 ZTGSJA(3) computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B

## SYNOPSIS

SUBROUTINE ZTGSJA(
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 )

CHARACTER JOBQ, JOBU, JOBV

INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, NCYCLE, P

DOUBLE PRECISION TOLA, TOLB

DOUBLE PRECISION ALPHA( * ), BETA( * )

COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

## PURPOSE

ZTGSJA 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 ZGGSVP 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'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),
where U, V and Q are unitary matrices, Z' denotes the conjugate transpose of Z, 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

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.

## ARGUMENTS

JOBU (input) 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 (input) 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 (input) 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 (input) INTEGER
The number of rows of the matrix A. M >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
K (input) INTEGER
L (input) 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 ZTGSJA. See Further Details. A (input/output) COMPLEX*16 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 (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) COMPLEX*16 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 (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TOLA (input) DOUBLE PRECISION
TOLB (input) 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.
ALPHA (output) DOUBLE PRECISION array, dimension (N)
BETA (output) 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 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 (input/output) COMPLEX*16 array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by ZGGSVP). 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 (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V (input/output) COMPLEX*16 array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by ZGGSVP). 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 (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by ZGGSVP). 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 (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK (workspace) COMPLEX*16 array, dimension (2*N)
NCYCLE (output) INTEGER
The number of cycles required for convergence.
INFO (output) 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.

## 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. Further Details =============== ZTGSJA 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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate transpose of Z. C1 and S1 are diagonal matrices satisfying C1**2 + S1**2 = I, and R1 is an L-by-L nonsingular upper triangular matrix.