ztgsja(3)

NAME

ZTGSJA - compute the generalized singular value decomposi

tion (GSVD) of two complex upper triangular (or trapezoidal) ma

trices 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 decomposi

tion (GSVD) of two complex upper triangular (or trapezoidal) ma

trices A and B. On entry, it is assumed that matrices A and B

have the following forms, which may be obtained by the prepro

cessing 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 non

singular 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 con

jugate transpose of Z, R is a nonsingular upper triangular ma

trix, 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 explic

itly, 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 en

try, 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 en

try, 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 en

try, 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 sub

blocks 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 neces

sary, 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 prepro

cessing 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 val

ue 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, AL

PHA(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 ille

gal 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.

LAPACK version 3.0 15 June 2000

ZTGSJA(3)