SRC/DEPRECATED/zggsvd.f(3)

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🌸 NAME 🌸

SRC/DEPRECATED/zggsvd.f — bestie, this is the page for that path in the Library Functions Manual. The page title is SRC/DEPRECATED/zggsvd.f(3), and it's the LAPACK zggsvd routine page — yes, the deprecated one, so keep that in mind, iykyk 💅.

✨ SYNOPSIS ✨

🎀 Functions/Subroutines

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

🫶 Function/Subroutine Documentation 🫶

💖 subroutine zggsvd 💖

subroutine zggsvd (character jobu, character jobv, character jobq, integer
    m, integer n, integer p, integer k, integer l, complex*16, dimension(
    lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb,
    double precision, dimension( * ) alpha, double precision, dimension( * )
    beta, complex*16, dimension( ldu, * ) u, integer ldu, complex*16,
    dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q,
    integer ldq, complex*16, dimension( * ) work, double precision,
    dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZGGSVD computes the singular value decomposition (SVD) for OTHER matrices.

🌈 Purpose 🌈

⚠️ Bestie, this routine is deprecated and has been replaced by routine ZGGSVD3. Like, it still does the thing, but the source is officially telling you to move on 💀.

ZGGSVD 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*16 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*16 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 DOUBLE PRECISION array, dimension (N).
BETA — BETA is DOUBLE PRECISION 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*16 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*16 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*16 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*16 array, dimension (max(3*N,M,P)+N).
RWORK — RWORK is DOUBLE PRECISION 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 ZTGSJA.

🧚 Internal Parameters 🧚

TOLA — DOUBLE PRECISION
TOLB — DOUBLE PRECISION

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)*MAZHEPS,
                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.

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 334 of file zggsvd.f. ⭐

💖 Author 💖

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Okay that's the whole page, bestie — LAPACK Version 3.12.0, source SRC/DEPRECATED/zggsvd.f(3). There's no separate date printed on this page, so I'm not inventing one; consider this your cute sign-off 🫶💖✨

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