# dtgsja (l) - Linux Manuals

## NAME

DTGSJA - computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices 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 decomposition (GSVD) of two real 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 DGGSVP from a general M-by-N matrix A and P-by-N matrix B:

N-K-L     L

A12  A13 if M-K-L >= 0;

A23 )

M-K-L         )

N-K-L     L

A12  A13 if M-K-L 0;

M-K       A23 )

N-K-L     L

B13 )

P-L         )
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,

Uaq*A*Q D1*( 0 R ),    Vaq*B*Q D2*( 0 R ),
where U, V and Q are orthogonal matrices, Zaq denotes the transpose of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are ``diagonalaqaq matrices, which are of the following structures: If M-K-L >= 0,

L

D1      )

)

M-K-L  )

L

D2    )

P-L  )

N-K-L     L

0 R    R11  R12 K

R22 L
where

diag( ALPHA(K+1), ... , ALPHA(K+L) ),

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         )

M-K       )

K M-K K+L-M

D2   M-K       )

K+L-M       )

P-L       )

N-K-L    M-K  K+L-M

M-K       R22  R23  )

K+L-M          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
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 explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1.

## ARGUMENTS

JOBU (input) CHARACTER*1
= aqUaq: U must contain an orthogonal matrix U1 on entry, and the product U1*U is returned; = aqIaq: U is initialized to the unit matrix, and the orthogonal matrix U is returned; = aqNaq: U is not computed.
JOBV (input) CHARACTER*1

= aqVaq: V must contain an orthogonal matrix V1 on entry, and the product V1*V is returned; = aqIaq: V is initialized to the unit matrix, and the orthogonal matrix V is returned; = aqNaq: V is not computed.
JOBQ (input) CHARACTER*1

= aqQaq: Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned; = aqIaq: Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = aqNaq: 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 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 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 and
BETA(K+L+1:N) = 0.
U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = aqUaq, U must contain a matrix U1 (usually the orthogonal matrix returned by DGGSVP). On exit, if JOBU = aqIaq, U contains the orthogonal matrix U; if JOBU = aqUaq, U contains the product U1*U. If JOBU = aqNaq, U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if JOBU = aqUaq; LDU >= 1 otherwise.
V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = aqVaq, V must contain a matrix V1 (usually the orthogonal matrix returned by DGGSVP). On exit, if JOBV = aqIaq, V contains the orthogonal matrix V; if JOBV = aqVaq, V contains the product V1*V. If JOBV = aqNaq, V is not referenced.
LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if JOBV = aqVaq; LDV >= 1 otherwise.
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = aqQaq, Q must contain a matrix Q1 (usually the orthogonal matrix returned by DGGSVP). On exit, if JOBQ = aqIaq, Q contains the orthogonal matrix Q; if JOBQ = aqQaq, Q contains the product Q1*Q. If JOBQ = aqNaq, Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = aqQaq; 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 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 =============== 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: U1aq*A13*Q1 = C1*R1; V1aq*B13*Q1 = S1*R1, where U1, V1 and Q1 are orthogonal matrix, and Zaq is the 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.