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DGGSVDRun Method

Purpose ======= DGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the effective numerical rank of the matrix (A',B')', 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 .GE. 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 .LT. 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 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'. If ( A',B')' 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'*A x = lambda* B'*B x. In some literature, the GSVD of A and B is presented in the form U'*A*X = ( 0 D1 ), V'*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) ).

Namespace: DotNumerics.LinearAlgebra.CSLapack
Assembly: DWSIM.MathOps.DotNumerics (in DWSIM.MathOps.DotNumerics.dll) Version: 1.0.0.0 (1.0.0.0)
Syntax
Copy
public void Run(
	string JOBU,
	string JOBV,
	string JOBQ,
	int M,
	int N,
	int P,
	ref int K,
	ref int L,
	ref double[] A,
	int offset_a,
	int LDA,
	ref double[] B,
	int offset_b,
	int LDB,
	ref double[] ALPHA,
	int offset_alpha,
	ref double[] BETA,
	int offset_beta,
	ref double[] U,
	int offset_u,
	int LDU,
	ref double[] V,
	int offset_v,
	int LDV,
	ref double[] Q,
	int offset_q,
	int LDQ,
	ref double[] WORK,
	int offset_work,
	ref int[] IWORK,
	int offset_iwork,
	ref int INFO
)
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Parameters

JOBU  String
(input) CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed.
JOBV  String
(input) CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed.
JOBQ  String
(input) CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed.
M  Int32
(input) INTEGER The number of rows of the matrix A. M .GE. 0.
N  Int32
(input) INTEGER The number of columns of the matrices A and B. N .GE. 0.
P  Int32
(input) INTEGER The number of rows of the matrix B. P .GE. 0.
K  Int32
L
L  Int32
( 0 C )
A  Double
(input/output) 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.
offset_a  Int32
 
LDA  Int32
(input) INTEGER The leading dimension of the array A. LDA .GE. max(1,M).
B  Double
(input/output) 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 .LT. 0. See Purpose for details.
offset_b  Int32
 
LDB  Int32
(input) INTEGER The leading dimension of the array B. LDB .GE. max(1,P).
ALPHA  Double
(output) DOUBLE PRECISION array, dimension (N)
offset_alpha  Int32
 
BETA  Double
(output) 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 .GE. 0, ALPHA(K+1:K+L) = C, BETA(K+1:K+L) = S, or if M-K-L .LT. 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
offset_beta  Int32
 
U  Double
(output) 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.
offset_u  Int32
 
LDU  Int32
(input) INTEGER The leading dimension of the array U. LDU .GE. max(1,M) if JOBU = 'U'; LDU .GE. 1 otherwise.
V  Double
(output) 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.
offset_v  Int32
 
LDV  Int32
(input) INTEGER The leading dimension of the array V. LDV .GE. max(1,P) if JOBV = 'V'; LDV .GE. 1 otherwise.
Q  Double
(output) 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.
offset_q  Int32
 
LDQ  Int32
(input) INTEGER The leading dimension of the array Q. LDQ .GE. max(1,N) if JOBQ = 'Q'; LDQ .GE. 1 otherwise.
WORK  Double
(workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P)+N)
offset_work  Int32
 
IWORK  Int32
(workspace/output) 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) .GE. ALPHA(2) .GE. ... .GE. ALPHA(N).
offset_iwork  Int32
 
INFO  Int32
(output) INTEGER = 0: successful exit .LT. 0: if INFO = -i, the i-th argument had an illegal value. .GT. 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine DTGSJA.
See Also