stgsja(3)

NAME

STGSJA - compute the generalized singular value decomposi

tion (GSVD) of two real upper triangular (or trapezoidal) matri

ces A and B

SYNOPSIS

SUBROUTINE STGSJA( 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
    REAL           TOLA, TOLB
    REAL            A(  LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

PURPOSE

STGSJA computes the generalized singular value decomposi

tion (GSVD) of two real upper triangular (or trapezoidal) matri

ces 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 ma

trix 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 orthogonal matrices, Z' denotes the

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 orthogonal transformation matrices

U, V or Q is optional. These matrices may either be formed ex

plicitly, or they may be postmultiplied into input matrices U1,

V1, or Q1.

ARGUMENTS

JOBU (input) 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 (input) 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 (input) 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 (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 STGSJA.

See Further details.

A (input/output) 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 (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,M).

B (input/output) REAL 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) REAL

TOLB (input) REAL TOLA and TOLB are the conver

gence 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 (output) REAL array, dimension (N)

BETA (output) REAL array, dimension (N) On ex

it, 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, 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 and
BETA(K+L+1:N) = 0.

U (input/output) 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 (input) INTEGER

The leading dimension of the array U. LDU >=

max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.

V (input/output) 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 (input) INTEGER

The leading dimension of the array V. LDV >=

max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.

Q (input/output) 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 (input) INTEGER

The leading dimension of the array Q. LDQ >=

max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK (workspace) REAL 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 ===============

STGSJA 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 orthogonal matrix, and Z'

is the transpose of Z. C1 and S1 are diagonal matrices satisfy

ing

C1**2 + S1**2 = I,

and R1 is an L-by-L nonsingular upper triangular

matrix.

LAPACK version 3.0 15 June 2000

STGSJA(3)