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DTGSJARun Method

Purpose ======= DTGSJA computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B. On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine DGGSVP from a general M-by-N matrix A and P-by-N matrix B: N-K-L K L A = K ( 0 A12 A13 ) if M-K-L .GE. 0; L ( 0 0 A23 ) M-K-L ( 0 0 0 ) N-K-L K L A = K ( 0 A12 A13 ) if M-K-L .LT. 0; M-K ( 0 0 A23 ) N-K-L K L B = L ( 0 0 B13 ) P-L ( 0 0 0 ) where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular upper triangular; A23 is L-by-L upper triangular if M-K-L .GE. 0, otherwise A23 is (M-K)-by-L upper trapezoidal. On exit, U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), where U, V and Q are orthogonal matrices, Z' denotes the transpose of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are ``diagonal'' matrices, which are of the following structures: 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 ) K L ( 0 0 R22 ) L 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. R = ( 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 computation of the orthogonal transformation matrices U, V or Q is optional. These matrices may either be formed explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1.

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 P,
	int N,
	int K,
	int L,
	ref double[] A,
	int offset_a,
	int LDA,
	ref double[] B,
	int offset_b,
	int LDB,
	double TOLA,
	double TOLB,
	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 NCYCLE,
	ref int INFO
)
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Parameters

JOBU  String
(input) CHARACTER*1 = 'U': U must contain an orthogonal matrix U1 on entry, and the product U1*U is returned; = 'I': U is initialized to the unit matrix, and the orthogonal matrix U is returned; = 'N': U is not computed.
JOBV  String
(input) CHARACTER*1 = 'V': V must contain an orthogonal matrix V1 on entry, and the product V1*V is returned; = 'I': V is initialized to the unit matrix, and the orthogonal matrix V is returned; = 'N': V is not computed.
JOBQ  String
(input) CHARACTER*1 = 'Q': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned; = 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = 'N': Q is not computed.
M  Int32
(input) INTEGER The number of rows of the matrix A. M .GE. 0.
P  Int32
(input) INTEGER The number of rows of the matrix B. P .GE. 0.
N  Int32
(input) INTEGER The number of columns of the matrices A and B. N .GE. 0.
K  Int32
L
L  Int32
( 0 0 A23 )
A  Double
(input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains a part of R. See Purpose for details.
offset_b  Int32
 
LDB  Int32
(input) INTEGER The leading dimension of the array B. LDB .GE. max(1,P).
TOLA  Double
(input) DOUBLE PRECISION
TOLB  Double
(input) DOUBLE PRECISION TOLA and TOLB are the convergence criteria for the Jacobi- Kogbetliantz iteration procedure. Generally, they are the same as used in the preprocessing step, say TOLA = max(M,N)*norm(A)*MAZHEPS, TOLB = max(P,N)*norm(B)*MAZHEPS.
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) = diag(C), BETA(K+1:K+L) = diag(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. Furthermore, if K+L .LT. N, ALPHA(K+L+1:N) = 0 and BETA(K+L+1:N) = 0.
offset_beta  Int32
 
U  Double
(input/output) DOUBLE PRECISION array, dimension (LDU,M) On entry, if JOBU = 'U', U must contain a matrix U1 (usually the orthogonal matrix returned by DGGSVP). On exit, if JOBU = 'I', U contains the orthogonal matrix U; if JOBU = 'U', U contains the product U1*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
(input/output) DOUBLE PRECISION array, dimension (LDV,P) On entry, if JOBV = 'V', V must contain a matrix V1 (usually the orthogonal matrix returned by DGGSVP). On exit, if JOBV = 'I', V contains the orthogonal matrix V; if JOBV = 'V', V contains the product V1*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
(input/output) DOUBLE PRECISION array, dimension (LDQ,N) On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the orthogonal matrix returned by DGGSVP). On exit, if JOBQ = 'I', Q contains the orthogonal matrix Q; if JOBQ = 'Q', Q contains the product Q1*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 (2*N)
offset_work  Int32
 
NCYCLE  Int32
(output) INTEGER The number of cycles required for convergence.
INFO  Int32
(output) INTEGER = 0: successful exit .LT. 0: if INFO = -i, the i-th argument had an illegal value. = 1: the procedure does not converge after MAXIT cycles.
See Also