DGGSVD (3)  Linux Manuals
NAME
dggsvd.f 
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)
DGGSVD computes the singular value decomposition (SVD) for OTHER matrices
Function/Subroutine Documentation
subroutine dggsvd (characterJOBU, characterJOBV, characterJOBQ, integerM, integerN, integerP, integerK, integerL, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( * )ALPHA, double precision, dimension( * )BETA, double precision, dimension( ldu, * )U, integerLDU, double precision, dimension( ldv, * )V, integerLDV, double precision, dimension( ldq, * )Q, integerLDQ, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
DGGSVD computes the singular value decomposition (SVD) for OTHER matrices
Purpose:

DGGSVD computes the generalized singular value decomposition (GSVD) of an MbyN real matrix A and PbyN 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+LbyK+L nonsingular upper triangular matrix, D1 and D2 are Mby(K+L) and Pby(K+L) "diagonal" matrices and of the following structures, respectively: If MKL >= 0, K L D1 = K ( I 0 ) L ( 0 C ) MKL ( 0 0 ) K L D2 = L ( 0 S ) PL ( 0 0 ) NKL 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,NKL+1:N) on exit. If MKL < 0, K MK K+LM D1 = K ( I 0 0 ) MK ( 0 C 0 ) K MK K+LM D2 = MK ( 0 S 0 ) K+LM ( 0 0 I ) PL ( 0 0 0 ) NKL K MK K+LM ( 0 R ) = K ( 0 R11 R12 R13 ) MK ( 0 0 R22 R23 ) K+LM ( 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, NKL+1:N), and R33 is stored ( 0 R22 R23 ) in B(MK+1:L,N+MKL+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 NbyN 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.
JOBVJOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed.
JOBQJOBQ is CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed.
MM is INTEGER The number of rows of the matrix A. M >= 0.
NN is INTEGER The number of columns of the matrices A and B. N >= 0.
PP is INTEGER The number of rows of the matrix B. P >= 0.
KK is INTEGER
LL 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.
AA is DOUBLE PRECISION array, dimension (LDA,N) On entry, the MbyN matrix A. On exit, A contains the triangular matrix R, or part of R. See Purpose for details.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
BB is DOUBLE PRECISION array, dimension (LDB,N) On entry, the PbyN matrix B. On exit, B contains the triangular matrix R if MKL < 0. See Purpose for details.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).
ALPHAALPHA is DOUBLE PRECISION array, dimension (N)
BETABETA 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 MKL >= 0, ALPHA(K+1:K+L) = C, BETA(K+1:K+L) = S, or if MKL < 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
UU is DOUBLE PRECISION array, dimension (LDU,M) If JOBU = 'U', U contains the MbyM orthogonal matrix U. If JOBU = 'N', U is not referenced.
LDULDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
VV is DOUBLE PRECISION array, dimension (LDV,P) If JOBV = 'V', V contains the PbyP orthogonal matrix V. If JOBV = 'N', V is not referenced.
LDVLDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
QQ is DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the NbyN orthogonal matrix Q. If JOBQ = 'N', Q is not referenced.
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORKWORK is DOUBLE PRECISION array, dimension (max(3*N,M,P)+N)
IWORKIWORK 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).
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, the Jacobitype 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.
Date:
 November 2011
Contributors:
 Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Definition at line 331 of file dggsvd.f.
Author
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