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realGEsing(3) LAPACK realGEsing(3)

NAME

realGEsing

SYNOPSIS

Functions


subroutine sgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
SGEJSV subroutine sgesdd (JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
SGESDD subroutine sgesvd (JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO)
SGESVD computes the singular value decomposition (SVD) for GE matrices subroutine sgesvdx (JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, IL, IU, NS, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
SGESVDX computes the singular value decomposition (SVD) for GE matrices subroutine sggsvd3 (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)
SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Detailed Description

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

Function Documentation

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

SGEJSV

Purpose:

 SGEJSV computes the singular value decomposition (SVD) of a real M-by-N
 matrix [A], where M >= N. The SVD of [A] is written as
              [A] = [U] * [SIGMA] * [V]^t,
 where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
 diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
 [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
 the singular values of [A]. The columns of [U] and [V] are the left and
 the right singular vectors of [A], respectively. The matrices [U] and [V]
 are computed and stored in the arrays U and V, respectively. The diagonal
 of [SIGMA] is computed and stored in the array SVA.
 SGEJSV can sometimes compute tiny singular values and their singular vectors much
 more accurately than other SVD routines, see below under Further Details.

Parameters:

JOBA

          JOBA is CHARACTER*1
         Specifies the level of accuracy:
       = 'C': This option works well (high relative accuracy) if A = B * D,
              with well-conditioned B and arbitrary diagonal matrix D.
              The accuracy cannot be spoiled by COLUMN scaling. The
              accuracy of the computed output depends on the condition of
              B, and the procedure aims at the best theoretical accuracy.
              The relative error max_{i=1:N}|d sigma_i| / sigma_i is
              bounded by f(M,N)*epsilon* cond(B), independent of D.
              The input matrix is preprocessed with the QRF with column
              pivoting. This initial preprocessing and preconditioning by
              a rank revealing QR factorization is common for all values of
              JOBA. Additional actions are specified as follows:
       = 'E': Computation as with 'C' with an additional estimate of the
              condition number of B. It provides a realistic error bound.
       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
              D1, D2, and well-conditioned matrix C, this option gives
              higher accuracy than the 'C' option. If the structure of the
              input matrix is not known, and relative accuracy is
              desirable, then this option is advisable. The input matrix A
              is preprocessed with QR factorization with FULL (row and
              column) pivoting.
       = 'G'  Computation as with 'F' with an additional estimate of the
              condition number of B, where A=D*B. If A has heavily weighted
              rows, then using this condition number gives too pessimistic
              error bound.
       = 'A': Small singular values are the noise and the matrix is treated
              as numerically rank deficient. The error in the computed
              singular values is bounded by f(m,n)*epsilon*||A||.
              The computed SVD A = U * S * V^t restores A up to
              f(m,n)*epsilon*||A||.
              This gives the procedure the licence to discard (set to zero)
              all singular values below N*epsilon*||A||.
       = 'R': Similar as in 'A'. Rank revealing property of the initial
              QR factorization is used do reveal (using triangular factor)
              a gap sigma_{r+1} < epsilon * sigma_r in which case the
              numerical RANK is declared to be r. The SVD is computed with
              absolute error bounds, but more accurately than with 'A'.

JOBU

          JOBU is CHARACTER*1
         Specifies whether to compute the columns of U:
       = 'U': N columns of U are returned in the array U.
       = 'F': full set of M left sing. vectors is returned in the array U.
       = 'W': U may be used as workspace of length M*N. See the description
              of U.
       = 'N': U is not computed.

JOBV

          JOBV is CHARACTER*1
         Specifies whether to compute the matrix V:
       = 'V': N columns of V are returned in the array V; Jacobi rotations
              are not explicitly accumulated.
       = 'J': N columns of V are returned in the array V, but they are
              computed as the product of Jacobi rotations. This option is
              allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
       = 'W': V may be used as workspace of length N*N. See the description
              of V.
       = 'N': V is not computed.

JOBR

          JOBR is CHARACTER*1
         Specifies the RANGE for the singular values. Issues the licence to
         set to zero small positive singular values if they are outside
         specified range. If A .NE. 0 is scaled so that the largest singular
         value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
         the licence to kill columns of A whose norm in c*A is less than
         SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
         where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
       = 'N': Do not kill small columns of c*A. This option assumes that
              BLAS and QR factorizations and triangular solvers are
              implemented to work in that range. If the condition of A
              is greater than BIG, use SGESVJ.
       = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
              (roughly, as described above). This option is recommended.
                                             ===========================
         For computing the singular values in the FULL range [SFMIN,BIG]
         use SGESVJ.

JOBT

          JOBT is CHARACTER*1
         If the matrix is square then the procedure may determine to use
         transposed A if A^t seems to be better with respect to convergence.
         If the matrix is not square, JOBT is ignored. This is subject to
         changes in the future.
         The decision is based on two values of entropy over the adjoint
         orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
       = 'T': transpose if entropy test indicates possibly faster
         convergence of Jacobi process if A^t is taken as input. If A is
         replaced with A^t, then the row pivoting is included automatically.
       = 'N': do not speculate.
         This option can be used to compute only the singular values, or the
         full SVD (U, SIGMA and V). For only one set of singular vectors
         (U or V), the caller should provide both U and V, as one of the
         matrices is used as workspace if the matrix A is transposed.
         The implementer can easily remove this constraint and make the
         code more complicated. See the descriptions of U and V.

JOBP

          JOBP is CHARACTER*1
         Issues the licence to introduce structured perturbations to drown
         denormalized numbers. This licence should be active if the
         denormals are poorly implemented, causing slow computation,
         especially in cases of fast convergence (!). For details see [1,2].
         For the sake of simplicity, this perturbations are included only
         when the full SVD or only the singular values are requested. The
         implementer/user can easily add the perturbation for the cases of
         computing one set of singular vectors.
       = 'P': introduce perturbation
       = 'N': do not perturb

M

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

N

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

A

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

LDA

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

SVA

          SVA is REAL array, dimension (N)
          On exit,
          - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
            computation SVA contains Euclidean column norms of the
            iterated matrices in the array A.
          - For WORK(1) .NE. WORK(2): The singular values of A are
            (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
            sigma_max(A) overflows or if small singular values have been
            saved from underflow by scaling the input matrix A.
          - If JOBR='R' then some of the singular values may be returned
            as exact zeros obtained by "set to zero" because they are
            below the numerical rank threshold or are denormalized numbers.

U

          U is REAL array, dimension ( LDU, N )
          If JOBU = 'U', then U contains on exit the M-by-N matrix of
                         the left singular vectors.
          If JOBU = 'F', then U contains on exit the M-by-M matrix of
                         the left singular vectors, including an ONB
                         of the orthogonal complement of the Range(A).
          If JOBU = 'W'  .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
                         then U is used as workspace if the procedure
                         replaces A with A^t. In that case, [V] is computed
                         in U as left singular vectors of A^t and then
                         copied back to the V array. This 'W' option is just
                         a reminder to the caller that in this case U is
                         reserved as workspace of length N*N.
          If JOBU = 'N'  U is not referenced, unless JOBT='T'.

LDU

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

V

          V is REAL array, dimension ( LDV, N )
          If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
                         the right singular vectors;
          If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
                         then V is used as workspace if the pprocedure
                         replaces A with A^t. In that case, [U] is computed
                         in V as right singular vectors of A^t and then
                         copied back to the U array. This 'W' option is just
                         a reminder to the caller that in this case V is
                         reserved as workspace of length N*N.
          If JOBV = 'N'  V is not referenced, unless JOBT='T'.

LDV

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

WORK

          WORK is REAL array, dimension (LWORK)
          On exit,
          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
                    that SCALE*SVA(1:N) are the computed singular values
                    of A. (See the description of SVA().)
          WORK(2) = See the description of WORK(1).
          WORK(3) = SCONDA is an estimate for the condition number of
                    column equilibrated A. (If JOBA .EQ. 'E' or 'G')
                    SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
                    It is computed using SPOCON. It holds
                    N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
                    where R is the triangular factor from the QRF of A.
                    However, if R is truncated and the numerical rank is
                    determined to be strictly smaller than N, SCONDA is
                    returned as -1, thus indicating that the smallest
                    singular values might be lost.
          If full SVD is needed, the following two condition numbers are
          useful for the analysis of the algorithm. They are provied for
          a developer/implementer who is familiar with the details of
          the method.
          WORK(4) = an estimate of the scaled condition number of the
                    triangular factor in the first QR factorization.
          WORK(5) = an estimate of the scaled condition number of the
                    triangular factor in the second QR factorization.
          The following two parameters are computed if JOBT .EQ. 'T'.
          They are provided for a developer/implementer who is familiar
          with the details of the method.
          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
                    of diag(A^t*A) / Trace(A^t*A) taken as point in the
                    probability simplex.
          WORK(7) = the entropy of A*A^t.

LWORK

          LWORK is INTEGER
          Length of WORK to confirm proper allocation of work space.
          LWORK depends on the job:
          If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
            -> .. no scaled condition estimate required (JOBE.EQ.'N'):
               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
               ->> For optimal performance (blocked code) the optimal value
               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
               block size for DGEQP3 and DGEQRF.
               In general, optimal LWORK is computed as
               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
            -> .. an estimate of the scaled condition number of A is
               required (JOBA='E', 'G'). In this case, LWORK is the maximum
               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
               ->> For optimal performance (blocked code) the optimal value
               is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
               In general, the optimal length LWORK is computed as
               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
                                                     N+N*N+LWORK(DPOCON),7).
          If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
            -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
               where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
               DORMLQ. In general, the optimal length LWORK is computed as
               LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
                       N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
          If SIGMA and the left singular vectors are needed
            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
            -> For optimal performance:
               if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
               if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
               where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
               In general, the optimal length LWORK is computed as
               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
                        2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
               Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or
               M*NB (for JOBU.EQ.'F').
          If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
            -> if JOBV.EQ.'V'
               the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
            -> if JOBV.EQ.'J' the minimal requirement is
               LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
            -> For optimal performance, LWORK should be additionally
               larger than N+M*NB, where NB is the optimal block size
               for DORMQR.

IWORK

          IWORK is INTEGER array, dimension (M+3*N).
          On exit,
          IWORK(1) = the numerical rank determined after the initial
                     QR factorization with pivoting. See the descriptions
                     of JOBA and JOBR.
          IWORK(2) = the number of the computed nonzero singular values
          IWORK(3) = if nonzero, a warning message:
                     If IWORK(3).EQ.1 then some of the column norms of A
                     were denormalized floats. The requested high accuracy
                     is not warranted by the data.

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Further Details:

  SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
  SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an
  additional row pivoting can be used as a preprocessor, which in some
  cases results in much higher accuracy. An example is matrix A with the
  structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
  diagonal matrices and C is well-conditioned matrix. In that case, complete
  pivoting in the first QR factorizations provides accuracy dependent on the
  condition number of C, and independent of D1, D2. Such higher accuracy is
  not completely understood theoretically, but it works well in practice.
  Further, if A can be written as A = B*D, with well-conditioned B and some
  diagonal D, then the high accuracy is guaranteed, both theoretically and
  in software, independent of D. For more details see [1], [2].
     The computational range for the singular values can be the full range
  ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
  & LAPACK routines called by SGEJSV are implemented to work in that range.
  If that is not the case, then the restriction for safe computation with
  the singular values in the range of normalized IEEE numbers is that the
  spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
  overflow. This code (SGEJSV) is best used in this restricted range,
  meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
  returned as zeros. See JOBR for details on this.
     Further, this implementation is somewhat slower than the one described
  in [1,2] due to replacement of some non-LAPACK components, and because
  the choice of some tuning parameters in the iterative part (SGESVJ) is
  left to the implementer on a particular machine.
     The rank revealing QR factorization (in this code: SGEQP3) should be
  implemented as in [3]. We have a new version of SGEQP3 under development
  that is more robust than the current one in LAPACK, with a cleaner cut in
  rank deficient cases. It will be available in the SIGMA library [4].
  If M is much larger than N, it is obvious that the initial QRF with
  column pivoting can be preprocessed by the QRF without pivoting. That
  well known trick is not used in SGEJSV because in some cases heavy row
  weighting can be treated with complete pivoting. The overhead in cases
  M much larger than N is then only due to pivoting, but the benefits in
  terms of accuracy have prevailed. The implementer/user can incorporate
  this extra QRF step easily. The implementer can also improve data movement
  (matrix transpose, matrix copy, matrix transposed copy) - this
  implementation of SGEJSV uses only the simplest, naive data movement.

Contributors:

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

References:

 [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
     LAPACK Working note 169.
 [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
     LAPACK Working note 170.
 [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
     factorization software - a case study.
     ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
     LAPACK Working note 176.
 [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
     QSVD, (H,K)-SVD computations.
     Department of Mathematics, University of Zagreb, 2008.

Bugs, examples and comments:

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

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

SGESDD

Purpose:

 SGESDD computes the singular value decomposition (SVD) of a real
 M-by-N matrix A, optionally computing the left and right singular
 vectors.  If singular vectors are desired, it uses a
 divide-and-conquer algorithm.
 The SVD is written
      A = U * SIGMA * transpose(V)
 where SIGMA is an M-by-N matrix which is zero except for its
 min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
 V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
 are the singular values of A; they are real and non-negative, and
 are returned in descending order.  The first min(m,n) columns of
 U and V are the left and right singular vectors of A.
 Note that the routine returns VT = V**T, not V.
 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

Parameters:

JOBZ

          JOBZ is CHARACTER*1
          Specifies options for computing all or part of the matrix U:
          = 'A':  all M columns of U and all N rows of V**T are
                  returned in the arrays U and VT;
          = 'S':  the first min(M,N) columns of U and the first
                  min(M,N) rows of V**T are returned in the arrays U
                  and VT;
          = 'O':  If M >= N, the first N columns of U are overwritten
                  on the array A and all rows of V**T are returned in
                  the array VT;
                  otherwise, all columns of U are returned in the
                  array U and the first M rows of V**T are overwritten
                  in the array A;
          = 'N':  no columns of U or rows of V**T are computed.

M

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

N

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

A

          A is REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBZ = 'O',  A is overwritten with the first N columns
                          of U (the left singular vectors, stored
                          columnwise) if M >= N;
                          A is overwritten with the first M rows
                          of V**T (the right singular vectors, stored
                          rowwise) otherwise.
          if JOBZ .ne. 'O', the contents of A are destroyed.

LDA

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

S

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

U

          U is REAL array, dimension (LDU,UCOL)
          UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
          UCOL = min(M,N) if JOBZ = 'S'.
          If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
          orthogonal matrix U;
          if JOBZ = 'S', U contains the first min(M,N) columns of U
          (the left singular vectors, stored columnwise);
          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

VT

          VT is REAL array, dimension (LDVT,N)
          If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
          N-by-N orthogonal matrix V**T;
          if JOBZ = 'S', VT contains the first min(M,N) rows of
          V**T (the right singular vectors, stored rowwise);
          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1;
          if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
          if JOBZ = 'S', LDVT >= min(M,N).

WORK

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

LWORK

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

IWORK

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

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  SBDSDC did not converge, updating process failed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

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

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

Purpose:

 SGESVD computes the singular value decomposition (SVD) of a real
 M-by-N matrix A, optionally computing the left and/or right singular
 vectors. The SVD is written
      A = U * SIGMA * transpose(V)
 where SIGMA is an M-by-N matrix which is zero except for its
 min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
 V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
 are the singular values of A; they are real and non-negative, and
 are returned in descending order.  The first min(m,n) columns of
 U and V are the left and right singular vectors of A.
 Note that the routine returns V**T, not V.

Parameters:

JOBU

          JOBU is CHARACTER*1
          Specifies options for computing all or part of the matrix U:
          = 'A':  all M columns of U are returned in array U:
          = 'S':  the first min(m,n) columns of U (the left singular
                  vectors) are returned in the array U;
          = 'O':  the first min(m,n) columns of U (the left singular
                  vectors) are overwritten on the array A;
          = 'N':  no columns of U (no left singular vectors) are
                  computed.

JOBVT

          JOBVT is CHARACTER*1
          Specifies options for computing all or part of the matrix
          V**T:
          = 'A':  all N rows of V**T are returned in the array VT;
          = 'S':  the first min(m,n) rows of V**T (the right singular
                  vectors) are returned in the array VT;
          = 'O':  the first min(m,n) rows of V**T (the right singular
                  vectors) are overwritten on the array A;
          = 'N':  no rows of V**T (no right singular vectors) are
                  computed.
          JOBVT and JOBU cannot both be 'O'.

M

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

N

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

A

          A is REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBU = 'O',  A is overwritten with the first min(m,n)
                          columns of U (the left singular vectors,
                          stored columnwise);
          if JOBVT = 'O', A is overwritten with the first min(m,n)
                          rows of V**T (the right singular vectors,
                          stored rowwise);
          if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
                          are destroyed.

LDA

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

S

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

U

          U is REAL array, dimension (LDU,UCOL)
          (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
          If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
          if JOBU = 'S', U contains the first min(m,n) columns of U
          (the left singular vectors, stored columnwise);
          if JOBU = 'N' or 'O', U is not referenced.

LDU

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

VT

          VT is REAL array, dimension (LDVT,N)
          If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
          V**T;
          if JOBVT = 'S', VT contains the first min(m,n) rows of
          V**T (the right singular vectors, stored rowwise);
          if JOBVT = 'N' or 'O', VT is not referenced.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
          if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
          superdiagonal elements of an upper bidiagonal matrix B
          whose diagonal is in S (not necessarily sorted). B
          satisfies A = U * B * VT, so it has the same singular values
          as A, and singular vectors related by U and VT.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
             - PATH 1  (M much larger than N, JOBU='N')
             - PATH 1t (N much larger than M, JOBVT='N')
          LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) for the other paths
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if SBDSQR did not converge, INFO specifies how many
                superdiagonals of an intermediate bidiagonal form B
                did not converge to zero. See the description of WORK
                above for details.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

subroutine sgesvdx (character JOBU, character JOBVT, character RANGE, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real VL, real VU, integer IL, integer IU, integer NS, real, dimension( * ) S, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldvt, * ) VT, integer LDVT, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

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

Purpose:

  SGESVDX computes the singular value decomposition (SVD) of a real
  M-by-N matrix A, optionally computing the left and/or right singular
  vectors. The SVD is written
      A = U * SIGMA * transpose(V)
  where SIGMA is an M-by-N matrix which is zero except for its
  min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
  V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA
  are the singular values of A; they are real and non-negative, and
  are returned in descending order.  The first min(m,n) columns of
  U and V are the left and right singular vectors of A.
  SGESVDX uses an eigenvalue problem for obtaining the SVD, which
  allows for the computation of a subset of singular values and
  vectors. See SBDSVDX for details.
  Note that the routine returns V**T, not V.

Parameters:

JOBU

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

JOBVT

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

RANGE

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

M

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

N

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

A

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

LDA

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

VL

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

VU

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

IL

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

IU

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

NS

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

S

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

U

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

LDU

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

VT

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

LDVT

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

WORK

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

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see
          comments inside the code):
             - PATH 1  (M much larger than N)
             - PATH 1t (N much larger than M)
          LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths.
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

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

INFO

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2016

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

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

Purpose:

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

Parameters:

JOBU

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

JOBV

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

JOBQ

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

M

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

N

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

P

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

K

          K is INTEGER

L

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

A

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

LDA

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

B

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

LDB

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

ALPHA

          ALPHA is REAL array, dimension (N)

BETA

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

U

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

LDU

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

V

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

LDV

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

Q

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

LDQ

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

WORK

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

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

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

INFO

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

Internal Parameters:

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

August 2015

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

SGGSVD3 replaces the deprecated subroutine SGGSVD.

Author

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