SRC/dggsvd3.f(3)

💖 NAME

SRC/dggsvd3.f — yes bestie, that's the name section, and we're in SRC/dggsvd3.f(3) from the Library Functions Manual. Buckle up, this one is a whole main-character LAPACK routine. ✨

✨ SYNOPSIS

🧁 Functions/Subroutines

subroutine dggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
    alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)
     DGGSVD3 computes the singular value decomposition (SVD) for OTHER
    matrices

🫶 Function/Subroutine Documentation

subroutine dggsvd3 (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 lwork, integer, dimension( * ) iwork,
    integer info)
 DGGSVD3 computes the singular value decomposition (SVD) for OTHER
matrices

🎀 Purpose

Okay so DGGSVD3 computes the generalized singular value decomposition, aka 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 literally these:

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) ).

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💅 Parameters

• JOBU — is CHARACTER*1. = 'U': orthogonal matrix U is computed; = 'N': U is not computed.
• JOBV — is CHARACTER*1. = 'V': orthogonal matrix V is computed; = 'N': V is not computed.
• JOBQ — is CHARACTER*1. = 'Q': orthogonal 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**T,B**T)**T.
• A — is 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 — is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• B — is 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 — 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 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 — is INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
• V — is 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 — is INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
• Q — is 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 — is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
• WORK — is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — is INTEGER. The dimension of the array WORK. LWORK >= 1.

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.

• 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 DTGSJA.

This is the part where you double-check your dimensions before yelling at your code, bestie. The routine is iconic, but it will absolutely notice an illegal argument. 💀

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🧚 Internal Parameters

• TOLA — DOUBLE PRECISION
• TOLB — DOUBLE PRECISION

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

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👑 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

DGGSVD3 replaces the deprecated subroutine DGGSVD.

Heads up, bestie: deprecated means don't start new stuff with the old one. This one is the replacement. Periodt. 💅

Definition at line 346 of file dggsvd3.f.

⭐ Author

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole page, bestie 💖 — this is LAPACK Version 3.12.0, page SRC/dggsvd3.f(3), and yes, the docs really did generate themselves via Doxygen. slay. ✨🎀🦋