
DTGSJA(l) ) DTGSJA(l)
NAME
DTGSJA  compute 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 tri
angular (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 MbyN matrix A and PbyN matrix B:
NKL K L
A = K ( 0 A12 A13 ) if MKL >= 0;
L ( 0 0 A23 )
MKL ( 0 0 0 )
NKL K L
A = K ( 0 A12 A13 ) if MKL < 0;
MK ( 0 0 A23 )
NKL K L
B = L ( 0 0 B13 )
PL ( 0 0 0 )
where the KbyK matrix A12 and LbyL matrix B13 are nonsingular upper triangular; A23 is
LbyL upper triangular if MKL >= 0, otherwise A23 is (MK)byL 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 nonsingu
lar upper triangular matrix, and D1 and D2 are ``diagonal'' matrices, which are of the
following structures:
If MKL >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
MKL ( 0 0 )
K L
D2 = L ( 0 S )
PL ( 0 0 )
NKL 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,NKL+1:N) on exit.
If MKL < 0,
K MK K+LM
D1 = K ( I 0 0 )
MK ( 0 C 0 )
K MK K+LM
D2 = MK ( 0 S 0 )
K+LM ( 0 0 I )
PL ( 0 0 0 )
NKL K MK K+LM
MK ( 0 0 R22 R23 )
K+LM ( 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, NKL+1:N) and R33 is stored
( 0 R22 R23 )
in B(MK+1:L,N+MKL+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 matri
ces 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 subblocks in the input matrices A and
B:
A23 = A(K+1:MIN(K+L,M),NL+1:N) and B13 = B(1:L,NL+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 MbyN matrix A. On exit, A(NK+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 PbyN matrix B. On exit, if necessary, B(MK+1:L,N+MKL+1:N) con
tains 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 MKL >= 0, ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(S), or if MKL < 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 = '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 oth
erwise.
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 oth
erwise.
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 oth
erwise.
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 ith 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)byL triangular (or trapezoidal) matrix A23 and LbyL 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 satisfying
C1**2 + S1**2 = I,
and R1 is an LbyL nonsingular upper triangular matrix.
LAPACK version 3.0 15 June 2000 DTGSJA(l) 
