LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

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Functions/Subroutines  
subroutine  dtgsja (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) 
DTGSJA 
subroutine dtgsja  (  character  JOBU, 
character  JOBV,  
character  JOBQ,  
integer  M,  
integer  P,  
integer  N,  
integer  K,  
integer  L,  
double precision, dimension( lda, * )  A,  
integer  LDA,  
double precision, dimension( ldb, * )  B,  
integer  LDB,  
double precision  TOLA,  
double precision  TOLB,  
double precision, dimension( * )  ALPHA,  
double precision, dimension( * )  BETA,  
double precision, dimension( ldu, * )  U,  
integer  LDU,  
double precision, dimension( ldv, * )  V,  
integer  LDV,  
double precision, dimension( ldq, * )  Q,  
integer  LDQ,  
double precision, dimension( * )  WORK,  
integer  NCYCLE,  
integer  INFO  
) 
DTGSJA
Download DTGSJA + dependencies [TGZ] [ZIP] [TXT]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 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.
[in]  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. 
[in]  JOBV  JOBV 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. 
[in]  JOBQ  JOBQ 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. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  P  P is INTEGER The number of rows of the matrix B. P >= 0. 
[in]  N  N is INTEGER The number of columns of the matrices A and B. N >= 0. 
[in]  K  K is INTEGER 
[in]  L  L 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 DTGSJA. See Further Details. 
[in,out]  A  A is DOUBLE PRECISION 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. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in,out]  B  B is DOUBLE PRECISION 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. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). 
[in]  TOLA  TOLA is DOUBLE PRECISION 
[in]  TOLB  TOLB is 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. 
[out]  ALPHA  ALPHA is DOUBLE PRECISION array, dimension (N) 
[out]  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) = 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. 
[in,out]  U  U is 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. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. 
[in,out]  V  V is 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. 
[in]  LDV  LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. 
[in,out]  Q  Q is 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. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. 
[out]  WORK  WORK is DOUBLE PRECISION array, dimension (2*N) 
[out]  NCYCLE  NCYCLE is INTEGER The number of cycles required for convergence. 
[out]  INFO  INFO 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.
DTGSJA 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 dtgsja.f.