
ZTGSJA(l) ) ZTGSJA(l)
NAME
ZTGSJA  compute the generalized singular value decomposition (GSVD) of two complex upper
triangular (or trapezoidal) matrices 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 decomposition (GSVD) of two complex 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 ZGGSVP
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 unitary matrices, Z' denotes the conjugate transpose of Z, R is a
nonsingular 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 unitary 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 a unitary matrix U1 on entry, 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 entry, 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 entry, 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 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 ZTGSJA. See Further details.
A (input/output) COMPLEX*16 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) COMPLEX*16 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
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 oth
erwise.
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 oth
erwise.
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 oth
erwise.
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 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 ===============
ZTGSJA 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 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 LbyL nonsingular upper triangular matrix.
LAPACK version 3.0 15 June 2000 ZTGSJA(l) 
