SRC/DEPRECATED/dggsvd.f(3)

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💅 NAME 💅

okay bestie, welcome to SRC/DEPRECATED/dggsvd.f(3) from the Library Functions Manual — this is the deprecated DGGSVD page, and it's literally here to serve you the generalized SVD tea. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine dggsvd (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
    alpha, beta, u, ldu, v, ldv, q, ldq, work, iwork, info)

The one-liner is literally: DGGSVD computes the singular value decomposition (SVD) for OTHER matrices — yes, "OTHER" is in all caps, and no, we are not asking questions. 🫶

🌸 Function/Subroutine Documentation 🌸

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

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

Purpose

⚠️ deprecated alert, bestie: This routine is deprecated and has been replaced by routine DGGSVD3.

DGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and a 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) matrices, and their structures are 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 — CHARACTER*1. = 'U': orthogonal matrix U is computed; = 'N': U is not computed.
JOBV — CHARACTER*1. = 'V': orthogonal matrix V is computed; = 'N': V is not computed.
JOBQ — CHARACTER*1. = 'Q': orthogonal matrix Q is computed; = 'N': Q is not computed.
M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrices A and B. N >= 0.
P — INTEGER. The number of rows of the matrix B. P >= 0.
K — INTEGER. (K's whole exit-era meaning is right below with L, bestie.)
L — 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 — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
B — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,P).
ALPHA — DOUBLE PRECISION array, dimension (N).
BETA — 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 — DOUBLE PRECISION array, dimension (LDU,M). If JOBU = 'U', U contains the M-by-M orthogonal matrix U. If JOBU = 'N', U is not referenced.
LDU — INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V — DOUBLE PRECISION array, dimension (LDV,P). If JOBV = 'V', V contains the P-by-P orthogonal matrix V. If JOBV = 'N', V is not referenced.
LDV — INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q — DOUBLE PRECISION array, dimension (LDQ,N). If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. If JOBQ = 'N', Q is not referenced.
LDQ — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK — DOUBLE PRECISION array, dimension (max(3*N,M,P)+N).
IWORK — 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 — 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 DTGSJA.

Internal Parameters

TOLA — DOUBLE PRECISION.
TOLB — DOUBLE PRECISION.

TOLA and TOLB are the thresholds to determine the effective rank of (A',B')**T. 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 331 of file dggsvd.f.

👑 Author (generated)

Generated automatically by Doxygen for LAPACK from the source code.

okay bestie, that's the whole page 💖✨ — this cute little sign-off is for LAPACK Version 3.12.0 and the file SRC/DEPRECATED/dggsvd.f(3), no cap. 🎀🦋

📦 where this page comes from

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