SRC/DEPRECATED/cggsvd.f(3)

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version 3.12.1 · 3 versions
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🌸 NAME 🌸

okay so this page is from SRC/DEPRECATED/cggsvd.f(3) in the Library Functions Manual, and the file it's serving is literally SRC/DEPRECATED/cggsvd.f. 🫶

✨ SYNOPSIS ✨

Functions/Subroutines

  • subroutine cggsvd — CGGSVD computes the singular value decomposition (SVD) for OTHER matrices
subroutine cggsvd (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
    alpha, beta, u, ldu, v, ldv, q, ldq, work, rwork, iwork, info)

🦋 Function/Subroutine Documentation 🦋

subroutine cggsvd — the full signature, no crumbs

subroutine cggsvd (character jobu, character jobv, character jobq, integer
    m, integer n, integer p, integer k, integer l, complex, dimension( lda,
    * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real,
    dimension( * ) alpha, real, dimension( * ) beta, complex, dimension(
    ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv,
    complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * )
    work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer
    info)

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

yes, the description really says “OTHER matrices,” and we are respecting the text. 💅

💖 Purpose 💖

First thing first: this routine is deprecated and has been replaced by routine CGGSVD3. That is not a drill, bestie. The deprecated label is right there.

CGGSVD 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) 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(3*N,M,P)+N)

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 is REAL
  • TOLB is 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

Definition at line 335 of file cggsvd.f.

📜 Author 📜

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