SRC/cggsvd3.f(3)

📦

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🌸 NAME

okay bestie, this page is SRC/cggsvd3.f(3), and the manual name is Library Functions Manual. Header tea folded in, no cap. ✨

✨ SYNOPSIS

🎀 Functions/Subroutines

subroutine cggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
    alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork,
    info)

cggsvd3 — CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

💖 Function/Subroutine Documentation

🌟 subroutine cggsvd3

subroutine cggsvd3 (character jobu, character jobv, character jobq, integer
    m, integer n, integer p, integer k, integer l, complex, dimension( lda,
    * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real,
    dimension( * ) alpha, real, dimension( * ) beta, complex, dimension(
    ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv,
    complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * )
    work, integer lwork, real, dimension( * ) rwork, integer, dimension( * )
    iwork, integer info)

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

🎀 Purpose

okay so CGGSVD3 computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B. The equations are literally:

      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 have 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. (That's the whole note in the original, bestie.)
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 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 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 REAL array, dimension (N)
BETA — BETA is REAL 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 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 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 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 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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.

RWORK — RWORK is REAL 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 CTGSJA.

🍓 Internal Parameters

TOLA — REAL
TOLB — REAL

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

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

⭐ Further Details

deprecated alert, bestie: CGGSVD3 replaces the deprecated subroutine CGGSVD.

🦋 Definition at line 351 of file cggsvd3.f

Definition at line 351 of file cggsvd3.f.

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📦 where this page comes from

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