ZTGSJA(3)

NAME
ZTGSJA - 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.