# ctgsja.f man page

ctgsja.f

## Synopsis

### Functions/Subroutines

subroutine **ctgsja** (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)**CTGSJA**

## Function/Subroutine Documentation

### subroutine ctgsja (character JOBU, character JOBV, character JOBQ, integer M, integer P, integer N, integer K, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, real TOLA, real TOLB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( * ) WORK, integer NCYCLE, integer INFO)

**CTGSJA**

**Purpose:**

CTGSJA computes the generalized singular value decomposition (GSVD) of two complex 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 CGGSVP 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**H *A*Q = D1*( 0 R ), V**H *B*Q = D2*( 0 R ), where U, V and Q are unitary 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 unitary 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 a unitary matrix U1 on entry, and the product U1*U is returned; = 'I': U is initialized to the unit matrix, and the unitary matrix U is returned; = 'N': U is not computed.

*JOBV*JOBV is CHARACTER*1 = 'V': V must contain a unitary matrix V1 on entry, and the product V1*V is returned; = 'I': V is initialized to the unit matrix, and the unitary matrix V is returned; = 'N': V is not computed.

*JOBQ*JOBQ is CHARACTER*1 = 'Q': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned; = 'I': Q is initialized to the unit matrix, and the unitary matrix Q is returned; = 'N': Q is not computed.

*M*M is INTEGER The number of rows of the matrix A. M >= 0.

*P*P is INTEGER The number of rows of the matrix B. P >= 0.

*N*N is INTEGER The number of columns of the matrices A and B. N >= 0.

*K*K is INTEGER

*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),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 CTGSJA. See Further Details.

*A*A is COMPLEX 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*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).

*B*B is COMPLEX 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*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).

*TOLA*TOLA is REAL

*TOLB*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 = MAX(M,N)*norm(A)*MACHEPS, TOLB = MAX(P,N)*norm(B)*MACHEPS.

*ALPHA*ALPHA is REAL array, dimension (N)

*BETA*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 BETA(K+L+1:N) = 0.

*U*U is COMPLEX array, dimension (LDU,M) On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by CGGSVP). On exit, if JOBU = 'I', U contains the unitary matrix U; if JOBU = 'U', U contains the product U1*U. If JOBU = 'N', U is not referenced.

*LDU*LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.

*V*V is COMPLEX array, dimension (LDV,P) On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by CGGSVP). On exit, if JOBV = 'I', V contains the unitary matrix V; if JOBV = 'V', V contains the product V1*V. If JOBV = 'N', V is not referenced.

*LDV*LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.

*Q*Q is COMPLEX array, dimension (LDQ,N) On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by CGGSVP). On exit, if JOBQ = 'I', Q contains the unitary matrix Q; if JOBQ = 'Q', Q contains the product Q1*Q. If JOBQ = 'N', Q is not referenced.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.

*WORK*WORK is COMPLEX array, dimension (2*N)

*NCYCLE*NCYCLE is INTEGER The number of cycles required for convergence.

*INFO*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.

**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.

**Author:**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**December 2016

**Further Details:**

CTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L matrix B13 to the form: U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1, where U1, V1 and Q1 are unitary matrix. C1 and S1 are diagonal matrices satisfying C1**2 + S1**2 = I, and R1 is an L-by-L nonsingular upper triangular matrix.

Definition at line 381 of file ctgsja.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page ctgsja(3) is an alias of ctgsja.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK