stgsja.f (3)  Linux Manuals
NAME
stgsja.f 
SYNOPSIS
Functions/Subroutines
subroutine stgsja (JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
STGSJA
Function/Subroutine Documentation
subroutine stgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN, integerK, integerL, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA, real, dimension( ldu, * )U, integerLDU, real, dimension( ldv, * )V, integerLDV, real, dimension( ldq, * )Q, integerLDQ, real, dimension( * )WORK, integerNCYCLE, integerINFO)
STGSJA
Purpose:

STGSJA 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 SGGSVP from a general MbyN matrix A and PbyN matrix B: NKL K L A = K ( 0 A12 A13 ) if MKL >= 0; L ( 0 0 A23 ) MKL ( 0 0 0 ) NKL K L A = K ( 0 A12 A13 ) if MKL < 0; MK ( 0 0 A23 ) NKL K L B = L ( 0 0 B13 ) PL ( 0 0 0 ) where the KbyK matrix A12 and LbyL matrix B13 are nonsingular upper triangular; A23 is LbyL upper triangular if MKL >= 0, otherwise A23 is (MK)byL upper trapezoidal. On exit, U**T *A*Q = D1*( 0 R ), V**T *B*Q = D2*( 0 R ), where U, V and Q are orthogonal matrices. R is a nonsingular upper triangular matrix, and D1 and D2 are ``diagonal'' matrices, which are of the following structures: 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 ) 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,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. R = ( 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 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.
Parameters:

JOBU
JOBU is 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.
JOBVJOBV is 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.
JOBQJOBQ is 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.
MM is INTEGER The number of rows of the matrix A. M >= 0.
PP is INTEGER The number of rows of the matrix B. P >= 0.
NN is INTEGER The number of columns of the matrices A and B. N >= 0.
KK is INTEGER
LL is INTEGER K and L specify the subblocks in the input matrices A and B: A23 = A(K+1:MIN(K+L,M),NL+1:N) and B13 = B(1:L,NL+1:N) of A and B, whose GSVD is going to be computed by STGSJA. See Further Details.
AA is REAL array, dimension (LDA,N) On entry, the MbyN matrix A. On exit, A(NK+1:N,1:MIN(K+L,M) ) 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 REAL array, dimension (LDB,N) On entry, the PbyN matrix B. On exit, if necessary, B(MK+1:L,N+MKL+1:N) contains a part of R. See Purpose for details.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).
TOLATOLA is REAL
TOLBTOLB is REAL 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)*MACHEPS, TOLB = max(P,N)*norm(B)*MACHEPS.
ALPHAALPHA is REAL array, dimension (N)
BETABETA 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 MKL >= 0, ALPHA(K+1:K+L) = diag(C), BETA(K+1:K+L) = diag(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. Furthermore, if K+L < N, ALPHA(K+L+1:N) = 0 and BETA(K+L+1:N) = 0.
UU is REAL array, dimension (LDU,M) On entry, if JOBU = 'U', U must contain a matrix U1 (usually the orthogonal matrix returned by SGGSVP). 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.
LDULDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
VV is REAL array, dimension (LDV,P) On entry, if JOBV = 'V', V must contain a matrix V1 (usually the orthogonal matrix returned by SGGSVP). 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.
LDVLDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
QQ is REAL array, dimension (LDQ,N) On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the orthogonal matrix returned by SGGSVP). 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.
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORKWORK is REAL array, dimension (2*N)
NCYCLENCYCLE is INTEGER The number of cycles required for convergence.
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. = 1: the procedure does not converge after MAXIT cycles.
Internal Parameters =================== MAXIT INTEGER MAXIT specifies the total loops that the iterative procedure may take. If after MAXIT cycles, the routine fails to converge, we return INFO = 1..fi Author:
 Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.
 November 2011
 STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce min(L,MK)byL triangular (or trapezoidal) matrix A23 and LbyL matrix B13 to the form: U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1, where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose of Z. C1 and S1 are diagonal matrices satisfying C1**2 + S1**2 = I, and R1 is an LbyL nonsingular upper triangular matrix.
Definition at line 377 of file stgsja.f.
Author
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