/home/abui...D/zggsvd.f(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 ZGGSVD — Library Functions Manual ✨

okay so this is the manual page for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/DEPRECATED/zggsvd.f — and yes bestie, DEPRECATED is literally right there in the path, we'll get to that 🫶 — living in the Library Functions Manual, no big deal.

🌸 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 (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🚨 bestie, listen: this routine is deprecated and has been replaced by routine ZGGSVD3. like, it still works, but it's giving "old icon still slaying at the reunion" — use the new one for new code, mwah 💅

Okay so 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 — is CHARACTER*1
    • = 'U': Unitary matrix U is computed;
    • = 'N': U is not computed.
  • JOBV — is CHARACTER*1
    • = 'V': Unitary matrix V is computed;
    • = 'N': V is not computed.
  • JOBQ — is CHARACTER*1
    • = 'Q': Unitary matrix Q is computed;
    • = 'N': Q is not computed.
  • M — is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — is INTEGER. The number of columns of the matrices A and B. N >= 0.
  • P — is INTEGER. The number of rows of the matrix B. P >= 0.
  • K — is INTEGER 🫶
  • 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 — 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 — is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • 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 — is INTEGER. The leading dimension of the array B. LDB >= max(1,P).
  • ALPHA — is DOUBLE PRECISION array, dimension (N)
  • 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 — 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 — is INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
  • 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 — is INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
  • 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 — is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
  • WORK — is COMPLEX*16 array, dimension (max(3*N,M,P)+N) ✨
  • RWORK — is DOUBLE PRECISION array, dimension (2*N)
  • 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 — 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 is DOUBLE PRECISION
  • TOLB is 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. 💀 (no cap, that's the actual warning)

👑 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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🎀 aaaaand that's the whole page bestie! LAPACK, Version 3.12.0, from /home/abui...D/zggsvd.f(3) — go forth and decompose those generalized singular values, you absolute icon 💅🔥
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