (S,P), where S is a quasi-triangular matrix
and P is upper triangular. Matrix pairs of this type are produced by
the generalized Schur factorization of a matrix pair (A,B):
A = Q*S*Z**T, B = Q*P*Z**T
as computed by SGGHRD + SHGEQZ.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate tranpose of y.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal blocks of S and P.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices.
If Q and Z are the orthogonal factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B). | [in] | SIDE | SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors. |
| [in] | HOWMNY | HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT. |
| [in] | SELECT | SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed. If w(j) is a real eigenvalue, the corresponding
real eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector
is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
set to .FALSE..
Not referenced if HOWMNY = 'A' or 'B'. |
| [in] | N | N is INTEGER
The order of the matrices S and P. N >= 0. |
| [in] | S | S is REAL array, dimension (LDS,N)
The upper quasi-triangular matrix S from a generalized Schur
factorization, as computed by SHGEQZ. |
| [in] | LDS | LDS is INTEGER
The leading dimension of array S. LDS >= max(1,N). |
| [in] | P | P is REAL array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by SHGEQZ.
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
of S must be in positive diagonal form. |
| [in] | LDP | LDP is INTEGER
The leading dimension of array P. LDP >= max(1,N). |
| [in,out] | VL | VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of left Schur vectors returned by SHGEQZ).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = 'R'. |
| [in] | LDVL | LDVL is INTEGER
The leading dimension of array VL. LDVL >= 1, and if
SIDE = 'L' or 'B', LDVL >= N. |
| [in,out] | VR | VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Z (usually the orthogonal matrix Z
of right Schur vectors returned by SHGEQZ).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B' or 'b', the matrix Z*X;
if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
specified by SELECT, stored consecutively in the
columns of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = 'L'. |
| [in] | LDVR | LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N. |
| [in] | MM | MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M. |
| [out] | M | M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
is set to N. Each selected real eigenvector occupies one
column and each selected complex eigenvector occupies two
columns. |
| [out] | WORK | WORK is REAL array, dimension (6*N) |
| [out] | INFO | INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
eigenvalue. |
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diagonal
WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
Rowwise vs. columnwise solution methods:
------- -- ---------- -------- -------
Finding a generalized eigenvector consists basically of solving the
singular triangular system
(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
Consider finding the i-th right eigenvector (assume all eigenvalues
are real). The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
k=j k=j
where C = (A - w B) (The components v(i+1:n) are 0.)
The "rowwise" method is:
(1) v(i) := 1
for j = i-1,. . .,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1
(3) v(j) := s / C(j,j)
Step 2 is sometimes called the "dot product" step, since it is an
inner product between the j-th row and the portion of the eigenvector
that has been computed so far.
The "columnwise" method consists basically in doing the sums
for all the rows in parallel. As each v(j) is computed, the
contribution of v(j) times the j-th column of C is added to the
partial sums. Since FORTRAN arrays are stored columnwise, this has
the advantage that at each step, the elements of C that are accessed
are adjacent to one another, whereas with the rowwise method, the
elements accessed at a step are spaced LDS (and LDP) words apart.
When finding left eigenvectors, the matrix in question is the
transpose of the one in storage, so the rowwise method then
actually accesses columns of A and B at each step, and so is the
preferred method. Definition at line 295 of file stgevc.f.
| subroutine stgexc | ( | logical | WANTQ, |
| logical | WANTZ, | ||
| integer | N, | ||
| real, dimension( lda, * ) | A, | ||
| integer | LDA, | ||
| real, dimension( ldb, * ) | B, | ||
| integer | LDB, | ||
| real, dimension( ldq, * ) | Q, | ||
| integer | LDQ, | ||
| real, dimension( ldz, * ) | Z, | ||
| integer | LDZ, | ||
| integer | IFST, | ||
| integer | ILST, | ||
| real, dimension( * ) | WORK, | ||
| integer | LWORK, | ||
| integer | INFO | ||
| ) |
STGEXC
Download STGEXC + dependencies [TGZ] [ZIP] [TXT] STGEXC reorders the generalized real Schur decomposition of a real
matrix pair (A,B) using an orthogonal equivalence transformation
(A, B) = Q * (A, B) * Z**T,
so that the diagonal block of (A, B) with row index IFST is moved
to row ILST.
(A, B) must be in generalized real Schur canonical form (as returned
by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
diagonal blocks. B is upper triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are
updated.
Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T
Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T | [in] | WANTQ | WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q. |
| [in] | WANTZ | WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z. |
| [in] | N | N is INTEGER
The order of the matrices A and B. N >= 0. |
| [in,out] | A | A is REAL array, dimension (LDA,N)
On entry, the matrix A in generalized real Schur canonical
form.
On exit, the updated matrix A, again in generalized
real Schur canonical form. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in,out] | B | B is REAL array, dimension (LDB,N)
On entry, the matrix B in generalized real Schur canonical
form (A,B).
On exit, the updated matrix B, again in generalized
real Schur canonical form (A,B). |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [in,out] | Q | Q is REAL array, dimension (LDZ,N)
On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
On exit, the updated matrix Q.
If WANTQ = .FALSE., Q is not referenced. |
| [in] | LDQ | LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N. |
| [in,out] | Z | Z is REAL array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., the orthogonal matrix Z.
On exit, the updated matrix Z.
If WANTZ = .FALSE., Z is not referenced. |
| [in] | LDZ | LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N. |
| [in,out] | IFST | IFST is INTEGER |
| [in,out] | ILST | ILST is INTEGER
Specify the reordering of the diagonal blocks of (A, B).
The block with row index IFST is moved to row ILST, by a
sequence of swapping between adjacent blocks.
On exit, if IFST pointed on entry to the second row of
a 2-by-2 block, it is changed to point to the first row;
ILST always points to the first row of the block in its
final position (which may differ from its input value by
+1 or -1). 1 <= IFST, ILST <= N. |
| [out] | WORK | WORK is REAL 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 >= 1 when N <= 1, otherwise LWORK >= 4*N + 16.
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.
=1: The transformed matrix pair (A, B) would be too far
from generalized Schur form; the problem is ill-
conditioned. (A, B) may have been partially reordered,
and ILST points to the first row of the current
position of the block being moved. |
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. Definition at line 220 of file stgexc.f.
1.8.1.1