LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
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Functions/Subroutines

subroutine sgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
SGEJSV
subroutine sgesdd (JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
SGESDD
subroutine sgesvd (JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO)
SGESVD computes the singular value decomposition (SVD) for GE matrices

Detailed Description

This is the group of real singular value driver functions for GE matrices

Function/Subroutine Documentation

 subroutine sgejsv ( character*1 JOBA, character*1 JOBU, character*1 JOBV, character*1 JOBR, character*1 JOBT, character*1 JOBP, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( n ) SVA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( lwork ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO )

SGEJSV

Purpose:
``` SGEJSV computes the singular value decomposition (SVD) of a real M-by-N
matrix [A], where M >= N. The SVD of [A] is written as

[A] = [U] * [SIGMA] * [V]^t,

where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
diagonal elements, [U] is an M-by-N (or M-by-M) 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. The matrices [U] and [V]
are computed and stored in the arrays U and V, respectively. The diagonal
of [SIGMA] is computed and stored in the array SVA.```
Parameters:
 [in] JOBA ``` JOBA is CHARACTER*1 Specifies the level of accuracy: = 'C': This option works well (high relative accuracy) if A = B * D, with well-conditioned B and arbitrary diagonal matrix D. The accuracy cannot be spoiled by COLUMN scaling. The accuracy of the computed output depends on the condition of B, and the procedure aims at the best theoretical accuracy. The relative error max_{i=1:N}|d sigma_i| / sigma_i is bounded by f(M,N)*epsilon* cond(B), independent of D. The input matrix is preprocessed with the QRF with column pivoting. This initial preprocessing and preconditioning by a rank revealing QR factorization is common for all values of JOBA. Additional actions are specified as follows: = 'E': Computation as with 'C' with an additional estimate of the condition number of B. It provides a realistic error bound. = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings D1, D2, and well-conditioned matrix C, this option gives higher accuracy than the 'C' option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable. The input matrix A is preprocessed with QR factorization with FULL (row and column) pivoting. = 'G' Computation as with 'F' with an additional estimate of the condition number of B, where A=D*B. If A has heavily weighted rows, then using this condition number gives too pessimistic error bound. = 'A': Small singular values are the noise and the matrix is treated as numerically rank defficient. The error in the computed singular values is bounded by f(m,n)*epsilon*||A||. The computed SVD A = U * S * V^t restores A up to f(m,n)*epsilon*||A||. This gives the procedure the licence to discard (set to zero) all singular values below N*epsilon*||A||. = 'R': Similar as in 'A'. Rank revealing property of the initial QR factorization is used do reveal (using triangular factor) a gap sigma_{r+1} < epsilon * sigma_r in which case the numerical RANK is declared to be r. The SVD is computed with absolute error bounds, but more accurately than with 'A'.``` [in] JOBU ``` JOBU is CHARACTER*1 Specifies whether to compute the columns of U: = 'U': N columns of U are returned in the array U. = 'F': full set of M left sing. vectors is returned in the array U. = 'W': U may be used as workspace of length M*N. See the description of U. = 'N': U is not computed.``` [in] JOBV ``` JOBV is CHARACTER*1 Specifies whether to compute the matrix V: = 'V': N columns of V are returned in the array V; Jacobi rotations are not explicitly accumulated. = 'J': N columns of V are returned in the array V, but they are computed as the product of Jacobi rotations. This option is allowed only if JOBU .NE. 'N', i.e. in computing the full SVD. = 'W': V may be used as workspace of length N*N. See the description of V. = 'N': V is not computed.``` [in] JOBR ``` JOBR is CHARACTER*1 Specifies the RANGE for the singular values. Issues the licence to set to zero small positive singular values if they are outside specified range. If A .NE. 0 is scaled so that the largest singular value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues the licence to kill columns of A whose norm in c*A is less than SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN, where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E'). = 'N': Do not kill small columns of c*A. This option assumes that BLAS and QR factorizations and triangular solvers are implemented to work in that range. If the condition of A is greater than BIG, use SGESVJ. = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)] (roughly, as described above). This option is recommended. =========================== For computing the singular values in the FULL range [SFMIN,BIG] use SGESVJ.``` [in] JOBT ``` JOBT is CHARACTER*1 If the matrix is square then the procedure may determine to use transposed A if A^t seems to be better with respect to convergence. If the matrix is not square, JOBT is ignored. This is subject to changes in the future. The decision is based on two values of entropy over the adjoint orbit of A^t * A. See the descriptions of WORK(6) and WORK(7). = 'T': transpose if entropy test indicates possibly faster convergence of Jacobi process if A^t is taken as input. If A is replaced with A^t, then the row pivoting is included automatically. = 'N': do not speculate. This option can be used to compute only the singular values, or the full SVD (U, SIGMA and V). For only one set of singular vectors (U or V), the caller should provide both U and V, as one of the matrices is used as workspace if the matrix A is transposed. The implementer can easily remove this constraint and make the code more complicated. See the descriptions of U and V.``` [in] JOBP ``` JOBP is CHARACTER*1 Issues the licence to introduce structured perturbations to drown denormalized numbers. This licence should be active if the denormals are poorly implemented, causing slow computation, especially in cases of fast convergence (!). For details see [1,2]. For the sake of simplicity, this perturbations are included only when the full SVD or only the singular values are requested. The implementer/user can easily add the perturbation for the cases of computing one set of singular vectors. = 'P': introduce perturbation = 'N': do not perturb``` [in] M ``` M is INTEGER The number of rows of the input matrix A. 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.``` [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, - For WORK(1)/WORK(2) = ONE: The singular values of A. During the computation SVA contains Euclidean column norms of the iterated matrices in the array A. - For WORK(1) .NE. WORK(2): The singular values of A are (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if sigma_max(A) overflows or if small singular values have been saved from underflow by scaling the input matrix A. - If JOBR='R' then some of the singular values may be returned as exact zeros obtained by "set to zero" because they are below the numerical rank threshold or are denormalized numbers.``` [out] U ``` U is REAL array, dimension ( LDU, N ) If JOBU = 'U', then U contains on exit the M-by-N matrix of the left singular vectors. If JOBU = 'F', then U contains on exit the M-by-M matrix of the left singular vectors, including an ONB of the orthogonal complement of the Range(A). If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N), then U is used as workspace if the procedure replaces A with A^t. In that case, [V] is computed in U as left singular vectors of A^t and then copied back to the V array. This 'W' option is just a reminder to the caller that in this case U is reserved as workspace of length N*N. If JOBU = 'N' U is not referenced.``` [in] LDU ``` LDU is INTEGER The leading dimension of the array U, LDU >= 1. IF JOBU = 'U' or 'F' or 'W', then LDU >= M.``` [out] V ``` V is REAL array, dimension ( LDV, N ) If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of the right singular vectors; If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N), then V is used as workspace if the pprocedure replaces A with A^t. In that case, [U] is computed in V as right singular vectors of A^t and then copied back to the U array. This 'W' option is just a reminder to the caller that in this case V is reserved as workspace of length N*N. If JOBV = 'N' V is not referenced.``` [in] LDV ``` LDV is INTEGER The leading dimension of the array V, LDV >= 1. If JOBV = 'V' or 'J' or 'W', then LDV >= N.``` [out] WORK ``` WORK is REAL array, dimension at least LWORK. On exit, WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such that SCALE*SVA(1:N) are the computed singular values of A. (See the description of SVA().) WORK(2) = See the description of WORK(1). WORK(3) = SCONDA is an estimate for the condition number of column equilibrated A. (If JOBA .EQ. 'E' or 'G') SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1). It is computed using SPOCON. It holds N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA where R is the triangular factor from the QRF of A. However, if R is truncated and the numerical rank is determined to be strictly smaller than N, SCONDA is returned as -1, thus indicating that the smallest singular values might be lost. If full SVD is needed, the following two condition numbers are useful for the analysis of the algorithm. They are provied for a developer/implementer who is familiar with the details of the method. WORK(4) = an estimate of the scaled condition number of the triangular factor in the first QR factorization. WORK(5) = an estimate of the scaled condition number of the triangular factor in the second QR factorization. The following two parameters are computed if JOBT .EQ. 'T'. They are provided for a developer/implementer who is familiar with the details of the method. WORK(6) = the entropy of A^t*A :: this is the Shannon entropy of diag(A^t*A) / Trace(A^t*A) taken as point in the probability simplex. WORK(7) = the entropy of A*A^t.``` [in] LWORK ``` LWORK is INTEGER Length of WORK to confirm proper allocation of work space. LWORK depends on the job: If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and -> .. no scaled condition estimate required (JOBE.EQ.'N'): LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement. ->> For optimal performance (blocked code) the optimal value is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal block size for DGEQP3 and DGEQRF. In general, optimal LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7). -> .. an estimate of the scaled condition number of A is required (JOBA='E', 'G'). In this case, LWORK is the maximum of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7). ->> For optimal performance (blocked code) the optimal value is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7). In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), N+N*N+LWORK(DPOCON),7). If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'), -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7), where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ, DORMLQ. In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)). If SIGMA and the left singular vectors are needed -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7). -> For optimal performance: if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7), if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7), where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR. In general, the optimal length LWORK is computed as LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON), 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or M*NB (for JOBU.EQ.'F'). If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and -> if JOBV.EQ.'V' the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N). -> if JOBV.EQ.'J' the minimal requirement is LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6). -> For optimal performance, LWORK should be additionally larger than N+M*NB, where NB is the optimal block size for DORMQR.``` [out] IWORK ``` IWORK is INTEGER array, dimension M+3*N. On exit, IWORK(1) = the numerical rank determined after the initial QR factorization with pivoting. See the descriptions of JOBA and JOBR. IWORK(2) = the number of the computed nonzero singular values IWORK(3) = if nonzero, a warning message: If IWORK(3).EQ.1 then some of the column norms of A were denormalized floats. The requested high accuracy is not warranted by the data.``` [out] INFO ``` INFO is INTEGER < 0 : if INFO = -i, then the i-th argument had an illegal value. = 0 : successfull exit; > 0 : SGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate.```
Date:
September 2012
Further Details:
```  SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an
additional row pivoting can be used as a preprocessor, which in some
cases results in much higher accuracy. An example is matrix A with the
structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
diagonal matrices and C is well-conditioned matrix. In that case, complete
pivoting in the first QR factorizations provides accuracy dependent on the
condition number of C, and independent of D1, D2. Such higher accuracy is
not completely understood theoretically, but it works well in practice.
Further, if A can be written as A = B*D, with well-conditioned B and some
diagonal D, then the high accuracy is guaranteed, both theoretically and
in software, independent of D. For more details see [1], [2].
The computational range for the singular values can be the full range
( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
& LAPACK routines called by SGEJSV are implemented to work in that range.
If that is not the case, then the restriction for safe computation with
the singular values in the range of normalized IEEE numbers is that the
spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
overflow. This code (SGEJSV) is best used in this restricted range,
meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
returned as zeros. See JOBR for details on this.
Further, this implementation is somewhat slower than the one described
in [1,2] due to replacement of some non-LAPACK components, and because
the choice of some tuning parameters in the iterative part (SGESVJ) is
left to the implementer on a particular machine.
The rank revealing QR factorization (in this code: SGEQP3) should be
implemented as in [3]. We have a new version of SGEQP3 under development
that is more robust than the current one in LAPACK, with a cleaner cut in
rank defficient cases. It will be available in the SIGMA library [4].
If M is much larger than N, it is obvious that the inital QRF with
column pivoting can be preprocessed by the QRF without pivoting. That
well known trick is not used in SGEJSV because in some cases heavy row
weighting can be treated with complete pivoting. The overhead in cases
M much larger than N is then only due to pivoting, but the benefits in
terms of accuracy have prevailed. The implementer/user can incorporate
this extra QRF step easily. The implementer can also improve data movement
(matrix transpose, matrix copy, matrix transposed copy) - this
implementation of SGEJSV uses only the simplest, naive data movement.```
Contributors:
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
References:
``` [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
factorization software - a case study.
ACM Trans. math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
LAPACK Working note 176.
[4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008.```
Please report all bugs and send interesting examples and/or comments to drmac.nosp@m.@mat.nosp@m.h.hr. Thank you.

Definition at line 473 of file sgejsv.f.

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 subroutine sgesdd ( character JOBZ, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldvt, * ) VT, integer LDVT, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO )

SGESDD

Purpose:
``` SGESDD computes the singular value decomposition (SVD) of a real
M-by-N matrix A, optionally computing the left and right singular
vectors.  If singular vectors are desired, it uses a
divide-and-conquer algorithm.

The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its
min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
are the singular values of A; they are real and non-negative, and
are returned in descending order.  The first min(m,n) columns of
U and V are the left and right singular vectors of A.

Note that the routine returns VT = V**T, not V.

The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.```
Parameters:
 [in] JOBZ ``` JOBZ is CHARACTER*1 Specifies options for computing all or part of the matrix U: = 'A': all M columns of U and all N rows of V**T are returned in the arrays U and VT; = 'S': the first min(M,N) columns of U and the first min(M,N) rows of V**T are returned in the arrays U and VT; = 'O': If M >= N, the first N columns of U are overwritten on the array A and all rows of V**T are returned in the array VT; otherwise, all columns of U are returned in the array U and the first M rows of V**T are overwritten in the array A; = 'N': no columns of U or rows of V**T are computed.``` [in] M ``` M is INTEGER The number of rows of the input matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the input 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 JOBZ = 'O', A is overwritten with the first N columns of U (the left singular vectors, stored columnwise) if M >= N; A is overwritten with the first M rows of V**T (the right singular vectors, stored rowwise) otherwise. if JOBZ .ne. 'O', the contents of A are destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] S ``` S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).``` [out] U ``` U is REAL array, dimension (LDU,UCOL) UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; UCOL = min(M,N) if JOBZ = 'S'. If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M orthogonal matrix U; if JOBZ = 'S', U contains the first min(M,N) columns of U (the left singular vectors, stored columnwise); if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.``` [in] LDU ``` LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.``` [out] VT ``` VT is REAL array, dimension (LDVT,N) If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the N-by-N orthogonal matrix V**T; if JOBZ = 'S', VT contains the first min(M,N) rows of V**T (the right singular vectors, stored rowwise); if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.``` [in] LDVT ``` LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; if JOBZ = 'S', LDVT >= min(M,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. If JOBZ = 'N', LWORK >= 3*min(M,N) + max(max(M,N),6*min(M,N)). If JOBZ = 'O', LWORK >= 3*min(M,N) + max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)). If JOBZ = 'S' or 'A' LWORK >= 3*min(M,N) + max(max(M,N),4*min(M,N)*min(M,N)+3*min(M,N)+max(M,N)). For good performance, LWORK should generally be larger. If LWORK = -1 but other input arguments are legal, WORK(1) returns the optimal LWORK.``` [out] IWORK ` IWORK is INTEGER array, dimension (8*min(M,N))` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: SBDSDC did not converge, updating process failed.```
Date:
September 2012
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 217 of file sgesdd.f.

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 subroutine sgesvd ( character JOBU, character JOBVT, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldvt, * ) VT, integer LDVT, real, dimension( * ) WORK, integer LWORK, integer INFO )

SGESVD computes the singular value decomposition (SVD) for GE matrices

Purpose:
``` SGESVD computes the singular value decomposition (SVD) of a real
M-by-N matrix A, optionally computing the left and/or right singular
vectors. The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its
min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
are the singular values of A; they are real and non-negative, and
are returned in descending order.  The first min(m,n) columns of
U and V are the left and right singular vectors of A.

Note that the routine returns V**T, not V.```
Parameters:
 [in] JOBU ``` JOBU is CHARACTER*1 Specifies options for computing all or part of the matrix U: = 'A': all M columns of U are returned in array U: = 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U; = 'O': the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A; = 'N': no columns of U (no left singular vectors) are computed.``` [in] JOBVT ``` JOBVT is CHARACTER*1 Specifies options for computing all or part of the matrix V**T: = 'A': all N rows of V**T are returned in the array VT; = 'S': the first min(m,n) rows of V**T (the right singular vectors) are returned in the array VT; = 'O': the first min(m,n) rows of V**T (the right singular vectors) are overwritten on the array A; = 'N': no rows of V**T (no right singular vectors) are computed. JOBVT and JOBU cannot both be 'O'.``` [in] M ``` M is INTEGER The number of rows of the input matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the input 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 JOBU = 'O', A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise); if JOBVT = 'O', A is overwritten with the first min(m,n) rows of V**T (the right singular vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A are destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] S ``` S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).``` [out] U ``` U is REAL array, dimension (LDU,UCOL) (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU = 'A', U contains the M-by-M orthogonal matrix U; if JOBU = 'S', U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise); if JOBU = 'N' or 'O', U is not referenced.``` [in] LDU ``` LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', LDU >= M.``` [out] VT ``` VT is REAL array, dimension (LDVT,N) If JOBVT = 'A', VT contains the N-by-N orthogonal matrix V**T; if JOBVT = 'S', VT contains the first min(m,n) rows of V**T (the right singular vectors, stored rowwise); if JOBVT = 'N' or 'O', VT is not referenced.``` [in] LDVT ``` LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK; if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix B whose diagonal is in S (not necessarily sorted). B satisfies A = U * B * VT, so it has the same singular values as A, and singular vectors related by U and VT.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code): - PATH 1 (M much larger than N, JOBU='N') - PATH 1t (N much larger than M, JOBVT='N') LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) for the other paths For good performance, LWORK should generally be larger. 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 SBDSQR did not converge, INFO specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero. See the description of WORK above for details.```
Date:
April 2012

Definition at line 211 of file sgesvd.f.

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