dtgsja(3)

NAME

DTGSJA - compute the generalized singular value decomposi

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

ces A and B

SYNOPSIS

SUBROUTINE DTGSJA( 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  A( LDA, * ), ALPHA( * ), B(
LDB, * ), BETA( * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK(
* )

PURPOSE

DTGSJA 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 DGGSVP 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 DTGSJA.

See Further details.

A (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 and
BETA(K+L+1:N) = 0.

U (input/output) DOUBLE PRECISION array, dimension

(LDU,M)

On entry, if JOBU = 'U', U must contain a matrix

U1 (usually the orthogonal matrix returned by DGGSVP). 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) DOUBLE PRECISION array, dimension

(LDV,P)

On entry, if JOBV = 'V', V must contain a matrix

V1 (usually the orthogonal matrix returned by DGGSVP). 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) DOUBLE PRECISION array, dimension

(LDQ,N)

On entry, if JOBQ = 'Q', Q must contain a matrix

Q1 (usually the orthogonal matrix returned by DGGSVP). 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) DOUBLE PRECISION 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 ===============

DTGSJA 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

DTGSJA(3)