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---
:name: stgsja
:md5sum: 604149ded91900c1bd601ad488246cc2
:category: :subroutine
:arguments:
- jobu:
:type: char
:intent: input
- jobv:
:type: char
:intent: input
- jobq:
:type: char
:intent: input
- m:
:type: integer
:intent: input
- p:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- k:
:type: integer
:intent: input
- l:
:type: integer
:intent: input
- a:
:type: real
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- b:
:type: real
:intent: input/output
:dims:
- ldb
- n
- ldb:
:type: integer
:intent: input
- tola:
:type: real
:intent: input
- tolb:
:type: real
:intent: input
- alpha:
:type: real
:intent: output
:dims:
- n
- beta:
:type: real
:intent: output
:dims:
- n
- u:
:type: real
:intent: input/output
:dims:
- ldu
- m
- ldu:
:type: integer
:intent: input
- v:
:type: real
:intent: input/output
:dims:
- ldv
- p
- ldv:
:type: integer
:intent: input
- q:
:type: real
:intent: input/output
:dims:
- ldq
- n
- ldq:
:type: integer
:intent: input
- work:
:type: real
:intent: workspace
:dims:
- 2*n
- ncycle:
:type: integer
:intent: output
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " 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 )\n\n\
* Purpose\n\
* =======\n\
*\n\
* STGSJA computes the generalized singular value decomposition (GSVD)\n\
* of two real upper triangular (or trapezoidal) matrices A and B.\n\
*\n\
* On entry, it is assumed that matrices A and B have the following\n\
* forms, which may be obtained by the preprocessing subroutine SGGSVP\n\
* from a general M-by-N matrix A and P-by-N matrix B:\n\
*\n\
* N-K-L K L\n\
* A = K ( 0 A12 A13 ) if M-K-L >= 0;\n\
* L ( 0 0 A23 )\n\
* M-K-L ( 0 0 0 )\n\
*\n\
* N-K-L K L\n\
* A = K ( 0 A12 A13 ) if M-K-L < 0;\n\
* M-K ( 0 0 A23 )\n\
*\n\
* N-K-L K L\n\
* B = L ( 0 0 B13 )\n\
* P-L ( 0 0 0 )\n\
*\n\
* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular\n\
* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,\n\
* otherwise A23 is (M-K)-by-L upper trapezoidal.\n\
*\n\
* On exit,\n\
*\n\
* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),\n\
*\n\
* where U, V and Q are orthogonal matrices, Z' denotes the transpose\n\
* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are\n\
* ``diagonal'' matrices, which are of the following structures:\n\
*\n\
* If M-K-L >= 0,\n\
*\n\
* K L\n\
* D1 = K ( I 0 )\n\
* L ( 0 C )\n\
* M-K-L ( 0 0 )\n\
*\n\
* K L\n\
* D2 = L ( 0 S )\n\
* P-L ( 0 0 )\n\
*\n\
* N-K-L K L\n\
* ( 0 R ) = K ( 0 R11 R12 ) K\n\
* L ( 0 0 R22 ) L\n\
*\n\
* where\n\
*\n\
* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),\n\
* S = diag( BETA(K+1), ... , BETA(K+L) ),\n\
* C**2 + S**2 = I.\n\
*\n\
* R is stored in A(1:K+L,N-K-L+1:N) on exit.\n\
*\n\
* If M-K-L < 0,\n\
*\n\
* K M-K K+L-M\n\
* D1 = K ( I 0 0 )\n\
* M-K ( 0 C 0 )\n\
*\n\
* K M-K K+L-M\n\
* D2 = M-K ( 0 S 0 )\n\
* K+L-M ( 0 0 I )\n\
* P-L ( 0 0 0 )\n\
*\n\
* N-K-L K M-K K+L-M\n\
* ( 0 R ) = K ( 0 R11 R12 R13 )\n\
* M-K ( 0 0 R22 R23 )\n\
* K+L-M ( 0 0 0 R33 )\n\
*\n\
* where\n\
* C = diag( ALPHA(K+1), ... , ALPHA(M) ),\n\
* S = diag( BETA(K+1), ... , BETA(M) ),\n\
* C**2 + S**2 = I.\n\
*\n\
* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored\n\
* ( 0 R22 R23 )\n\
* in B(M-K+1:L,N+M-K-L+1:N) on exit.\n\
*\n\
* The computation of the orthogonal transformation matrices U, V or Q\n\
* is optional. These matrices may either be formed explicitly, or they\n\
* may be postmultiplied into input matrices U1, V1, or Q1.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOBU (input) CHARACTER*1\n\
* = 'U': U must contain an orthogonal matrix U1 on entry, and\n\
* the product U1*U is returned;\n\
* = 'I': U is initialized to the unit matrix, and the\n\
* orthogonal matrix U is returned;\n\
* = 'N': U is not computed.\n\
*\n\
* JOBV (input) CHARACTER*1\n\
* = 'V': V must contain an orthogonal matrix V1 on entry, and\n\
* the product V1*V is returned;\n\
* = 'I': V is initialized to the unit matrix, and the\n\
* orthogonal matrix V is returned;\n\
* = 'N': V is not computed.\n\
*\n\
* JOBQ (input) CHARACTER*1\n\
* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and\n\
* the product Q1*Q is returned;\n\
* = 'I': Q is initialized to the unit matrix, and the\n\
* orthogonal matrix Q is returned;\n\
* = 'N': Q is not computed.\n\
*\n\
* M (input) INTEGER\n\
* The number of rows of the matrix A. M >= 0.\n\
*\n\
* P (input) INTEGER\n\
* The number of rows of the matrix B. P >= 0.\n\
*\n\
* N (input) INTEGER\n\
* The number of columns of the matrices A and B. N >= 0.\n\
*\n\
* K (input) INTEGER\n\
* L (input) INTEGER\n\
* K and L specify the subblocks in the input matrices A and B:\n\
* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)\n\
* of A and B, whose GSVD is going to be computed by STGSJA.\n\
* See Further Details.\n\
*\n\
* A (input/output) REAL array, dimension (LDA,N)\n\
* On entry, the M-by-N matrix A.\n\
* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular\n\
* matrix R or part of R. See Purpose for details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,M).\n\
*\n\
* B (input/output) REAL array, dimension (LDB,N)\n\
* On entry, the P-by-N matrix B.\n\
* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains\n\
* a part of R. See Purpose for details.\n\
*\n\
* LDB (input) INTEGER\n\
* The leading dimension of the array B. LDB >= max(1,P).\n\
*\n\
* TOLA (input) REAL\n\
* TOLB (input) REAL\n\
* TOLA and TOLB are the convergence criteria for the Jacobi-\n\
* Kogbetliantz iteration procedure. Generally, they are the\n\
* same as used in the preprocessing step, say\n\
* TOLA = max(M,N)*norm(A)*MACHEPS,\n\
* TOLB = max(P,N)*norm(B)*MACHEPS.\n\
*\n\
* ALPHA (output) REAL array, dimension (N)\n\
* BETA (output) REAL array, dimension (N)\n\
* On exit, ALPHA and BETA contain the generalized singular\n\
* value pairs of A and B;\n\
* ALPHA(1:K) = 1,\n\
* BETA(1:K) = 0,\n\
* and if M-K-L >= 0,\n\
* ALPHA(K+1:K+L) = diag(C),\n\
* BETA(K+1:K+L) = diag(S),\n\
* or if M-K-L < 0,\n\
* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0\n\
* BETA(K+1:M) = S, BETA(M+1:K+L) = 1.\n\
* Furthermore, if K+L < N,\n\
* ALPHA(K+L+1:N) = 0 and\n\
* BETA(K+L+1:N) = 0.\n\
*\n\
* U (input/output) REAL array, dimension (LDU,M)\n\
* On entry, if JOBU = 'U', U must contain a matrix U1 (usually\n\
* the orthogonal matrix returned by SGGSVP).\n\
* On exit,\n\
* if JOBU = 'I', U contains the orthogonal matrix U;\n\
* if JOBU = 'U', U contains the product U1*U.\n\
* If JOBU = 'N', U is not referenced.\n\
*\n\
* LDU (input) INTEGER\n\
* The leading dimension of the array U. LDU >= max(1,M) if\n\
* JOBU = 'U'; LDU >= 1 otherwise.\n\
*\n\
* V (input/output) REAL array, dimension (LDV,P)\n\
* On entry, if JOBV = 'V', V must contain a matrix V1 (usually\n\
* the orthogonal matrix returned by SGGSVP).\n\
* On exit,\n\
* if JOBV = 'I', V contains the orthogonal matrix V;\n\
* if JOBV = 'V', V contains the product V1*V.\n\
* If JOBV = 'N', V is not referenced.\n\
*\n\
* LDV (input) INTEGER\n\
* The leading dimension of the array V. LDV >= max(1,P) if\n\
* JOBV = 'V'; LDV >= 1 otherwise.\n\
*\n\
* Q (input/output) REAL array, dimension (LDQ,N)\n\
* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually\n\
* the orthogonal matrix returned by SGGSVP).\n\
* On exit,\n\
* if JOBQ = 'I', Q contains the orthogonal matrix Q;\n\
* if JOBQ = 'Q', Q contains the product Q1*Q.\n\
* If JOBQ = 'N', Q is not referenced.\n\
*\n\
* LDQ (input) INTEGER\n\
* The leading dimension of the array Q. LDQ >= max(1,N) if\n\
* JOBQ = 'Q'; LDQ >= 1 otherwise.\n\
*\n\
* WORK (workspace) REAL array, dimension (2*N)\n\
*\n\
* NCYCLE (output) INTEGER\n\
* The number of cycles required for convergence.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value.\n\
* = 1: the procedure does not converge after MAXIT cycles.\n\
*\n\
* Internal Parameters\n\
* ===================\n\
*\n\
* MAXIT INTEGER\n\
* MAXIT specifies the total loops that the iterative procedure\n\
* may take. If after MAXIT cycles, the routine fails to\n\
* converge, we return INFO = 1.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce\n\
* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L\n\
* matrix B13 to the form:\n\
*\n\
* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,\n\
*\n\
* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose\n\
* of Z. C1 and S1 are diagonal matrices satisfying\n\
*\n\
* C1**2 + S1**2 = I,\n\
*\n\
* and R1 is an L-by-L nonsingular upper triangular matrix.\n\
*\n\
* =====================================================================\n\
*\n"
|
