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 computes the singular value decomposition (SVD) for OTHER matrices
CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
Purpose:
CGGSVD3 computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N 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 M-by-(K+L) and P-by-(K+L) "diagonal" 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. (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 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
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. 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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
Further Details:
Definition at line 356 of file cggsvd3.f.
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