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template<typename T>
void tgsja(char *jobu, char *jobv, char *jobq, integer *m, integer *p, integer *n, integer *k, integer *l, T *a, integer *lda, T *b, integer *ldb, T *tola, T *tolb, T *alpha, T *beta, T *u, integer *ldu, T *v, integer *ldv, T *q, integer *ldq, T *work, integer *ncycle, integer *info)# TGSJA computes the generalized singular value decomposition (GSVD)
of two real upper triangular (or trapezoidal) matrices A and B.
Purpose:
TGSJA 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 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 >= 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 < 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 >= 0, otherwise A23 is (M-K)-by-L 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 M-K-L >= 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 < 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.- Parameters:
JOBU – [in]
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.
JOBV – [in]
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.
JOBQ – [in]
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.
M – [in]
M is INTEGER
The number of rows of the matrix A. M >= 0.
P – [in]
P is INTEGER
The number of rows of the matrix B. P >= 0.
N – [in]
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
K – [in] K is INTEGER
L – [in]
L is INTEGER
K and L specify the subblocks in the input matrices A and B: A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) of A and B, whose GSVD is going to be computed by STGSJA. See Further Details.
A – [inout]
A is REAL 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.
LDA – [in]
LDA is INTEGER
The leading dimension of the array A. LDA >= fla_max(1,M).
B – [inout]
B is REAL 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.
LDB – [in]
LDB is INTEGER
The leading dimension of the array B. LDB >= fla_max(1,P).
TOLA – [in] TOLA is REAL
TOLB – [in]
TOLB 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 = fla_max(M,N)*norm(A)*MACHEPS,
TOLB = fla_max(P,N)*norm(B)*MACHEPS.
ALPHA – [out] ALPHA is REAL array, dimension (N)
BETA – [out]
BETA 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 M-K-L >= 0,
ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(S),
or if M-K-L < 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.U – [inout]
U 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.
LDU – [in]
LDU is INTEGER
The leading dimension of the array U. LDU >= fla_max(1,M) if JOBU = ‘U’; LDU >= 1 otherwise.
V – [inout]
V 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.
LDV – [in]
LDV is INTEGER
The leading dimension of the array V. LDV >= fla_max(1,P) if JOBV = ‘V’; LDV >= 1 otherwise.
Q – [inout]
Q 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.
LDQ – [in]
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= fla_max(1,N) if JOBQ = ‘Q’; LDQ >= 1 otherwise.
WORK – [out] WORK is REAL array, dimension (2*N)
NCYCLE – [out]
NCYCLE is INTEGER
The number of cycles required for convergence.
INFO – [out]
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles.
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template<typename T, typename Ta>
void tgsja(char *jobu, char *jobv, char *jobq, integer *m, integer *p, integer *n, integer *k, integer *l, T *a, integer *lda, T *b, integer *ldb, Ta *tola, Ta *tolb, Ta *alpha, Ta *beta, T *u, integer *ldu, T *v, integer *ldv, T *q, integer *ldq, T *work, integer *ncycle, integer *info)#