complexGEsing(3) LAPACK complexGEsing(3)

complexGEsing - complex


subroutine cgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, CWORK, LWORK, RWORK, LRWORK, IWORK, INFO)
CGEJSV subroutine cgesdd (JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, RWORK, IWORK, INFO)
CGESDD subroutine cgesvd (JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, RWORK, INFO)
CGESVD computes the singular value decomposition (SVD) for GE matrices subroutine cgesvdq (JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA, S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK, CWORK, LCWORK, RWORK, LRWORK, INFO)
CGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices subroutine cgesvdx (JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, IL, IU, NS, S, U, LDU, VT, LDVT, WORK, LWORK, RWORK, IWORK, INFO)
CGESVDX computes the singular value decomposition (SVD) for GE matrices subroutine cggsvd3 (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, RWORK, IWORK, INFO)
CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

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

CGEJSV

Purpose:

CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
matrix [A], where M >= N. The SVD of [A] is written as
             [A] = [U] * [SIGMA] * [V]^*,
where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
diagonal elements, [U] is an M-by-N (or M-by-M) unitary 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. The matrices [U] and [V]
are computed and stored in the arrays U and V, respectively. The diagonal
of [SIGMA] is computed and stored in the array SVA.

Parameters
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=B*D. If A has heavily weighted
       rows, then using this condition number gives too pessimistic
       error bound.
= 'A': Small singular values are not well determined by the data 
       and are considered as noisy; 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^* 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, if JOBT = 'N'.
= 'W': V may be used as workspace of length N*N. See the description
       of V.
= 'N': V is not computed.

JOBR

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

JOBT

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

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 COMPLEX 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 COMPLEX array, dimension ( LDU, N ) or ( LDU, M )
If JOBU = 'U', then U contains on exit the M-by-N matrix of
               the left singular vectors.
If JOBU = 'F', then U contains on exit the M-by-M matrix of
               the left singular vectors, including an ONB
               of the orthogonal complement of the Range(A).
If JOBU = 'W'  .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N),
               then U is used as workspace if the procedure
               replaces A with A^*. In that case, [V] is computed
               in U as left singular vectors of A^* 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 COMPLEX array, dimension ( LDV, N )
If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
               the right singular vectors;
If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N),
               then V is used as workspace if the pprocedure
               replaces A with A^*. In that case, [U] is computed
               in V as right singular vectors of A^* 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.

CWORK

CWORK is COMPLEX array, dimension (MAX(2,LWORK))
If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
LRWORK=-1), then on exit CWORK(1) contains the required length of 
CWORK for the job parameters used in the call.

LWORK

LWORK is INTEGER
Length of CWORK to confirm proper allocation of workspace.
LWORK depends on the job:
1. If only SIGMA is needed ( JOBU = 'N', JOBV = 'N' ) and
  1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
     LWORK >= 2*N+1. This is the minimal requirement.
     ->> For optimal performance (blocked code) the optimal value
     is LWORK >= N + (N+1)*NB. Here NB is the optimal
     block size for CGEQP3 and CGEQRF.
     In general, optimal LWORK is computed as
     LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ)).        
  1.2. .. an estimate of the scaled condition number of A is
     required (JOBA='E', or 'G'). In this case, LWORK the minimal
     requirement is LWORK >= N*N + 2*N.
     ->> For optimal performance (blocked code) the optimal value
     is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
     In general, the optimal length LWORK is computed as
     LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ),
                  N*N+LWORK(CPOCON)).
2. If SIGMA and the right singular vectors are needed (JOBV = 'V'),
   (JOBU = 'N')
  2.1   .. no scaled condition estimate requested (JOBE = 'N'):    
  -> the minimal requirement is LWORK >= 3*N.
  -> For optimal performance, 
     LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
     where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
     CUNMLQ. In general, the optimal length LWORK is computed as
     LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ),
             N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
  2.2 .. an estimate of the scaled condition number of A is
     required (JOBA='E', or 'G').
  -> the minimal requirement is LWORK >= 3*N.      
  -> For optimal performance, 
     LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
     where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
     CUNMLQ. In general, the optimal length LWORK is computed as
     LWORK >= max(N+LWORK(CGEQP3), LWORK(CPOCON), N+LWORK(CGESVJ),
             N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).   
3. If SIGMA and the left singular vectors are needed
  3.1  .. no scaled condition estimate requested (JOBE = 'N'):
  -> the minimal requirement is LWORK >= 3*N.
  -> For optimal performance:
     if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
     where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
     In general, the optimal length LWORK is computed as
     LWORK >= max(N+LWORK(CGEQP3), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)). 
  3.2  .. an estimate of the scaled condition number of A is
     required (JOBA='E', or 'G').
  -> the minimal requirement is LWORK >= 3*N.
  -> For optimal performance:
     if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
     where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
     In general, the optimal length LWORK is computed as
     LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON),
              2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
4. If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
  4.1. if JOBV = 'V'
     the minimal requirement is LWORK >= 5*N+2*N*N.
  4.2. if JOBV = 'J' the minimal requirement is
     LWORK >= 4*N+N*N.
  In both cases, the allocated CWORK can accommodate blocked runs
  of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ.

If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
minimal length of CWORK for the job parameters used in the call.        

RWORK

RWORK is REAL array, dimension (MAX(7,LWORK))
On exit,
RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
          such that SCALE*SVA(1:N) are the computed singular values
          of A. (See the description of SVA().)
RWORK(2) = See the description of RWORK(1).
RWORK(3) = SCONDA is an estimate for the condition number of
          column equilibrated A. (If JOBA = 'E' or 'G')
          SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
          It is computed using CPOCON. 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 provided for
a developer/implementer who is familiar with the details of
the method.
RWORK(4) = an estimate of the scaled condition number of the
          triangular factor in the first QR factorization.
RWORK(5) = an estimate of the scaled condition number of the
          triangular factor in the second QR factorization.
The following two parameters are computed if JOBT = 'T'.
They are provided for a developer/implementer who is familiar
with the details of the method.
RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
          of diag(A^* * A) / Trace(A^* * A) taken as point in the
          probability simplex.
RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
LRWORK=-1), then on exit RWORK(1) contains the required length of
RWORK for the job parameters used in the call.

LRWORK

   LRWORK is INTEGER
   Length of RWORK to confirm proper allocation of workspace.
   LRWORK depends on the job:
1. If only the singular values are requested i.e. if
   LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
   then:
   1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
        then: LRWORK = max( 7, 2 * M ).
   1.2. Otherwise, LRWORK  = max( 7,  N ).
2. If singular values with the right singular vectors are requested
   i.e. if
   (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
   .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
   then:
   2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
   then LRWORK = max( 7, 2 * M ).
   2.2. Otherwise, LRWORK  = max( 7,  N ).
3. If singular values with the left singular vectors are requested, i.e. if
   (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
   .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
   then:
   3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
   then LRWORK = max( 7, 2 * M ).
   3.2. Otherwise, LRWORK  = max( 7,  N ).
4. If singular values with both the left and the right singular vectors
   are requested, i.e. if
   (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
   (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
   then:
   4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
   then LRWORK = max( 7, 2 * M ).
   4.2. Otherwise, LRWORK  = max( 7, N ).

   If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and 
   the length of RWORK is returned in RWORK(1). 

IWORK

IWORK is INTEGER array, of dimension at least 4, that further depends
on the job:

1. If only the singular values are requested then:
   If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 
   then the length of IWORK is N+M; otherwise the length of IWORK is N.
2. If the singular values and the right singular vectors are requested then:
   If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 
   then the length of IWORK is N+M; otherwise the length of IWORK is N. 
3. If the singular values and the left singular vectors are requested then:
   If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 
   then the length of IWORK is N+M; otherwise the length of IWORK is N. 
4. If the singular values with both the left and the right singular vectors
   are requested, then:      
   4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
        If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 
        then the length of IWORK is N+M; otherwise the length of IWORK is N. 
   4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
        If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 
        then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.

On exit,
IWORK(1) = the numerical rank determined after the initial
           QR factorization with pivoting. See the descriptions
           of JOBA and JOBR.
IWORK(2) = the number of the computed nonzero singular values
IWORK(3) = if nonzero, a warning message:
           If IWORK(3) = 1 then some of the column norms of A
           were denormalized floats. The requested high accuracy
           is not warranted by the data.
IWORK(4) = 1 or -1. If IWORK(4) = 1, then the procedure used A^* to
           do the job as specified by the JOB parameters.
If the call to CGEJSV is a workspace query (indicated by LWORK = -1 and 
LRWORK = -1), then on exit IWORK(1) contains the required length of 
IWORK for the job parameters used in the call.

INFO

INFO is INTEGER
 < 0:  if INFO = -i, then the i-th argument had an illegal value.
 = 0:  successful exit;
 > 0:  CGEJSV  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.

Further Details:

CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3,
CGEQRF, and CGELQF 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 CGEJSV 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 (CGEJSV) 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 (CGESVJ) is
left to the implementer on a particular machine.
   The rank revealing QR factorization (in this code: CGEQP3) should be
implemented as in [3]. We have a new version of CGEQP3 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 CGEJSV 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 CGEJSV uses only the simplest, naive data movement.

Contributor:

Zlatko Drmac (Zagreb, Croatia)

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, 2016.

Bugs, examples and comments:

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

Definition at line 565 of file cgejsv.f.

CGESDD

Purpose:

CGESDD computes the singular value decomposition (SVD) of a complex
M-by-N matrix A, optionally computing the left and/or right singular
vectors, by using divide-and-conquer method. The SVD is written
     A = U * SIGMA * conjugate-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 unitary matrix, and
V is an N-by-N unitary 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**H, 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**H 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**H are returned in the arrays U
        and VT;
= 'O':  If M >= N, the first N columns of U are overwritten
        in the array A and all rows of V**H are returned in
        the array VT;
        otherwise, all columns of U are returned in the
        array U and the first M rows of V**H are overwritten
        in the array A;
= 'N':  no columns of U or rows of V**H 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 COMPLEX 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**H (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 COMPLEX 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
unitary 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 COMPLEX array, dimension (LDVT,N)
If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
N-by-N unitary matrix V**H;
if JOBZ = 'S', VT contains the first min(M,N) rows of
V**H (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 COMPLEX 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 >= 2*mn + mx.
If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
If JOBZ = 'S', LWORK >=   mn*mn + 3*mn.
If JOBZ = 'A', LWORK >=   mn*mn + 2*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.

RWORK

RWORK is REAL array, dimension (MAX(1,LRWORK))
Let mx = max(M,N) and mn = min(M,N).
If JOBZ = 'N',    LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
else              LRWORK >= max( 5*mn*mn + 5*mn,
                                 2*mx*mn + 2*mn*mn + mn ).

IWORK

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

INFO

INFO is INTEGER
<  0:  if INFO = -i, the i-th argument had an illegal value.
= -4:  if A had a NAN entry.
>  0:  The updating process of SBDSDC did not converge.
=  0:  successful exit.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Definition at line 225 of file cgesdd.f.

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

Purpose:

CGESVD computes the singular value decomposition (SVD) of a complex
M-by-N matrix A, optionally computing the left and/or right singular
vectors. The SVD is written
     A = U * SIGMA * conjugate-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 unitary matrix, and
V is an N-by-N unitary 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**H, 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**H:
= 'A':  all N rows of V**H are returned in the array VT;
= 'S':  the first min(m,n) rows of V**H (the right singular
        vectors) are returned in the array VT;
= 'O':  the first min(m,n) rows of V**H (the right singular
        vectors) are overwritten on the array A;
= 'N':  no rows of V**H (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 COMPLEX 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**H (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 COMPLEX 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 unitary 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 COMPLEX array, dimension (LDVT,N)
If JOBVT = 'A', VT contains the N-by-N unitary matrix
V**H;
if JOBVT = 'S', VT contains the first min(m,n) rows of
V**H (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 COMPLEX 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(1,2*MIN(M,N)+MAX(M,N)).
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.

RWORK

RWORK is REAL array, dimension (5*min(M,N))
On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) 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.

INFO

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 212 of file cgesvd.f.

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

Purpose:

CGESVDQ computes the singular value decomposition (SVD) of a complex
M-by-N matrix A, where M >= N. The SVD of A is written as
                                   [++]   [xx]   [x0]   [xx]
             A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
                                   [++]   [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N unitary matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.

Parameters

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

JOBP

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

JOBR

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

JOBU

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

JOBV

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

M

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

N

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

A

A is COMPLEX array of dimensions LDA x N
On entry, the input matrix A.
On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
the Householder vectors as stored by CGEQP3. If JOBU = 'F', these Householder
vectors together with CWORK(1:N) can be used to restore the Q factors from
the initial pivoted QR factorization of A. See the description of U.

LDA

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

S

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

U

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

LDU

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

V

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

LDV

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

NUMRANK

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

IWORK

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

LIWORK

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

CWORK

CWORK is COMPLEX array, dimension (max(2, LCWORK)), used as a workspace.
On exit, if, on entry, LCWORK.NE.-1, CWORK(1:N) contains parameters
needed to recover the Q factor from the QR factorization computed by
CGEQP3.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
CWORK(1) returns the optimal LCWORK, and
CWORK(2) returns the minimal LCWORK.

LCWORK

 LCWORK is INTEGER
 The dimension of the array CWORK. It is determined as follows:
 Let  LWQP3 = N+1,  LWCON = 2*N, and let
 LWUNQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
         { MAX( M, 1 ),  if JOBU = 'A'
 LWSVD = MAX( 3*N, 1 )
 LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
 LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
 Then the minimal value of LCWORK is:
 = MAX( N + LWQP3, LWSVD )        if only the singular values are needed;
 = MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
                          and a scaled condition estimate requested;
 = N + MAX( LWQP3, LWSVD, LWUNQ ) if the singular values and the left
                          singular vectors are requested;
 = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the singular values and the left
                          singular vectors are requested, and also
                          a scaled condition estimate requested;
 = N + MAX( LWQP3, LWSVD )        if the singular values and the right
                          singular vectors are requested;
 = N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
                          singular vectors are requested, and also
                          a scaled condition etimate requested;
 = N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
                          independent of JOBR;
 = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
                          JOBV = 'R' and, also a scaled condition
                          estimate requested; independent of JOBR;
 = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
                full SVD is requested with JOBV = 'A' or 'V', and
                JOBR ='N'
 = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
                if the full SVD is requested with JOBV = 'A' or 'V', and
                JOBR ='N', and also a scaled condition number estimate
                requested.
 = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
                full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
 = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
                if the full SVD is requested with JOBV = 'A', 'V' and
                JOBR ='T', and also a scaled condition number estimate
                requested.
 Finally, LCWORK must be at least two: LCWORK = MAX( 2, LCWORK ).
 If LCWORK = -1, then a workspace query is assumed; the routine
 only calculates and returns the optimal and minimal sizes
 for the CWORK, IWORK, and RWORK arrays, and no error
 message related to LCWORK is issued by XERBLA.

RWORK

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

LRWORK

LRWORK is INTEGER.
The dimension of the array RWORK.
If JOBP ='P', then LRWORK >= MAX(2, M, 5*N);
Otherwise, LRWORK >= MAX(2, 5*N).
If LRWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.

INFO

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

Further Details:

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

Bugs, examples and comments

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

References

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

Contributors:

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 410 of file cgesvdq.f.

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

Purpose:

CGESVDX computes the singular value decomposition (SVD) of a complex
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 unitary matrix, and
V is an N-by-N unitary 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.
CGESVDX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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.

RWORK

RWORK is REAL array, dimension (MAX(1,LRWORK))
LRWORK >= MIN(M,N)*(MIN(M,N)*2+15*MIN(M,N)).

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.

Definition at line 267 of file cgesvdx.f.

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

CGGSVD3 computes the generalized singular value decomposition (GSVD)
of an M-by-N complex matrix A and P-by-N complex matrix B:
      U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )
where U, V and Q are unitary matrices.
Let K+L = the effective numerical rank of the
matrix (A**H,B**H)**H, 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 unitary
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**H.
If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x.
In some literature, the GSVD of A and B is presented in the form
                 U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and 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':  Unitary matrix U is computed;
= 'N':  U is not computed.

JOBV

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

JOBQ

JOBQ is CHARACTER*1
= 'Q':  Unitary 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**H,B**H)**H.

A

A is COMPLEX 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 COMPLEX array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of 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 COMPLEX array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary 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 COMPLEX array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P unitary 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 COMPLEX array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N unitary 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 COMPLEX 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.

RWORK

RWORK is REAL array, dimension (2*N)

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 CTGSJA.

Internal Parameters:

TOLA    REAL
TOLB    REAL
        TOLA and TOLB are the thresholds to determine the effective
        rank of (A**H,B**H)**H. 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.

Contributors:

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

Further Details:

CGGSVD3 replaces the deprecated subroutine CGGSVD.

Definition at line 351 of file cggsvd3.f.

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