gesvj(3)

NAME
gesvj - gesvj: SVD, Jacobi, low-level

SYNOPSIS

  Functions

subroutine cgesvj (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, cwork, lwork, rwork, lrwork, info)
CGESVJ
subroutine dgesvj (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, work, lwork, info)
DGESVJ
subroutine sgesvj (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, work, lwork, info)
SGESVJ
subroutine zgesvj (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, cwork, lwork, rwork, lrwork, info)
ZGESVJ

Detailed Description

Function Documentation

  subroutine cgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( n ) sva, integer mv, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( lwork ) cwork, integer lwork, real, dimension( lrwork ) rwork, integer lrwork, integer info)

CGESVJ

Purpose:

    !>
    !> CGESVJ 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.
    !>

Parameters
    JOBA

    !>          JOBA is CHARACTER*1
    !>          Specifies the structure of A.
    !>          = 'L': The input matrix A is lower triangular;
    !>          = 'U': The input matrix A is upper triangular;
    !>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
    !>

    JOBU

    !>          JOBU is CHARACTER*1
    !>          Specifies whether to compute the left singular vectors
    !>          (columns of U):
    !>          = 'U' or 'F': The left singular vectors corresponding to the nonzero
    !>                 singular values are computed and returned in the leading
    !>                 columns of A. See more details in the description of A.
    !>                 The default numerical orthogonality threshold is set to
    !>                 approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
    !>          = 'C': Analogous to JOBU='U', except that user can control the
    !>                 level of numerical orthogonality of the computed left
    !>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
    !>                 CTOL is given on input in the array WORK.
    !>                 No CTOL smaller than ONE is allowed. CTOL greater
    !>                 than 1 / EPS is meaningless. The option 'C'
    !>                 can be used if M*EPS is satisfactory orthogonality
    !>                 of the computed left singular vectors, so CTOL=M could
    !>                 save few sweeps of Jacobi rotations.
    !>                 See the descriptions of A and WORK(1).
    !>          = 'N': The matrix U is not computed. However, see the
    !>                 description of A.
    !>

    JOBV

    !>          JOBV is CHARACTER*1
    !>          Specifies whether to compute the right singular vectors, that
    !>          is, the matrix V:
    !>          = 'V' or 'J': the matrix V is computed and returned in the array V
    !>          = 'A':  the Jacobi rotations are applied to the MV-by-N
    !>                  array V. In other words, the right singular vector
    !>                  matrix V is not computed explicitly; instead it is
    !>                  applied to an MV-by-N matrix initially stored in the
    !>                  first MV rows of V.
    !>          = 'N':  the matrix V is not computed and the array V is not
    !>                  referenced
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A. 1/SLAMCH('E') > M >= 0.
    !>

    N

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

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit,
    !>          If JOBU = 'U' .OR. JOBU = 'C':
    !>                 If INFO = 0 :
    !>                 RANKA orthonormal columns of U are returned in the
    !>                 leading RANKA columns of the array A. Here RANKA <= N
    !>                 is the number of computed singular values of A that are
    !>                 above the underflow threshold SLAMCH('S'). The singular
    !>                 vectors corresponding to underflowed or zero singular
    !>                 values are not computed. The value of RANKA is returned
    !>                 in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
    !>                 descriptions of SVA and RWORK. The computed columns of U
    !>                 are mutually numerically orthogonal up to approximately
    !>                 TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
    !>                 see the description of JOBU.
    !>                 If INFO > 0,
    !>                 the procedure CGESVJ did not converge in the given number
    !>                 of iterations (sweeps). In that case, the computed
    !>                 columns of U may not be orthogonal up to TOL. The output
    !>                 U (stored in A), SIGMA (given by the computed singular
    !>                 values in SVA(1:N)) and V is still a decomposition of the
    !>                 input matrix A in the sense that the residual
    !>                 || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
    !>          If JOBU = 'N':
    !>                 If INFO = 0 :
    !>                 Note that the left singular vectors are 'for free' in the
    !>                 one-sided Jacobi SVD algorithm. However, if only the
    !>                 singular values are needed, the level of numerical
    !>                 orthogonality of U is not an issue and iterations are
    !>                 stopped when the columns of the iterated matrix are
    !>                 numerically orthogonal up to approximately M*EPS. Thus,
    !>                 on exit, A contains the columns of U scaled with the
    !>                 corresponding singular values.
    !>                 If INFO > 0 :
    !>                 the procedure CGESVJ did not converge in the given number
    !>                 of iterations (sweeps).
    !>

    LDA

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

    SVA

    !>          SVA is REAL array, dimension (N)
    !>          On exit,
    !>          If INFO = 0 :
    !>          depending on the value SCALE = RWORK(1), we have:
    !>                 If SCALE = ONE:
    !>                 SVA(1:N) contains the computed singular values of A.
    !>                 During the computation SVA contains the Euclidean column
    !>                 norms of the iterated matrices in the array A.
    !>                 If SCALE .NE. ONE:
    !>                 The singular values of A are SCALE*SVA(1:N), and this
    !>                 factored representation is due to the fact that some of the
    !>                 singular values of A might underflow or overflow.
    !>
    !>          If INFO > 0 :
    !>          the procedure CGESVJ did not converge in the given number of
    !>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
    !>

    MV

    !>          MV is INTEGER
    !>          If JOBV = 'A', then the product of Jacobi rotations in CGESVJ
    !>          is applied to the first MV rows of V. See the description of JOBV.
    !>

    V

    !>          V is COMPLEX array, dimension (LDV,N)
    !>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
    !>                         the right singular vectors;
    !>          If JOBV = 'A', then V contains the product of the computed right
    !>                         singular vector matrix and the initial matrix in
    !>                         the array V.
    !>          If JOBV = 'N', then V is not referenced.
    !>

    LDV

    !>          LDV is INTEGER
    !>          The leading dimension of the array V, LDV >= 1.
    !>          If JOBV = 'V', then LDV >= max(1,N).
    !>          If JOBV = 'A', then LDV >= max(1,MV) .
    !>

    CWORK

    !>          CWORK is COMPLEX array, dimension (max(1,LWORK))
    !>          Used as workspace.
    !>

    LWORK

    !>          LWORK is INTEGER.
    !>          Length of CWORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M+N, otherwise.
    !>
    !>          If on entry LWORK = -1, then a workspace query is assumed and
    !>          no computation is done; CWORK(1) is set to the minial (and optimal)
    !>          length of CWORK.
    !>

    RWORK

    !>          RWORK is REAL array, dimension (max(6,LRWORK))
    !>          On entry,
    !>          If JOBU = 'C' :
    !>          RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
    !>                    The process stops if all columns of A are mutually
    !>                    orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
    !>                    It is required that CTOL >= ONE, i.e. it is not
    !>                    allowed to force the routine to obtain orthogonality
    !>                    below EPSILON.
    !>          On exit,
    !>          RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
    !>                    are the computed singular values of A.
    !>                    (See description of SVA().)
    !>          RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
    !>                    singular values.
    !>          RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
    !>                    values that are larger than the underflow threshold.
    !>          RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
    !>                    rotations needed for numerical convergence.
    !>          RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
    !>                    This is useful information in cases when CGESVJ did
    !>                    not converge, as it can be used to estimate whether
    !>                    the output is still useful and for post festum analysis.
    !>          RWORK(6) = the largest absolute value over all sines of the
    !>                    Jacobi rotation angles in the last sweep. It can be
    !>                    useful for a post festum analysis.
    !>

    LRWORK

    !>         LRWORK is INTEGER
    !>         Length of RWORK.
    !>         LRWORK >= 1, if MIN(M,N) = 0, and LRWORK >= MAX(6,N), otherwise
    !>
    !>         If on entry LRWORK = -1, then a workspace query is assumed and
    !>         no computation is done; RWORK(1) is set to the minial (and optimal)
    !>         length of RWORK.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, then the i-th argument had an illegal value
    !>          > 0:  CGESVJ did not converge in the maximal allowed number
    !>                (NSWEEP=30) of sweeps. The output may still be useful.
    !>                See the description of RWORK.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
    !> rotations. In the case of underflow of the tangent of the Jacobi angle, a
    !> modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
    !> column interchanges of de Rijk [1]. The relative accuracy of the computed
    !> singular values and the accuracy of the computed singular vectors (in
    !> angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
    !> The condition number that determines the accuracy in the full rank case
    !> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
    !> spectral condition number. The best performance of this Jacobi SVD
    !> procedure is achieved if used in an  accelerated version of Drmac and
    !> Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
    !> Some tuning parameters (marked with [TP]) are available for the
    !> implementer.
    !> The computational range for the nonzero singular values is the  machine
    !> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
    !> denormalized singular values can be computed with the corresponding
    !> gradual loss of accurate digits.
    !>

Contributor:

    !>
    !>  ============
    !>
    !>  Zlatko Drmac (Zagreb, Croatia)
    !>
    !>

References:

    !>
    !> [1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
    !>    singular value decomposition on a vector computer.
    !>    SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
    !> [2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
    !> [3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
    !>    value computation in floating point arithmetic.
    !>    SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
    !> [4] 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.
    !> [5] 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.
    !> [6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
    !>    QSVD, (H,K)-SVD computations.
    !>    Department of Mathematics, University of Zagreb, 2008, 2015.
    !>

Bugs, examples and comments:

    !>  ===========================
    !>  Please report all bugs and send interesting test examples and comments to
    !>  drmac@math.hr. Thank you.
    !>

  subroutine dgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( n ) sva, integer mv, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( lwork ) work, integer lwork, integer info)

DGESVJ

Purpose:

    !>
    !> DGESVJ computes the singular value decomposition (SVD) of a real
    !> M-by-N matrix A, where M >= N. The SVD of A is written as
    !>                                    [++]   [xx]   [x0]   [xx]
    !>              A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
    !>                                    [++]   [xx]
    !> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
    !> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
    !> of SIGMA are the singular values of A. The columns of U and V are the
    !> left and the right singular vectors of A, respectively.
    !> DGESVJ 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 structure of A.
    !>          = 'L': The input matrix A is lower triangular;
    !>          = 'U': The input matrix A is upper triangular;
    !>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
    !>

    JOBU

    !>          JOBU is CHARACTER*1
    !>          Specifies whether to compute the left singular vectors
    !>          (columns of U):
    !>          = 'U': The left singular vectors corresponding to the nonzero
    !>                 singular values are computed and returned in the leading
    !>                 columns of A. See more details in the description of A.
    !>                 The default numerical orthogonality threshold is set to
    !>                 approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
    !>          = 'C': Analogous to JOBU='U', except that user can control the
    !>                 level of numerical orthogonality of the computed left
    !>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
    !>                 CTOL is given on input in the array WORK.
    !>                 No CTOL smaller than ONE is allowed. CTOL greater
    !>                 than 1 / EPS is meaningless. The option 'C'
    !>                 can be used if M*EPS is satisfactory orthogonality
    !>                 of the computed left singular vectors, so CTOL=M could
    !>                 save few sweeps of Jacobi rotations.
    !>                 See the descriptions of A and WORK(1).
    !>          = 'N': The matrix U is not computed. However, see the
    !>                 description of A.
    !>

    JOBV

    !>          JOBV is CHARACTER*1
    !>          Specifies whether to compute the right singular vectors, that
    !>          is, the matrix V:
    !>          = 'V':  the matrix V is computed and returned in the array V
    !>          = 'A':  the Jacobi rotations are applied to the MV-by-N
    !>                  array V. In other words, the right singular vector
    !>                  matrix V is not computed explicitly, instead it is
    !>                  applied to an MV-by-N matrix initially stored in the
    !>                  first MV rows of V.
    !>          = 'N':  the matrix V is not computed and the array V is not
    !>                  referenced
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
    !>

    N

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

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit :
    !>          If JOBU = 'U' .OR. JOBU = 'C' :
    !>                 If INFO = 0 :
    !>                 RANKA orthonormal columns of U are returned in the
    !>                 leading RANKA columns of the array A. Here RANKA <= N
    !>                 is the number of computed singular values of A that are
    !>                 above the underflow threshold DLAMCH('S'). The singular
    !>                 vectors corresponding to underflowed or zero singular
    !>                 values are not computed. The value of RANKA is returned
    !>                 in the array WORK as RANKA=NINT(WORK(2)). Also see the
    !>                 descriptions of SVA and WORK. The computed columns of U
    !>                 are mutually numerically orthogonal up to approximately
    !>                 TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
    !>                 see the description of JOBU.
    !>                 If INFO > 0 :
    !>                 the procedure DGESVJ did not converge in the given number
    !>                 of iterations (sweeps). In that case, the computed
    !>                 columns of U may not be orthogonal up to TOL. The output
    !>                 U (stored in A), SIGMA (given by the computed singular
    !>                 values in SVA(1:N)) and V is still a decomposition of the
    !>                 input matrix A in the sense that the residual
    !>                 ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
    !>
    !>          If JOBU = 'N' :
    !>                 If INFO = 0 :
    !>                 Note that the left singular vectors are 'for free' in the
    !>                 one-sided Jacobi SVD algorithm. However, if only the
    !>                 singular values are needed, the level of numerical
    !>                 orthogonality of U is not an issue and iterations are
    !>                 stopped when the columns of the iterated matrix are
    !>                 numerically orthogonal up to approximately M*EPS. Thus,
    !>                 on exit, A contains the columns of U scaled with the
    !>                 corresponding singular values.
    !>                 If INFO > 0 :
    !>                 the procedure DGESVJ did not converge in the given number
    !>                 of iterations (sweeps).
    !>

    LDA

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

    SVA

    !>          SVA is DOUBLE PRECISION array, dimension (N)
    !>          On exit :
    !>          If INFO = 0 :
    !>          depending on the value SCALE = WORK(1), we have:
    !>                 If SCALE = ONE :
    !>                 SVA(1:N) contains the computed singular values of A.
    !>                 During the computation SVA contains the Euclidean column
    !>                 norms of the iterated matrices in the array A.
    !>                 If SCALE .NE. ONE :
    !>                 The singular values of A are SCALE*SVA(1:N), and this
    !>                 factored representation is due to the fact that some of the
    !>                 singular values of A might underflow or overflow.
    !>          If INFO > 0 :
    !>          the procedure DGESVJ did not converge in the given number of
    !>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
    !>

    MV

    !>          MV is INTEGER
    !>          If JOBV = 'A', then the product of Jacobi rotations in DGESVJ
    !>          is applied to the first MV rows of V. See the description of JOBV.
    !>

    V

    !>          V is DOUBLE PRECISION array, dimension (LDV,N)
    !>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
    !>                         the right singular vectors;
    !>          If JOBV = 'A', then V contains the product of the computed right
    !>                         singular vector matrix and the initial matrix in
    !>                         the array V.
    !>          If JOBV = 'N', then V is not referenced.
    !>

    LDV

    !>          LDV is INTEGER
    !>          The leading dimension of the array V, LDV >= 1.
    !>          If JOBV = 'V', then LDV >= max(1,N).
    !>          If JOBV = 'A', then LDV >= max(1,MV) .
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On entry :
    !>          If JOBU = 'C' :
    !>          WORK(1) = CTOL, where CTOL defines the threshold for convergence.
    !>                    The process stops if all columns of A are mutually
    !>                    orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
    !>                    It is required that CTOL >= ONE, i.e. it is not
    !>                    allowed to force the routine to obtain orthogonality
    !>                    below EPS.
    !>          On exit :
    !>          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
    !>                    are the computed singular values of A.
    !>                    (See description of SVA().)
    !>          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
    !>                    singular values.
    !>          WORK(3) = NINT(WORK(3)) is the number of the computed singular
    !>                    values that are larger than the underflow threshold.
    !>          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
    !>                    rotations needed for numerical convergence.
    !>          WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
    !>                    This is useful information in cases when DGESVJ did
    !>                    not converge, as it can be used to estimate whether
    !>                    the output is still useful and for post festum analysis.
    !>          WORK(6) = the largest absolute value over all sines of the
    !>                    Jacobi rotation angles in the last sweep. It can be
    !>                    useful for a post festum analysis.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(6,M+N), otherwise.
    !>
    !>          If on entry LWORK = -1, then a workspace query is assumed and
    !>          no computation is done; WORK(1) is set to the minial (and optimal)
    !>          length of WORK.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, then the i-th argument had an illegal value
    !>          > 0:  DGESVJ did not converge in the maximal allowed number (30)
    !>                of sweeps. The output may still be useful. See the
    !>                description of WORK.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
    !>  rotations. The rotations are implemented as fast scaled rotations of
    !>  Anda and Park [1]. In the case of underflow of the Jacobi angle, a
    !>  modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
    !>  column interchanges of de Rijk [2]. The relative accuracy of the computed
    !>  singular values and the accuracy of the computed singular vectors (in
    !>  angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
    !>  The condition number that determines the accuracy in the full rank case
    !>  is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
    !>  spectral condition number. The best performance of this Jacobi SVD
    !>  procedure is achieved if used in an  accelerated version of Drmac and
    !>  Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
    !>  Some tuning parameters (marked with [TP]) are available for the
    !>  implementer.
    !>  The computational range for the nonzero singular values is the  machine
    !>  number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
    !>  denormalized singular values can be computed with the corresponding
    !>  gradual loss of accurate digits.
    !>

Contributors:

    !>
    !>  ============
    !>
    !>  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
    !>

References:

    !>
    !> [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
    !>     SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
    !> [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
    !>     singular value decomposition on a vector computer.
    !>     SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
    !> [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
    !> [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
    !>     value computation in floating point arithmetic.
    !>     SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
    !> [5] 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.
    !> [6] 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.
    !> [7] 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:

    !>  ===========================
    !>  Please report all bugs and send interesting test examples and comments to
    !>  drmac@math.hr. Thank you.
    !>

  subroutine sgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( n ) sva, integer mv, real, dimension( ldv, * ) v, integer ldv, real, dimension( lwork ) work, integer lwork, integer info)

SGESVJ

Purpose:

    !>
    !> SGESVJ computes the singular value decomposition (SVD) of a real
    !> M-by-N matrix A, where M >= N. The SVD of A is written as
    !>                                    [++]   [xx]   [x0]   [xx]
    !>              A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
    !>                                    [++]   [xx]
    !> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
    !> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
    !> of SIGMA are the singular values of A. The columns of U and V are the
    !> left and the right singular vectors of A, respectively.
    !> SGESVJ 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 structure of A.
    !>          = 'L': The input matrix A is lower triangular;
    !>          = 'U': The input matrix A is upper triangular;
    !>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
    !>

    JOBU

    !>          JOBU is CHARACTER*1
    !>          Specifies whether to compute the left singular vectors
    !>          (columns of U):
    !>          = 'U': The left singular vectors corresponding to the nonzero
    !>                 singular values are computed and returned in the leading
    !>                 columns of A. See more details in the description of A.
    !>                 The default numerical orthogonality threshold is set to
    !>                 approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
    !>          = 'C': Analogous to JOBU='U', except that user can control the
    !>                 level of numerical orthogonality of the computed left
    !>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
    !>                 CTOL is given on input in the array WORK.
    !>                 No CTOL smaller than ONE is allowed. CTOL greater
    !>                 than 1 / EPS is meaningless. The option 'C'
    !>                 can be used if M*EPS is satisfactory orthogonality
    !>                 of the computed left singular vectors, so CTOL=M could
    !>                 save few sweeps of Jacobi rotations.
    !>                 See the descriptions of A and WORK(1).
    !>          = 'N': The matrix U is not computed. However, see the
    !>                 description of A.
    !>

    JOBV

    !>          JOBV is CHARACTER*1
    !>          Specifies whether to compute the right singular vectors, that
    !>          is, the matrix V:
    !>          = 'V':  the matrix V is computed and returned in the array V
    !>          = 'A':  the Jacobi rotations are applied to the MV-by-N
    !>                  array V. In other words, the right singular vector
    !>                  matrix V is not computed explicitly; instead it is
    !>                  applied to an MV-by-N matrix initially stored in the
    !>                  first MV rows of V.
    !>          = 'N':  the matrix V is not computed and the array V is not
    !>                  referenced
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A. 1/SLAMCH('E') > 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.
    !>          On exit,
    !>          If JOBU = 'U' .OR. JOBU = 'C':
    !>                 If INFO = 0:
    !>                 RANKA orthonormal columns of U are returned in the
    !>                 leading RANKA columns of the array A. Here RANKA <= N
    !>                 is the number of computed singular values of A that are
    !>                 above the underflow threshold SLAMCH('S'). The singular
    !>                 vectors corresponding to underflowed or zero singular
    !>                 values are not computed. The value of RANKA is returned
    !>                 in the array WORK as RANKA=NINT(WORK(2)). Also see the
    !>                 descriptions of SVA and WORK. The computed columns of U
    !>                 are mutually numerically orthogonal up to approximately
    !>                 TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
    !>                 see the description of JOBU.
    !>                 If INFO > 0,
    !>                 the procedure SGESVJ did not converge in the given number
    !>                 of iterations (sweeps). In that case, the computed
    !>                 columns of U may not be orthogonal up to TOL. The output
    !>                 U (stored in A), SIGMA (given by the computed singular
    !>                 values in SVA(1:N)) and V is still a decomposition of the
    !>                 input matrix A in the sense that the residual
    !>                 ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
    !>          If JOBU = 'N':
    !>                 If INFO = 0:
    !>                 Note that the left singular vectors are 'for free' in the
    !>                 one-sided Jacobi SVD algorithm. However, if only the
    !>                 singular values are needed, the level of numerical
    !>                 orthogonality of U is not an issue and iterations are
    !>                 stopped when the columns of the iterated matrix are
    !>                 numerically orthogonal up to approximately M*EPS. Thus,
    !>                 on exit, A contains the columns of U scaled with the
    !>                 corresponding singular values.
    !>                 If INFO > 0:
    !>                 the procedure SGESVJ did not converge in the given number
    !>                 of iterations (sweeps).
    !>

    LDA

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

    SVA

    !>          SVA is REAL array, dimension (N)
    !>          On exit,
    !>          If INFO = 0 :
    !>          depending on the value SCALE = WORK(1), we have:
    !>                 If SCALE = ONE:
    !>                 SVA(1:N) contains the computed singular values of A.
    !>                 During the computation SVA contains the Euclidean column
    !>                 norms of the iterated matrices in the array A.
    !>                 If SCALE .NE. ONE:
    !>                 The singular values of A are SCALE*SVA(1:N), and this
    !>                 factored representation is due to the fact that some of the
    !>                 singular values of A might underflow or overflow.
    !>
    !>          If INFO > 0 :
    !>          the procedure SGESVJ did not converge in the given number of
    !>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
    !>

    MV

    !>          MV is INTEGER
    !>          If JOBV = 'A', then the product of Jacobi rotations in SGESVJ
    !>          is applied to the first MV rows of V. See the description of JOBV.
    !>

    V

    !>          V is REAL array, dimension (LDV,N)
    !>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
    !>                         the right singular vectors;
    !>          If JOBV = 'A', then V contains the product of the computed right
    !>                         singular vector matrix and the initial matrix in
    !>                         the array V.
    !>          If JOBV = 'N', then V is not referenced.
    !>

    LDV

    !>          LDV is INTEGER
    !>          The leading dimension of the array V, LDV >= 1.
    !>          If JOBV = 'V', then LDV >= max(1,N).
    !>          If JOBV = 'A', then LDV >= max(1,MV) .
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK))
    !>          On entry,
    !>          If JOBU = 'C' :
    !>          WORK(1) = CTOL, where CTOL defines the threshold for convergence.
    !>                    The process stops if all columns of A are mutually
    !>                    orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
    !>                    It is required that CTOL >= ONE, i.e. it is not
    !>                    allowed to force the routine to obtain orthogonality
    !>                    below EPSILON.
    !>          On exit,
    !>          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
    !>                    are the computed singular vcalues of A.
    !>                    (See description of SVA().)
    !>          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
    !>                    singular values.
    !>          WORK(3) = NINT(WORK(3)) is the number of the computed singular
    !>                    values that are larger than the underflow threshold.
    !>          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
    !>                    rotations needed for numerical convergence.
    !>          WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
    !>                    This is useful information in cases when SGESVJ did
    !>                    not converge, as it can be used to estimate whether
    !>                    the output is still useful and for post festum analysis.
    !>          WORK(6) = the largest absolute value over all sines of the
    !>                    Jacobi rotation angles in the last sweep. It can be
    !>                    useful for a post festum analysis.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          Length of WORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(6,M+N), otherwise.
    !>
    !>          If on entry LWORK = -1, then a workspace query is assumed and
    !>          no computation is done; WORK(1) is set to the minial (and optimal)
    !>          length of WORK.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, then the i-th argument had an illegal value
    !>          > 0:  SGESVJ did not converge in the maximal allowed number (30)
    !>                of sweeps. The output may still be useful. See the
    !>                description of WORK.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:
    The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane rotations. The rotations are implemented as fast scaled rotations of Anda and Park [1]. In the case of underflow of the Jacobi angle, a modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses column interchanges of de Rijk [2]. The relative accuracy of the computed singular values and the accuracy of the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. The condition number that determines the accuracy in the full rank case is essentially min_{D=diag} kappa(A*D), where kappa(.) is the spectral condition number. The best performance of this Jacobi SVD procedure is achieved if used in an accelerated version of Drmac and Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. Some tuning parameters (marked with [TP]) are available for the implementer.
    The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits.

Contributors:
    Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

References:
    [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
     SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.

    [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer.
     SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.

    [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
    [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic.
     SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.

    [5] 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.

    [6] 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.

    [7] 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:
    Please report all bugs and send interesting test examples and comments to drmac@math.hr. Thank you.

  subroutine zgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( n ) sva, integer mv, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( lwork ) cwork, integer lwork, double precision, dimension( lrwork ) rwork, integer lrwork, integer info)

ZGESVJ

Purpose:

    !>
    !> ZGESVJ 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.
    !>

Parameters
    JOBA

    !>          JOBA is CHARACTER*1
    !>          Specifies the structure of A.
    !>          = 'L': The input matrix A is lower triangular;
    !>          = 'U': The input matrix A is upper triangular;
    !>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
    !>

    JOBU

    !>          JOBU is CHARACTER*1
    !>          Specifies whether to compute the left singular vectors
    !>          (columns of U):
    !>          = 'U' or 'F': The left singular vectors corresponding to the nonzero
    !>                 singular values are computed and returned in the leading
    !>                 columns of A. See more details in the description of A.
    !>                 The default numerical orthogonality threshold is set to
    !>                 approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=DLAMCH('E').
    !>          = 'C': Analogous to JOBU='U', except that user can control the
    !>                 level of numerical orthogonality of the computed left
    !>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
    !>                 CTOL is given on input in the array WORK.
    !>                 No CTOL smaller than ONE is allowed. CTOL greater
    !>                 than 1 / EPS is meaningless. The option 'C'
    !>                 can be used if M*EPS is satisfactory orthogonality
    !>                 of the computed left singular vectors, so CTOL=M could
    !>                 save few sweeps of Jacobi rotations.
    !>                 See the descriptions of A and WORK(1).
    !>          = 'N': The matrix U is not computed. However, see the
    !>                 description of A.
    !>

    JOBV

    !>          JOBV is CHARACTER*1
    !>          Specifies whether to compute the right singular vectors, that
    !>          is, the matrix V:
    !>          = 'V' or 'J': the matrix V is computed and returned in the array V
    !>          = 'A':  the Jacobi rotations are applied to the MV-by-N
    !>                  array V. In other words, the right singular vector
    !>                  matrix V is not computed explicitly; instead it is
    !>                  applied to an MV-by-N matrix initially stored in the
    !>                  first MV rows of V.
    !>          = 'N':  the matrix V is not computed and the array V is not
    !>                  referenced
    !>

    M

    !>          M is INTEGER
    !>          The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
    !>

    N

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

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit,
    !>          If JOBU = 'U' .OR. JOBU = 'C':
    !>                 If INFO = 0 :
    !>                 RANKA orthonormal columns of U are returned in the
    !>                 leading RANKA columns of the array A. Here RANKA <= N
    !>                 is the number of computed singular values of A that are
    !>                 above the underflow threshold DLAMCH('S'). The singular
    !>                 vectors corresponding to underflowed or zero singular
    !>                 values are not computed. The value of RANKA is returned
    !>                 in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
    !>                 descriptions of SVA and RWORK. The computed columns of U
    !>                 are mutually numerically orthogonal up to approximately
    !>                 TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
    !>                 see the description of JOBU.
    !>                 If INFO > 0,
    !>                 the procedure ZGESVJ did not converge in the given number
    !>                 of iterations (sweeps). In that case, the computed
    !>                 columns of U may not be orthogonal up to TOL. The output
    !>                 U (stored in A), SIGMA (given by the computed singular
    !>                 values in SVA(1:N)) and V is still a decomposition of the
    !>                 input matrix A in the sense that the residual
    !>                 || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
    !>          If JOBU = 'N':
    !>                 If INFO = 0 :
    !>                 Note that the left singular vectors are 'for free' in the
    !>                 one-sided Jacobi SVD algorithm. However, if only the
    !>                 singular values are needed, the level of numerical
    !>                 orthogonality of U is not an issue and iterations are
    !>                 stopped when the columns of the iterated matrix are
    !>                 numerically orthogonal up to approximately M*EPS. Thus,
    !>                 on exit, A contains the columns of U scaled with the
    !>                 corresponding singular values.
    !>                 If INFO > 0:
    !>                 the procedure ZGESVJ did not converge in the given number
    !>                 of iterations (sweeps).
    !>

    LDA

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

    SVA

    !>          SVA is DOUBLE PRECISION array, dimension (N)
    !>          On exit,
    !>          If INFO = 0 :
    !>          depending on the value SCALE = RWORK(1), we have:
    !>                 If SCALE = ONE:
    !>                 SVA(1:N) contains the computed singular values of A.
    !>                 During the computation SVA contains the Euclidean column
    !>                 norms of the iterated matrices in the array A.
    !>                 If SCALE .NE. ONE:
    !>                 The singular values of A are SCALE*SVA(1:N), and this
    !>                 factored representation is due to the fact that some of the
    !>                 singular values of A might underflow or overflow.
    !>
    !>          If INFO > 0:
    !>          the procedure ZGESVJ did not converge in the given number of
    !>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
    !>

    MV

    !>          MV is INTEGER
    !>          If JOBV = 'A', then the product of Jacobi rotations in ZGESVJ
    !>          is applied to the first MV rows of V. See the description of JOBV.
    !>

    V

    !>          V is COMPLEX*16 array, dimension (LDV,N)
    !>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
    !>                         the right singular vectors;
    !>          If JOBV = 'A', then V contains the product of the computed right
    !>                         singular vector matrix and the initial matrix in
    !>                         the array V.
    !>          If JOBV = 'N', then V is not referenced.
    !>

    LDV

    !>          LDV is INTEGER
    !>          The leading dimension of the array V, LDV >= 1.
    !>          If JOBV = 'V', then LDV >= MAX(1,N).
    !>          If JOBV = 'A', then LDV >= MAX(1,MV) .
    !>

    CWORK

    !>          CWORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          Used as workspace.
    !>

    LWORK

    !>          LWORK is INTEGER.
    !>          Length of CWORK.
    !>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M+N, otherwise.
    !>
    !>          If on entry LWORK = -1, then a workspace query is assumed and
    !>          no computation is done; CWORK(1) is set to the minial (and optimal)
    !>          length of CWORK.
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (max(6,LRWORK))
    !>          On entry,
    !>          If JOBU = 'C' :
    !>          RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
    !>                    The process stops if all columns of A are mutually
    !>                    orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
    !>                    It is required that CTOL >= ONE, i.e. it is not
    !>                    allowed to force the routine to obtain orthogonality
    !>                    below EPSILON.
    !>          On exit,
    !>          RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
    !>                    are the computed singular values of A.
    !>                    (See description of SVA().)
    !>          RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
    !>                    singular values.
    !>          RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
    !>                    values that are larger than the underflow threshold.
    !>          RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
    !>                    rotations needed for numerical convergence.
    !>          RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
    !>                    This is useful information in cases when ZGESVJ did
    !>                    not converge, as it can be used to estimate whether
    !>                    the output is still useful and for post festum analysis.
    !>          RWORK(6) = the largest absolute value over all sines of the
    !>                    Jacobi rotation angles in the last sweep. It can be
    !>                    useful for a post festum analysis.
    !>

    LRWORK

    !>         LRWORK is INTEGER
    !>         Length of RWORK.
    !>         LRWORK >= 1, if MIN(M,N) = 0, and LRWORK >= MAX(6,N), otherwise.
    !>
    !>         If on entry LRWORK = -1, then a workspace query is assumed and
    !>         no computation is done; RWORK(1) is set to the minial (and optimal)
    !>         length of RWORK.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, then the i-th argument had an illegal value
    !>          > 0:  ZGESVJ did not converge in the maximal allowed number
    !>                (NSWEEP=30) of sweeps. The output may still be useful.
    !>                See the description of RWORK.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
    !> rotations. In the case of underflow of the tangent of the Jacobi angle, a
    !> modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
    !> column interchanges of de Rijk [1]. The relative accuracy of the computed
    !> singular values and the accuracy of the computed singular vectors (in
    !> angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
    !> The condition number that determines the accuracy in the full rank case
    !> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
    !> spectral condition number. The best performance of this Jacobi SVD
    !> procedure is achieved if used in an  accelerated version of Drmac and
    !> Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
    !> Some tuning parameters (marked with [TP]) are available for the
    !> implementer.
    !> The computational range for the nonzero singular values is the  machine
    !> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
    !> denormalized singular values can be computed with the corresponding
    !> gradual loss of accurate digits.
    !>

Contributor:

    !>
    !>  ============
    !>
    !>  Zlatko Drmac (Zagreb, Croatia)
    !>
    !>

References:

    !>
    !> [1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
    !>    singular value decomposition on a vector computer.
    !>    SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
    !> [2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
    !> [3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
    !>    value computation in floating point arithmetic.
    !>    SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
    !> [4] 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.
    !> [5] 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.
    !> [6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
    !>    QSVD, (H,K)-SVD computations.
    !>    Department of Mathematics, University of Zagreb, 2008, 2015.
    !>

Bugs, examples and comments:

    !>  ===========================
    !>  Please report all bugs and send interesting test examples and comments to
    !>  drmac@math.hr. Thank you.
    !>

Author

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