
NAMEzggsvd.f SYNOPSISFunctions/Subroutinessubroutine zggsvd (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK, IWORK, INFO) Function/Subroutine Documentationsubroutine zggsvd (characterJOBU, characterJOBV, characterJOBQ, integerM, integerN, integerP, integerK, integerL, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldb, * )B, integerLDB, double precision, dimension( * )ALPHA, double precision, dimension( * )BETA, complex*16, dimension( ldu, * )U, integerLDU, complex*16, dimension( ldv, * )V, integerLDV, complex*16, dimension( ldq, * )Q, integerLDQ, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integer, dimension( * )IWORK, integerINFO)ZGGSVD computes the singular value decomposition (SVD) for OTHER matrices Purpose:ZGGSVD computes the generalized singular value decomposition (GSVD) of an MbyN complex matrix A and PbyN complex matrix B: 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 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 unitary 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**H. If ( A**H,B**H)**H has orthnormal 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) ) JOBU
Internal Parameters:
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 INTEGERL 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*16 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.LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).B B is COMPLEX*16 array, dimension (LDB,N) On entry, the PbyN matrix B. On exit, B contains part of the triangular matrix R if MKL < 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 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) = 0U U is COMPLEX*16 array, dimension (LDU,M) If JOBU = 'U', U contains the MbyM 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*16 array, dimension (LDV,P) If JOBV = 'V', V contains the PbyP 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*16 array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the NbyN 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*16 array, dimension (max(3*N,M,P)+N)RWORK RWORK is DOUBLE PRECISION 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 ith argument had an illegal value. > 0: if INFO = 1, the Jacobitype procedure failed to converge. For further details, see subroutine ZTGSJA. TOLA DOUBLE PRECISION TOLB DOUBLE PRECISION 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)*MAZHEPS, TOLB = MAX(P,N)*norm(B)*MAZHEPS. The size of TOLA and TOLB may affect the size of backward errors of the decomposition. 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 334 of file zggsvd.f.
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