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

okay so welcome to /home/abui...D/dggsvd.f(3), bestie — this is the Library Functions Manual page for dggsvd.f, and it is deprecated but iconic. 💅

💅 NAME 💅

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/DEPRECATED/dggsvd.f

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🦋 SYNOPSIS 🦋

🧚 Functions/Subroutines 🧚

okay so the function list has one main character, and it is literally:

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

and the one-line summary is literally: DGGSVD computes the singular value decomposition (SVD) for OTHER matrices — yes, all-caps OTHER, we do not question the source, iykyk. 😭

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🌈 Function/Subroutine Documentation 🌈

here is the full signature, no cap:

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)

and the summary comes back like: DGGSVD computes the singular value decomposition (SVD) for OTHER matrices. slay. 💅

🎯 Purpose 🎯

🚨 Deprecated alert, bestie: this routine is deprecated and has been replaced by routine DGGSVD3. keep that in mind, fr fr.

DGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and 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 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.

and

(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 — JOBU is CHARACTER*1; = 'U': orthogonal matrix U is computed; = 'N': U is not computed. 💖
• JOBV — JOBV is CHARACTER*1; = 'V': orthogonal matrix V is computed; = 'N': V is not computed.
• JOBQ — JOBQ is CHARACTER*1; = 'Q': orthogonal 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. that's the whole line, bestie, no extra description in the source. 🫶
• 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**T,B**T)**T.
• A — 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 — LDA is INTEGER; the leading dimension of the array A. LDA >= max(1,M).
• B — 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 — 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 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 — LDU is INTEGER; the leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
• V — 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 — LDV is INTEGER; the leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
• Q — 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 — LDQ is INTEGER; the leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
• WORK — WORK is DOUBLE PRECISION array, dimension (max(3*N,M,P)+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 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.

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🌟 Author 🌟

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