## table of contents

realGEsing(3) | LAPACK | realGEsing(3) |

# NAME¶

realGEsing# SYNOPSIS¶

## Functions¶

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**subroutine

**sgesvdq**(JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA, S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK, WORK, LWORK, RWORK, LRWORK, INFO)

**SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices**subroutine

**sgesvdx**(JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, IL, IU, NS, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)

**SGESVDX computes the singular value decomposition (SVD) for GE matrices**subroutine

**sggsvd3**(JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)

**SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices**

# Detailed Description¶

This is the group of real singular value driver functions for GE matrices# Function 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. SGEJSV can sometimes compute tiny singular values and their singular vectors much more accurately than other SVD routines, see below under Further Details.

**Parameters**

*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 deficient. 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'.

*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.

*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.

*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 = '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.

*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.

*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

*M*

M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the input matrix A. M >= N >= 0.

*A*

A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*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.

*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 = 'V' .AND. JOBT = 'T' .AND. M = 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, unless JOBT='T'.

*LDU*

LDU is INTEGER The leading dimension of the array U, LDU >= 1. IF JOBU = 'U' or 'F' or 'W', then LDU >= M.

*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 = 'U' AND JOBT = 'T' AND M = 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, unless JOBT='T'.

*LDV*

LDV is INTEGER The leading dimension of the array V, LDV >= 1. If JOBV = 'V' or 'J' or 'W', then LDV >= N.

*WORK*

WORK is REAL array, dimension (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 = '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 = '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.

*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 = 'N', JOBV = 'N' ) and -> .. no scaled condition estimate required (JOBE = '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 = '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 = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7), if JOBU = '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 = 'U') or M*NB (for JOBU = 'F'). If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and -> if JOBV = 'V' the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N). -> if JOBV = '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.

*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) = 1 then some of the column norms of A were denormalized floats. The requested high accuracy is not warranted by the data.

*INFO*

INFO is INTEGER < 0: if INFO = -i, then the i-th argument had an illegal value. = 0: successful exit; > 0: SGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 deficient cases. It will be available in the SIGMA library [4]. If M is much larger than N, it is obvious that the initial 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:**

**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.

**Bugs, examples and comments:**

## 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**

*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.

*M*

M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the input matrix A. N >= 0.

*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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*S*

S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*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.

*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.

*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.

*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).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= 1. If LWORK = -1, a workspace query is assumed. The optimal size for the WORK array is calculated and stored in WORK(1), and no other work except argument checking is performed. Let mx = max(M,N) and mn = min(M,N). If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ). If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ). If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn. If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx. These are not tight minimums in all cases; see comments inside code. For good performance, LWORK should generally be larger; a query is recommended.

*IWORK*

IWORK is INTEGER array, dimension (8*min(M,N))

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

## 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**

*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.

*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'.

*M*

M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the input matrix A. N >= 0.

*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.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*S*

S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*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.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'S' or 'A', LDU >= M.

*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.

*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).

*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.

*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.

*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.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgesvdq (character JOBA, character JOBP, character JOBR, character JOBU, character JOBV, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, integer NUMRANK, integer, dimension( * ) IWORK, integer LIWORK, real, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer INFO)¶

**SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices**

**Purpose:**

SGESVDQ 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^*, [++] = [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.

**Parameters**

*JOBA*

JOBA is CHARACTER*1 Specifies the level of accuracy in the computed SVD = 'A' The requested accuracy corresponds to having the backward error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F, where EPS = SLAMCH('Epsilon'). This authorises CGESVDQ to truncate the computed triangular factor in a rank revealing QR factorization whenever the truncated part is below the threshold of the order of EPS * ||A||_F. This is aggressive truncation level. = 'M' Similarly as with 'A', but the truncation is more gentle: it is allowed only when there is a drop on the diagonal of the triangular factor in the QR factorization. This is medium truncation level. = 'H' High accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted. = 'E' Same as 'H', and in addition the condition number of column scaled A is estimated and returned in RWORK(1). N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)

*JOBP*

JOBP is CHARACTER*1 = 'P' The rows of A are ordered in decreasing order with respect to ||A(i,:)||_infty. This enhances numerical accuracy at the cost of extra data movement. Recommended for numerical robustness. = 'N' No row pivoting.

*JOBR*

JOBR is CHARACTER*1 = 'T' After the initial pivoted QR factorization, SGESVD is applied to the transposed R**T of the computed triangular factor R. This involves some extra data movement (matrix transpositions). Useful for experiments, research and development. = 'N' The triangular factor R is given as input to SGESVD. This may be preferred as it involves less data movement.

*JOBU*

JOBU is CHARACTER*1 = 'A' All M left singular vectors are computed and returned in the matrix U. See the description of U. = 'S' or 'U' N = min(M,N) left singular vectors are computed and returned in the matrix U. See the description of U. = 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular vectors are computed and returned in the matrix U. = 'F' The N left singular vectors are returned in factored form as the product of the Q factor from the initial QR factorization and the N left singular vectors of (R**T , 0)**T. If row pivoting is used, then the necessary information on the row pivoting is stored in IWORK(N+1:N+M-1). = 'N' The left singular vectors are not computed.

*JOBV*

JOBV is CHARACTER*1 = 'A', 'V' All N right singular vectors are computed and returned in the matrix V. = 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular vectors are computed and returned in the matrix V. This option is allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal. = 'N' The right singular vectors are not computed.

*M*

M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the input matrix A. M >= N >= 0.

*A*

A is REAL array of dimensions LDA x N On entry, the input matrix A. On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains the Householder vectors as stored by SGEQP3. If JOBU = 'F', these Householder vectors together with WORK(1:N) can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U.

*LDA*

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

*S*

S is REAL array of dimension N. The singular values of A, ordered so that S(i) >= S(i+1).

*U*

U is REAL array, dimension LDU x M if JOBU = 'A'; see the description of LDU. In this case, on exit, U contains the M left singular vectors. LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this case, U contains the leading N or the leading NUMRANK left singular vectors. LDU x N if JOBU = 'F' ; see the description of LDU. In this case U contains N x N orthogonal matrix that can be used to form the left singular vectors. If JOBU = 'N', U is not referenced.

*LDU*

LDU is INTEGER. The leading dimension of the array U. If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M). If JOBU = 'F', LDU >= max(1,N). Otherwise, LDU >= 1.

*V*

V is REAL array, dimension LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' . If JOBV = 'A', or 'V', V contains the N-by-N orthogonal matrix V**T; If JOBV = 'R', V contains the first NUMRANK rows of V**T (the right singular vectors, stored rowwise, of the NUMRANK largest singular values). If JOBV = 'N' and JOBA = 'E', V is used as a workspace. If JOBV = 'N', and JOBA.NE.'E', V is not referenced.

*LDV*

LDV is INTEGER The leading dimension of the array V. If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N). Otherwise, LDV >= 1.

*NUMRANK*

NUMRANK is INTEGER NUMRANK is the numerical rank first determined after the rank revealing QR factorization, following the strategy specified by the value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK leading singular values and vectors are then requested in the call of SGESVD. The final value of NUMRANK might be further reduced if some singular values are computed as zeros.

*IWORK*

IWORK is INTEGER array, dimension (max(1, LIWORK)). On exit, IWORK(1:N) contains column pivoting permutation of the rank revealing QR factorization. If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence of row swaps used in row pivoting. These can be used to restore the left singular vectors in the case JOBU = 'F'. If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, LIWORK(1) returns the minimal LIWORK.

*LIWORK*

LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= N + M - 1, if JOBP = 'P' and JOBA .NE. 'E'; LIWORK >= N if JOBP = 'N' and JOBA .NE. 'E'; LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E'; LIWORK >= N + N if JOBP = 'N' and JOBA = 'E'. If LIWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA.

*WORK*

WORK is REAL array, dimension (max(2, LWORK)), used as a workspace. On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters needed to recover the Q factor from the QR factorization computed by SGEQP3. If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, WORK(1) returns the optimal LWORK, and WORK(2) returns the minimal LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. It is determined as follows: Let LWQP3 = 3*N+1, LWCON = 3*N, and let LWORQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U' { MAX( M, 1 ), if JOBU = 'A' LWSVD = MAX( 5*N, 1 ) LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ), LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 ) Then the minimal value of LWORK is: = MAX( N + LWQP3, LWSVD ) if only the singular values are needed; = MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed, and a scaled condition estimate requested; = N + MAX( LWQP3, LWSVD, LWORQ ) if the singular values and the left singular vectors are requested; = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the singular values and the left singular vectors are requested, and also a scaled condition estimate requested; = N + MAX( LWQP3, LWSVD ) if the singular values and the right singular vectors are requested; = N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right singular vectors are requested, and also a scaled condition etimate requested; = N + MAX( LWQP3, LWSVD, LWORQ ) if the full SVD is requested with JOBV = 'R'; independent of JOBR; = N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested, JOBV = 'R' and, also a scaled condition estimate requested; independent of JOBR; = MAX( N + MAX( LWQP3, LWSVD, LWORQ ), N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the full SVD is requested with JOBV = 'A' or 'V', and JOBR ='N' = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ), N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) ) if the full SVD is requested with JOBV = 'A' or 'V', and JOBR ='N', and also a scaled condition number estimate requested. = MAX( N + MAX( LWQP3, LWSVD, LWORQ ), N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the full SVD is requested with JOBV = 'A', 'V', and JOBR ='T' = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ), N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the full SVD is requested with JOBV = 'A' or 'V', and JOBR ='T', and also a scaled condition number estimate requested. Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ). If LWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA.

*RWORK*

RWORK is REAL array, dimension (max(1, LRWORK)). On exit, 1. If JOBA = 'E', RWORK(1) contains an estimate of the condition number of column scaled A. If A = C * D where D is diagonal and C has unit columns in the Euclidean norm, then, assuming full column rank, N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1). Otherwise, RWORK(1) = -1. 2. RWORK(2) contains the number of singular values computed as exact zeros in SGESVD applied to the upper triangular or trapeziodal R (from the initial QR factorization). In case of early exit (no call to SGESVD, such as in the case of zero matrix) RWORK(2) = -1. If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0, RWORK(1) returns the minimal LRWORK.

*LRWORK*

LRWORK is INTEGER. The dimension of the array RWORK. If JOBP ='P', then LRWORK >= MAX(2, M). Otherwise, LRWORK >= 2 If LRWORK = -1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the WORK, IWORK, and RWORK arrays, and no error message related to LWORK is issued by XERBLA.

*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 (computed in SGESVD) did not converge to zero.

**Further Details:**

1. The data movement (matrix transpose) is coded using simple nested DO-loops because BLAS and LAPACK do not provide corresponding subroutines. Those DO-loops are easily identified in this source code - by the CONTINUE statements labeled with 11**. In an optimized version of this code, the nested DO loops should be replaced with calls to an optimized subroutine. 2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause column norm overflow. This is the minial precaution and it is left to the SVD routine (CGESVD) to do its own preemptive scaling if potential over- or underflows are detected. To avoid repeated scanning of the array A, an optimal implementation would do all necessary scaling before calling CGESVD and the scaling in CGESVD can be switched off. 3. Other comments related to code optimization are given in comments in the code, enlosed in [[double brackets]].

**Bugs, examples and comments**

Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you.

**References**

[1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for Computing the SVD with High Accuracy. ACM Trans. Math. Softw. 44(1): 11:1-11:30 (2017) SIGMA library, xGESVDQ section updated February 2016. Developed and coded by Zlatko Drmac, Department of Mathematics University of Zagreb, Croatia, drmac@math.hr

**Contributors:**

Developed and coded by Zlatko Drmac, Department of Mathematics University of Zagreb, Croatia, drmac@math.hr

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgesvdx (character JOBU, character JOBVT, character RANGE, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real VL, real VU, integer IL, integer IU, integer NS, 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)¶

**SGESVDX computes the singular value decomposition (SVD) for GE matrices**

**Purpose:**

SGESVDX 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. SGESVDX uses an eigenvalue problem for obtaining the SVD, which allows for the computation of a subset of singular values and vectors. See SBDSVDX for details. Note that the routine returns V**T, not V.

**Parameters**

*JOBU*

JOBU is CHARACTER*1 Specifies options for computing all or part of the matrix U: = 'V': the first min(m,n) columns of U (the left singular vectors) or as specified by RANGE are returned in the array U; = 'N': no columns of U (no left singular vectors) are computed.

*JOBVT*

JOBVT is CHARACTER*1 Specifies options for computing all or part of the matrix V**T: = 'V': the first min(m,n) rows of V**T (the right singular vectors) or as specified by RANGE are returned in the array VT; = 'N': no rows of V**T (no right singular vectors) are computed.

*RANGE*

RANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th singular values will be found.

*M*

M is INTEGER The number of rows of the input matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the input matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the contents of A are destroyed.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*VL*

VL is REAL If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*VU*

VU is REAL If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.

*IL*

IL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*IU*

IU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.

*NS*

NS is INTEGER The total number of singular values found, 0 <= NS <= min(M,N). If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

*S*

S is REAL array, dimension (min(M,N)) The singular values of A, sorted so that S(i) >= S(i+1).

*U*

U is REAL array, dimension (LDU,UCOL) If JOBU = 'V', U contains columns of U (the left singular vectors, stored columnwise) as specified by RANGE; if JOBU = 'N', U is not referenced. Note: The user must ensure that UCOL >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= 1; if JOBU = 'V', LDU >= M.

*VT*

VT is REAL array, dimension (LDVT,N) If JOBVT = 'V', VT contains the rows of V**T (the right singular vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N', VT is not referenced. Note: The user must ensure that LDVT >= NS; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.

*LDVT*

LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1; if JOBVT = 'V', LDVT >= NS (see above).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

*LWORK*

LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see comments inside the code): - PATH 1 (M much larger than N) - PATH 1t (N much larger than M) LWORK >= MAX(1,MIN(M,N)*2+MAX(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.

*IWORK*

IWORK is INTEGER array, dimension (12*MIN(M,N)) If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in SBDSVDX/SSTEVX.

*INFO*

INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, then i eigenvectors failed to converge in SBDSVDX/SSTEVX. if INFO = N*2 + 1, an internal error occurred in SBDSVDX

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sggsvd3 (character JOBU, character JOBV, character JOBQ, integer M, integer N, integer P, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

**SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices**

**Purpose:**

SGGSVD3 computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B: U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) where U, V and Q are orthogonal matrices. Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T, then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respectively: If M-K-L >= 0, K L D1 = K ( I 0 ) L ( 0 C ) M-K-L ( 0 0 ) K L D2 = L ( 0 S ) P-L ( 0 0 ) N-K-L K L ( 0 R ) = K ( 0 R11 R12 ) L ( 0 0 R22 ) where C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), S = diag( BETA(K+1), ... , BETA(K+L) ), C**2 + S**2 = I. R is stored in A(1:K+L,N-K-L+1:N) on exit. If M-K-L < 0, K M-K K+L-M D1 = K ( I 0 0 ) M-K ( 0 C 0 ) K M-K K+L-M D2 = M-K ( 0 S 0 ) K+L-M ( 0 0 I ) P-L ( 0 0 0 ) N-K-L K M-K K+L-M ( 0 R ) = K ( 0 R11 R12 R13 ) M-K ( 0 0 R22 R23 ) K+L-M ( 0 0 0 R33 ) where C = diag( ALPHA(K+1), ... , ALPHA(M) ), S = diag( BETA(K+1), ... , BETA(M) ), C**2 + S**2 = I. (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored ( 0 R22 R23 ) in B(M-K+1:L,N+M-K-L+1:N) on exit. The routine computes C, S, R, and optionally the orthogonal transformation matrices U, V and Q. In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B): A*inv(B) = U*(D1*inv(D2))*V**T. If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem: A**T*A x = lambda* B**T*B x. In some literature, the GSVD of A and B is presented in the form U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 ) where U and V are orthogonal and X is nonsingular, D1 and D2 are ``diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as X = Q*( I 0 ) ( 0 inv(R) ).

**Parameters**

*JOBU*

JOBU is CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed.

*JOBV*

JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed.

*JOBQ*

JOBQ is CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed.

*M*

M is INTEGER The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER The number of columns of the matrices A and B. N >= 0.

*P*

P is INTEGER The number of rows of the matrix B. P >= 0.

*K*

K is INTEGER

*L*

L is INTEGER On exit, K and L specify the dimension of the subblocks described in Purpose. K + L = effective numerical rank of (A**T,B**T)**T.

*A*

A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A contains the triangular matrix R, or part of R. See Purpose for details.

*LDA*

LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*B*

B is REAL array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, B contains the triangular matrix R if M-K-L < 0. See Purpose for details.

*LDB*

LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).

*ALPHA*

ALPHA is REAL array, dimension (N)

*BETA*

BETA is REAL array, dimension (N) On exit, ALPHA and BETA contain the generalized singular value pairs of A and B; ALPHA(1:K) = 1, BETA(1:K) = 0, and if M-K-L >= 0, ALPHA(K+1:K+L) = C, BETA(K+1:K+L) = S, or if M-K-L < 0, ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 BETA(K+1:M) =S, BETA(M+1:K+L) =1 and ALPHA(K+L+1:N) = 0 BETA(K+L+1:N) = 0

*U*

U is REAL array, dimension (LDU,M) If JOBU = 'U', U contains the M-by-M orthogonal matrix U. If JOBU = 'N', U is not referenced.

*LDU*

LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.

*V*

V is REAL array, dimension (LDV,P) If JOBV = 'V', V contains the P-by-P orthogonal matrix V. If JOBV = 'N', V is not referenced.

*LDV*

LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.

*Q*

Q is REAL array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. If JOBQ = 'N', Q is not referenced.

*LDQ*

LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER The dimension of the array WORK. 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.

*IWORK*

IWORK is INTEGER array, dimension (N) On exit, IWORK stores the sorting information. More precisely, the following loop will sort ALPHA for I = K+1, min(M,K+L) swap ALPHA(I) and ALPHA(IWORK(I)) endfor such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

*INFO*

INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine STGSJA.

**Internal Parameters:**

TOLA REAL TOLB REAL TOLA and TOLB are the thresholds to determine the effective rank of (A**T,B**T)**T. Generally, they are set to TOLA = MAX(M,N)*norm(A)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Contributors:**

**Further Details:**

# Author¶

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