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SUBROUTINE <a name="CTGSJA.1"></a><a href="ctgsja.f.html#CTGSJA.1">CTGSJA</a>( 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 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER JOBQ, JOBU, JOBV
INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
$ NCYCLE, P
REAL TOLA, TOLB
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> REAL ALPHA( * ), BETA( * )
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ U( LDU, * ), V( LDV, * ), WORK( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="CTGSJA.24"></a><a href="ctgsja.f.html#CTGSJA.1">CTGSJA</a> computes the generalized singular value decomposition (GSVD)
</span><span class="comment">*</span><span class="comment"> of two complex upper triangular (or trapezoidal) matrices A and B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On entry, it is assumed that matrices A and B have the following
</span><span class="comment">*</span><span class="comment"> forms, which may be obtained by the preprocessing subroutine <a name="CGGSVP.28"></a><a href="cggsvp.f.html#CGGSVP.1">CGGSVP</a>
</span><span class="comment">*</span><span class="comment"> from a general M-by-N matrix A and P-by-N matrix B:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N-K-L K L
</span><span class="comment">*</span><span class="comment"> A = K ( 0 A12 A13 ) if M-K-L >= 0;
</span><span class="comment">*</span><span class="comment"> L ( 0 0 A23 )
</span><span class="comment">*</span><span class="comment"> M-K-L ( 0 0 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N-K-L K L
</span><span class="comment">*</span><span class="comment"> A = K ( 0 A12 A13 ) if M-K-L < 0;
</span><span class="comment">*</span><span class="comment"> M-K ( 0 0 A23 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N-K-L K L
</span><span class="comment">*</span><span class="comment"> B = L ( 0 0 B13 )
</span><span class="comment">*</span><span class="comment"> P-L ( 0 0 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
</span><span class="comment">*</span><span class="comment"> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
</span><span class="comment">*</span><span class="comment"> otherwise A23 is (M-K)-by-L upper trapezoidal.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where U, V and Q are unitary matrices, Z' denotes the conjugate
</span><span class="comment">*</span><span class="comment"> transpose of Z, R is a nonsingular upper triangular matrix, and D1
</span><span class="comment">*</span><span class="comment"> and D2 are ``diagonal'' matrices, which are of the following
</span><span class="comment">*</span><span class="comment"> structures:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If M-K-L >= 0,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K L
</span><span class="comment">*</span><span class="comment"> D1 = K ( I 0 )
</span><span class="comment">*</span><span class="comment"> L ( 0 C )
</span><span class="comment">*</span><span class="comment"> M-K-L ( 0 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K L
</span><span class="comment">*</span><span class="comment"> D2 = L ( 0 S )
</span><span class="comment">*</span><span class="comment"> P-L ( 0 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N-K-L K L
</span><span class="comment">*</span><span class="comment"> ( 0 R ) = K ( 0 R11 R12 ) K
</span><span class="comment">*</span><span class="comment"> L ( 0 0 R22 ) L
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
</span><span class="comment">*</span><span class="comment"> S = diag( BETA(K+1), ... , BETA(K+L) ),
</span><span class="comment">*</span><span class="comment"> C**2 + S**2 = I.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> R is stored in A(1:K+L,N-K-L+1:N) on exit.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If M-K-L < 0,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K M-K K+L-M
</span><span class="comment">*</span><span class="comment"> D1 = K ( I 0 0 )
</span><span class="comment">*</span><span class="comment"> M-K ( 0 C 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K M-K K+L-M
</span><span class="comment">*</span><span class="comment"> D2 = M-K ( 0 S 0 )
</span><span class="comment">*</span><span class="comment"> K+L-M ( 0 0 I )
</span><span class="comment">*</span><span class="comment"> P-L ( 0 0 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N-K-L K M-K K+L-M
</span><span class="comment">*</span><span class="comment"> ( 0 R ) = K ( 0 R11 R12 R13 )
</span><span class="comment">*</span><span class="comment"> M-K ( 0 0 R22 R23 )
</span><span class="comment">*</span><span class="comment"> K+L-M ( 0 0 0 R33 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where
</span><span class="comment">*</span><span class="comment"> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
</span><span class="comment">*</span><span class="comment"> S = diag( BETA(K+1), ... , BETA(M) ),
</span><span class="comment">*</span><span class="comment"> C**2 + S**2 = I.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
</span><span class="comment">*</span><span class="comment"> ( 0 R22 R23 )
</span><span class="comment">*</span><span class="comment"> in B(M-K+1:L,N+M-K-L+1:N) on exit.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The computation of the unitary transformation matrices U, V or Q
</span><span class="comment">*</span><span class="comment"> is optional. These matrices may either be formed explicitly, or they
</span><span class="comment">*</span><span class="comment"> may be postmultiplied into input matrices U1, V1, or Q1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBU (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'U': U must contain a unitary matrix U1 on entry, and
</span><span class="comment">*</span><span class="comment"> the product U1*U is returned;
</span><span class="comment">*</span><span class="comment"> = 'I': U is initialized to the unit matrix, and the
</span><span class="comment">*</span><span class="comment"> unitary matrix U is returned;
</span><span class="comment">*</span><span class="comment"> = 'N': U is not computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBV (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'V': V must contain a unitary matrix V1 on entry, and
</span><span class="comment">*</span><span class="comment"> the product V1*V is returned;
</span><span class="comment">*</span><span class="comment"> = 'I': V is initialized to the unit matrix, and the
</span><span class="comment">*</span><span class="comment"> unitary matrix V is returned;
</span><span class="comment">*</span><span class="comment"> = 'N': V is not computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBQ (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'Q': Q must contain a unitary matrix Q1 on entry, and
</span><span class="comment">*</span><span class="comment"> the product Q1*Q is returned;
</span><span class="comment">*</span><span class="comment"> = 'I': Q is initialized to the unit matrix, and the
</span><span class="comment">*</span><span class="comment"> unitary matrix Q is returned;
</span><span class="comment">*</span><span class="comment"> = 'N': Q is not computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> M (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of the matrix A. M >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> P (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of the matrix B. P >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of columns of the matrices A and B. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K (input) INTEGER
</span><span class="comment">*</span><span class="comment"> L (input) INTEGER
</span><span class="comment">*</span><span class="comment"> K and L specify the subblocks in the input matrices A and B:
</span><span class="comment">*</span><span class="comment"> A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
</span><span class="comment">*</span><span class="comment"> of A and B, whose GSVD is going to be computed by <a name="CTGSJA.146"></a><a href="ctgsja.f.html#CTGSJA.1">CTGSJA</a>.
</span><span class="comment">*</span><span class="comment"> See Further details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) COMPLEX array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment"> On entry, the M-by-N matrix A.
</span><span class="comment">*</span><span class="comment"> On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
</span><span class="comment">*</span><span class="comment"> matrix R or part of R. See Purpose for details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input/output) COMPLEX array, dimension (LDB,N)
</span><span class="comment">*</span><span class="comment"> On entry, the P-by-N matrix B.
</span><span class="comment">*</span><span class="comment"> On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
</span><span class="comment">*</span><span class="comment"> a part of R. See Purpose for details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array B. LDB >= max(1,P).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TOLA (input) REAL
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