gesvdq#

sgesvdq

Functions

void sgesvdq(
    const char*          joba,
    const char*          jobp,
    const char*          jobr,
    const char*          jobu,
    const char*          jobv,
    const INT            m,
    const INT            n,
          f32*  restrict A,
    const INT            lda,
          f32*  restrict S,
          f32*  restrict U,
    const INT            ldu,
          f32*  restrict V,
    const INT            ldv,
          INT*           numrank,
          INT*  restrict iwork,
    const INT            liwork,
          f32*  restrict work,
    const INT            lwork,
          f32*  restrict rwork,
    const INT            lrwork,
          INT*           info
);

void sgesvdq(const char *joba, const char *jobp, const char *jobr, const char *jobu, const char *jobv, const INT m, const INT n, f32 *restrict A, const INT lda, f32 *restrict S, f32 *restrict U, const INT ldu, f32 *restrict V, const INT ldv, INT *numrank, INT *restrict iwork, const INT liwork, f32 *restrict work, const INT lwork, f32 *restrict rwork, const INT lrwork, INT *info)#

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

         A = U * SIGMA * V**T

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. 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.

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 minimal precaution and it is left to the SVD routine (SGESVD) 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 SGESVD and the scaling in SGESVD can be switched off.

3. Other comments related to code optimization are given in comments in the code, enclosed 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

Parameters

in

joba

Specifies the level of accuracy in the computed SVD. joba='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 SGESVDQ 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. joba='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. joba='H': High accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted. joba='E': Same as 'H', and in addition the condition number of column scaled A is estimated and returned in rwork[0]. N^(-1/4)*rwork[0] <= ||pinv(A_scaled)||_2 <= N^(1/4)*rwork[0].

in

jobp

jobp='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. jobp='N': No row pivoting.

in

jobr

jobr='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. jobr='N': The triangular factor R is given as input to SGESVD. This may be preferred as it involves less data movement.

in

jobu

jobu='A': All m left singular vectors are computed and returned in the matrix U. See the description of U. jobu='S' or jobu='U': n = min(m,n) left singular vectors are computed and returned in the matrix U. See the description of U. jobu='R': Numerical rank numrank is determined and only numrank left singular vectors are computed and returned in the matrix U. jobu='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:n+m-2]. jobu='N': The left singular vectors are not computed.

in

jobv

jobv='A' or jobv='V': All n right singular vectors are computed and returned in the matrix V. jobv='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. jobv='N': The right singular vectors are not computed.

in

m

The number of rows of the input matrix A. m>=0.

in

n

The number of columns of the input matrix A. m>=n>=0.

inout

A

Array of dimensions lda x n. On entry, the input matrix A. On exit, if jobu!='N' or jobv!='N', the lower triangle of A contains the Householder vectors as stored by SGEQP3. If jobu='F', these Householder vectors together with work[0:n-1] can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U.

in

lda

The leading dimension of the array A. lda>=max(1,m).

out

S

Array of dimension n. The singular values of A, ordered so that S[i]>=S[i+1].

out

U

Array of 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.

in

ldu

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.

out

V

Array of 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!='E', V is not referenced.

in

ldv

The leading dimension of the array V. If jobv='A', 'V', 'R', or joba='E', ldv>=max(1,n). Otherwise, ldv>=1.

out

numrank

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.

out

iwork

Integer array of dimension (max(1,liwork)). On exit, iwork[0:n-1] contains column pivoting permutation of the rank revealing QR factorization. If jobp='P', iwork[n:n+m-2] 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, iwork[0] returns the minimal liwork.

in

liwork

The dimension of the array iwork. liwork >= n+m-1, if jobp='P' and joba!='E'; liwork >= n, if jobp='N' and joba!='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.

out

work

Array of dimension (max(2,lwork)), used as a workspace. On exit, if, on entry, lwork!=-1, work[0:n-1] 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[0] returns the optimal lwork, and work[1] returns the minimal lwork.

inout

lwork

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 estimate 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.

out

rwork

Array of dimension (max(1,lrwork)). On exit,

If joba='E', rwork[0] 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[0] <= ||pinv(C)||_2 <= N^(1/4) * rwork[0]. Otherwise, rwork[0] = -1.
rwork[1] contains the number of singular values computed as exact zeros in SGESVD applied to the upper triangular or trapezoidal R (from the initial QR factorization). In case of early exit (no call to SGESVD, such as in the case of zero matrix) rwork[1] = -1. If liwork, lwork, or lrwork = -1, then on exit, if info=0, rwork[0] returns the minimal lrwork.

in

lrwork

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.

out

info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value. info>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.

dgesvdq

Functions

void dgesvdq(
    const char*          joba,
    const char*          jobp,
    const char*          jobr,
    const char*          jobu,
    const char*          jobv,
    const INT            m,
    const INT            n,
          f64*  restrict A,
    const INT            lda,
          f64*  restrict S,
          f64*  restrict U,
    const INT            ldu,
          f64*  restrict V,
    const INT            ldv,
          INT*           numrank,
          INT*  restrict iwork,
    const INT            liwork,
          f64*  restrict work,
    const INT            lwork,
          f64*  restrict rwork,
    const INT            lrwork,
          INT*           info
);

void dgesvdq(const char *joba, const char *jobp, const char *jobr, const char *jobu, const char *jobv, const INT m, const INT n, f64 *restrict A, const INT lda, f64 *restrict S, f64 *restrict U, const INT ldu, f64 *restrict V, const INT ldv, INT *numrank, INT *restrict iwork, const INT liwork, f64 *restrict work, const INT lwork, f64 *restrict rwork, const INT lrwork, INT *info)#

DGESVDQ 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 diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N orthogonal matrix. DGESVDQ 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.

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 minimal precaution and it is left to the SVD routine (DGESVD) 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 DGESVD and the scaling in DGESVD can be switched off.

3. Other comments related to code optimization are given in comments in the code, enclosed 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

Parameters

in

joba

Specifies the level of accuracy in the computed SVD. joba='A': The requested accuracy corresponds to having the backward error bounded by || delta A ||_F <= f(m,n)*EPS*|| A ||_F, where EPS = dlamch("Epsilon"). This authorises DGESVDQ 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. joba='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. joba='H': High accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted. joba='E': Same as 'H', and in addition the condition number of column scaled A is estimated and returned in rwork[0]. N^(-1/4)*rwork[0] <= ||pinv(A_scaled)||_2 <= N^(1/4)*rwork[0].

in

jobp

jobp='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. jobp='N': No row pivoting.

in

jobr

jobr='T': After the initial pivoted QR factorization, DGESVD 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. jobr='N': The triangular factor R is given as input to DGESVD. This may be preferred as it involves less data movement.

in

jobu

jobu='A': All m left singular vectors are computed and returned in the matrix U. See the description of U. jobu='S' or jobu='U': n = min(m,n) left singular vectors are computed and returned in the matrix U. See the description of U. jobu='R': Numerical rank numrank is determined and only numrank left singular vectors are computed and returned in the matrix U. jobu='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:n+m-2]. jobu='N': The left singular vectors are not computed.

in

jobv

jobv='A' or jobv='V': All n right singular vectors are computed and returned in the matrix V. jobv='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. jobv='N': The right singular vectors are not computed.

in

m

The number of rows of the input matrix A. m>=0.

in

n

The number of columns of the input matrix A. m>=n>=0.

inout

A

Array of dimensions lda x n. On entry, the input matrix A. On exit, if jobu!='N' or jobv!='N', the lower triangle of A contains the Householder vectors as stored by DGEQP3. If jobu='F', these Householder vectors together with work[0:n-1] can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U.

in

lda

The leading dimension of the array A. lda>=max(1,m).

out

S

Array of dimension n. The singular values of A, ordered so that S[i]>=S[i+1].

out

U

Array of 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.

in

ldu

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.

out

V

Array of 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!='E', V is not referenced.

in

ldv

The leading dimension of the array V. If jobv='A', 'V', 'R', or joba='E', ldv>=max(1,n). Otherwise, ldv>=1.

out

numrank

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 DGESVD. The final value of numrank might be further reduced if some singular values are computed as zeros.

out

iwork

Integer array of dimension (max(1,liwork)). On exit, iwork[0:n-1] contains column pivoting permutation of the rank revealing QR factorization. If jobp='P', iwork[n:n+m-2] 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, iwork[0] returns the minimal liwork.

in

liwork

The dimension of the array iwork. liwork >= n+m-1, if jobp='P' and joba!='E'; liwork >= n, if jobp='N' and joba!='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.

out

work

Array of dimension (max(2,lwork)), used as a workspace. On exit, if, on entry, lwork!=-1, work[0:n-1] contains parameters needed to recover the Q factor from the QR factorization computed by DGEQP3. If liwork, lwork, or lrwork = -1, then on exit, if info=0, work[0] returns the optimal lwork, and work[1] returns the minimal lwork.

inout

lwork

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 estimate 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.

out

rwork

Array of dimension (max(1,lrwork)). On exit,

If joba='E', rwork[0] 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[0] <= ||pinv(C)||_2 <= N^(1/4) * rwork[0]. Otherwise, rwork[0] = -1.
rwork[1] contains the number of singular values computed as exact zeros in DGESVD applied to the upper triangular or trapezoidal R (from the initial QR factorization). In case of early exit (no call to DGESVD, such as in the case of zero matrix) rwork[1] = -1. If liwork, lwork, or lrwork = -1, then on exit, if info=0, rwork[0] returns the minimal lrwork.

in

lrwork

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.

out

info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value. info>0: if DBDSQR did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form B (computed in DGESVD) did not converge to zero.

cgesvdq

Functions

void cgesvdq(
    const char*          joba,
    const char*          jobp,
    const char*          jobr,
    const char*          jobu,
    const char*          jobv,
    const INT            m,
    const INT            n,
          c64*  restrict A,
    const INT            lda,
          f32*  restrict S,
          c64*  restrict U,
    const INT            ldu,
          c64*  restrict V,
    const INT            ldv,
          INT*           numrank,
          INT*  restrict iwork,
    const INT            liwork,
          c64*  restrict cwork,
    const INT            lcwork,
          f32*  restrict rwork,
    const INT            lrwork,
          INT*           info
);

void cgesvdq(const char *joba, const char *jobp, const char *jobr, const char *jobu, const char *jobv, const INT m, const INT n, c64 *restrict A, const INT lda, f32 *restrict S, c64 *restrict U, const INT ldu, c64 *restrict V, const INT ldv, INT *numrank, INT *restrict iwork, const INT liwork, c64 *restrict cwork, const INT lcwork, f32 *restrict rwork, const INT lrwork, INT *info)#

CGESVDQ computes the singular value decomposition (SVD) of a complex M-by-N matrix A, where M >= N.

The SVD of A is written as

         A = U * SIGMA * V^*

where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N unitary matrix. CGESVDQ computes the singular value decomposition (SVD) of a complex 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 unitary 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.

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 minimal 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, enclosed 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

Parameters

in

joba

Specifies the level of accuracy in the computed SVD. joba='A': The requested accuracy corresponds to having the backward error bounded by || delta A ||_F <= f(m,n)*EPS*|| A ||_F, where EPS = clamch("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. joba='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. joba='H': High accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted. joba='E': Same as 'H', and in addition the condition number of column scaled A is estimated and returned in rwork[0]. N^(-1/4)*rwork[0] <= ||pinv(A_scaled)||_2 <= N^(1/4)*rwork[0].

in

jobp

jobp='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. jobp='N': No row pivoting.

in

jobr

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

in

jobu

jobu='A': All m left singular vectors are computed and returned in the matrix U. See the description of U. jobu='S' or jobu='U': n = min(m,n) left singular vectors are computed and returned in the matrix U. See the description of U. jobu='R': Numerical rank numrank is determined and only numrank left singular vectors are computed and returned in the matrix U. jobu='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**H, 0)**H. If row pivoting is used, then the necessary information on the row pivoting is stored in iwork[n:n+m-2]. jobu='N': The left singular vectors are not computed.

in

jobv

jobv='A' or jobv='V': All n right singular vectors are computed and returned in the matrix V. jobv='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. jobv='N': The right singular vectors are not computed.

in

m

The number of rows of the input matrix A. m>=0.

in

n

The number of columns of the input matrix A. m>=n>=0.

inout

A

Array of dimensions lda x n. On entry, the input matrix A. On exit, if jobu!='N' or jobv!='N', the lower triangle of A contains the Householder vectors as stored by CGEQP3. If jobu='F', these Householder vectors together with cwork[0:n-1] can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U.

in

lda

The leading dimension of the array A. lda>=max(1,m).

out

S

Array of dimension n. The singular values of A, ordered so that S[i]>=S[i+1].

out

U

Array of 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 unitary matrix that can be used to form the left singular vectors. If jobu='N', U is not referenced.

in

ldu

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.

out

V

Array of dimension ldv x n if jobv='A', 'V', 'R' or if joba='E'. If jobv='A' or 'V', V contains the n-by-n unitary matrix V**H; If jobv='R', V contains the first numrank rows of V**H (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!='E', V is not referenced.

in

ldv

The leading dimension of the array V. If jobv='A', 'V', 'R', or joba='E', ldv>=max(1,n). Otherwise, ldv>=1.

out

numrank

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 CGESVD. The final value of numrank might be further reduced if some singular values are computed as zeros.

out

iwork

Integer array of dimension (max(1,liwork)). On exit, iwork[0:n-1] contains column pivoting permutation of the rank revealing QR factorization. If jobp='P', iwork[n:n+m-2] 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, lcwork, or lrwork = -1, then on exit, if info=0, iwork[0] returns the minimal liwork.

in

liwork

The dimension of the array iwork. liwork >= n+m-1, if jobp='P' and joba!='E'; liwork >= n, if jobp='N' and joba!='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 cwork, iwork, and rwork arrays, and no error message related to lcwork is issued by XERBLA.

out

cwork

Array of dimension (max(2,lcwork)), used as a workspace. On exit, if, on entry, lcwork!=-1, cwork[0:n-1] contains parameters needed to recover the Q factor from the QR factorization computed by CGEQP3. If liwork, lcwork, or lrwork = -1, then on exit, if info=0, cwork[0] returns the optimal lcwork, and cwork[1] returns the minimal lcwork.

inout

lcwork

The dimension of the array cwork. It is determined as follows:

 Let  LWQP3 = N+1,  LWCON = 2*N, and let
 LWUNQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
         { MAX( M, 1 ),  if JOBU = 'A'
 LWSVD = MAX( 3*N, 1 )
 LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
 LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
 Then the minimal value of LCWORK 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, LWUNQ ) if the singular values and the left
                          singular vectors are requested;
 = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) 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 estimate requested;

 = N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
                          independent of JOBR;
 = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
                          JOBV = 'R' and, also a scaled condition
                          estimate requested; independent of JOBR;
 = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
                full SVD is requested with JOBV = 'A' or 'V', and
                JOBR ='N'
 = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
                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, LWUNQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
                full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
 = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
                if the full SVD is requested with JOBV = 'A', 'V' and
                JOBR ='T', and also a scaled condition number estimate
                requested.

Finally, lcwork must be at least two: lcwork = max(2,lcwork). If lcwork=-1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the cwork, iwork, and rwork arrays, and no error message related to lcwork is issued by XERBLA.

out

rwork

Array of dimension (max(1,lrwork)). On exit,

If joba='E', rwork[0] 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[0] <= ||pinv(C)||_2 <= N^(1/4) * rwork[0]. Otherwise, rwork[0] = -1.
rwork[1] contains the number of singular values computed as exact zeros in CGESVD applied to the upper triangular or trapezoidal R (from the initial QR factorization). In case of early exit (no call to CGESVD, such as in the case of zero matrix) rwork[1] = -1. If liwork, lcwork, or lrwork = -1, then on exit, if info=0, rwork[0] returns the minimal lrwork.

in

lrwork

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 cwork, iwork, and rwork arrays, and no error message related to lcwork is issued by XERBLA.

out

info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value. info>0: if CBDSQR did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form B (computed in CGESVD) did not converge to zero.

zgesvdq

Functions

void zgesvdq(
    const char*          joba,
    const char*          jobp,
    const char*          jobr,
    const char*          jobu,
    const char*          jobv,
    const INT            m,
    const INT            n,
          c128* restrict A,
    const INT            lda,
          f64*  restrict S,
          c128* restrict U,
    const INT            ldu,
          c128* restrict V,
    const INT            ldv,
          INT*           numrank,
          INT*  restrict iwork,
    const INT            liwork,
          c128* restrict cwork,
    const INT            lcwork,
          f64*  restrict rwork,
    const INT            lrwork,
          INT*           info
);

void zgesvdq(const char *joba, const char *jobp, const char *jobr, const char *jobu, const char *jobv, const INT m, const INT n, c128 *restrict A, const INT lda, f64 *restrict S, c128 *restrict U, const INT ldu, c128 *restrict V, const INT ldv, INT *numrank, INT *restrict iwork, const INT liwork, c128 *restrict cwork, const INT lcwork, f64 *restrict rwork, const INT lrwork, INT *info)#

ZGESVDQ computes the singular value decomposition (SVD) of a complex M-by-N matrix A, where M >= N.

The SVD of A is written as

         A = U * SIGMA * V^*

where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal matrix, and V is an N-by-N unitary matrix. ZGESVDQ computes the singular value decomposition (SVD) of a complex 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 unitary 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.

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 minimal precaution and it is left to the SVD routine (ZGESVD) 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 ZGESVD and the scaling in ZGESVD can be switched off.

3. Other comments related to code optimization are given in comments in the code, enclosed 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

Parameters

in

joba

Specifies the level of accuracy in the computed SVD. joba='A': The requested accuracy corresponds to having the backward error bounded by || delta A ||_F <= f(m,n)*EPS*|| A ||_F, where EPS = zlamch("Epsilon"). This authorises ZGESVDQ 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. joba='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. joba='H': High accuracy requested. No numerical rank determination based on the rank revealing QR factorization is attempted. joba='E': Same as 'H', and in addition the condition number of column scaled A is estimated and returned in rwork[0]. N^(-1/4)*rwork[0] <= ||pinv(A_scaled)||_2 <= N^(1/4)*rwork[0].

in

jobp

jobp='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. jobp='N': No row pivoting.

in

jobr

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

in

jobu

jobu='A': All m left singular vectors are computed and returned in the matrix U. See the description of U. jobu='S' or jobu='U': n = min(m,n) left singular vectors are computed and returned in the matrix U. See the description of U. jobu='R': Numerical rank numrank is determined and only numrank left singular vectors are computed and returned in the matrix U. jobu='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**H, 0)**H. If row pivoting is used, then the necessary information on the row pivoting is stored in iwork[n:n+m-2]. jobu='N': The left singular vectors are not computed.

in

jobv

jobv='A' or jobv='V': All n right singular vectors are computed and returned in the matrix V. jobv='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. jobv='N': The right singular vectors are not computed.

in

m

The number of rows of the input matrix A. m>=0.

in

n

The number of columns of the input matrix A. m>=n>=0.

inout

A

Array of dimensions lda x n. On entry, the input matrix A. On exit, if jobu!='N' or jobv!='N', the lower triangle of A contains the Householder vectors as stored by ZGEQP3. If jobu='F', these Householder vectors together with cwork[0:n-1] can be used to restore the Q factors from the initial pivoted QR factorization of A. See the description of U.

in

lda

The leading dimension of the array A. lda>=max(1,m).

out

S

Array of dimension n. The singular values of A, ordered so that S[i]>=S[i+1].

out

U

Array of 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 unitary matrix that can be used to form the left singular vectors. If jobu='N', U is not referenced.

in

ldu

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.

out

V

Array of dimension ldv x n if jobv='A', 'V', 'R' or if joba='E'. If jobv='A' or 'V', V contains the n-by-n unitary matrix V**H; If jobv='R', V contains the first numrank rows of V**H (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!='E', V is not referenced.

in

ldv

The leading dimension of the array V. If jobv='A', 'V', 'R', or joba='E', ldv>=max(1,n). Otherwise, ldv>=1.

out

numrank

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 ZGESVD. The final value of numrank might be further reduced if some singular values are computed as zeros.

out

iwork

Integer array of dimension (max(1,liwork)). On exit, iwork[0:n-1] contains column pivoting permutation of the rank revealing QR factorization. If jobp='P', iwork[n:n+m-2] 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, lcwork, or lrwork = -1, then on exit, if info=0, iwork[0] returns the minimal liwork.

in

liwork

The dimension of the array iwork. liwork >= n+m-1, if jobp='P' and joba!='E'; liwork >= n, if jobp='N' and joba!='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 cwork, iwork, and rwork arrays, and no error message related to lcwork is issued by XERBLA.

out

cwork

Array of dimension (max(2,lcwork)), used as a workspace. On exit, if, on entry, lcwork!=-1, cwork[0:n-1] contains parameters needed to recover the Q factor from the QR factorization computed by ZGEQP3. If liwork, lcwork, or lrwork = -1, then on exit, if info=0, cwork[0] returns the optimal lcwork, and cwork[1] returns the minimal lcwork.

inout

lcwork

The dimension of the array cwork. It is determined as follows:

 Let  LWQP3 = N+1,  LWCON = 2*N, and let
 LWUNQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
         { MAX( M, 1 ),  if JOBU = 'A'
 LWSVD = MAX( 3*N, 1 )
 LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
 LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
 Then the minimal value of LCWORK 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, LWUNQ ) if the singular values and the left
                          singular vectors are requested;
 = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) 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 estimate requested;

 = N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
                          independent of JOBR;
 = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
                          JOBV = 'R' and, also a scaled condition
                          estimate requested; independent of JOBR;
 = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
                full SVD is requested with JOBV = 'A' or 'V', and
                JOBR ='N'
 = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
                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, LWUNQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
                full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
 = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
                if the full SVD is requested with JOBV = 'A', 'V' and
                JOBR ='T', and also a scaled condition number estimate
                requested.

Finally, lcwork must be at least two: lcwork = max(2,lcwork). If lcwork=-1, then a workspace query is assumed; the routine only calculates and returns the optimal and minimal sizes for the cwork, iwork, and rwork arrays, and no error message related to lcwork is issued by XERBLA.

out

rwork

Array of dimension (max(1,lrwork)). On exit,

If joba='E', rwork[0] 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[0] <= ||pinv(C)||_2 <= N^(1/4) * rwork[0]. Otherwise, rwork[0] = -1.
rwork[1] contains the number of singular values computed as exact zeros in ZGESVD applied to the upper triangular or trapezoidal R (from the initial QR factorization). In case of early exit (no call to ZGESVD, such as in the case of zero matrix) rwork[1] = -1. If liwork, lcwork, or lrwork = -1, then on exit, if info=0, rwork[0] returns the minimal lrwork.

in

lrwork

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 cwork, iwork, and rwork arrays, and no error message related to lcwork is issued by XERBLA.

out

info

info=0: successful exit. info<0: if info=-i, the i-th argument had an illegal value. info>0: if ZBDSQR did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form B (computed in ZGESVD) did not converge to zero.