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SUBROUTINE <a name="DGGSVP.1"></a><a href="dggsvp.f.html#DGGSVP.1">DGGSVP</a>( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
$ IWORK, TAU, WORK, 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, P
DOUBLE PRECISION TOLA, TOLB
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ TAU( * ), 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="DGGSVP.23"></a><a href="dggsvp.f.html#DGGSVP.1">DGGSVP</a> computes orthogonal matrices U, V and Q such that
</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"> U'*A*Q = 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"> = 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"> V'*B*Q = 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. K+L = the effective
</span><span class="comment">*</span><span class="comment"> numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the
</span><span class="comment">*</span><span class="comment"> transpose of Z.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This decomposition is the preprocessing step for computing the
</span><span class="comment">*</span><span class="comment"> Generalized Singular Value Decomposition (GSVD), see subroutine
</span><span class="comment">*</span><span class="comment"> <a name="DGGSVD.46"></a><a href="dggsvd.f.html#DGGSVD.1">DGGSVD</a>.
</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': Orthogonal matrix U is computed;
</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': Orthogonal matrix V is computed;
</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': Orthogonal matrix Q is computed;
</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"> A (input/output) DOUBLE PRECISION 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 contains the triangular (or trapezoidal) matrix
</span><span class="comment">*</span><span class="comment"> described in the Purpose section.
</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) DOUBLE PRECISION 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, B contains the triangular matrix described in
</span><span class="comment">*</span><span class="comment"> the Purpose section.
</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) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> TOLB (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> TOLA and TOLB are the thresholds to determine the effective
</span><span class="comment">*</span><span class="comment"> numerical rank of matrix B and a subblock of A. Generally,
</span><span class="comment">*</span><span class="comment"> they are set to
</span><span class="comment">*</span><span class="comment"> TOLA = MAX(M,N)*norm(A)*MAZHEPS,
</span><span class="comment">*</span><span class="comment"> TOLB = MAX(P,N)*norm(B)*MAZHEPS.
</span><span class="comment">*</span><span class="comment"> The size of TOLA and TOLB may affect the size of backward
</span><span class="comment">*</span><span class="comment"> errors of the decomposition.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K (output) INTEGER
</span><span class="comment">*</span><span class="comment"> L (output) INTEGER
</span><span class="comment">*</span><span class="comment"> On exit, K and L specify the dimension of the subblocks
</span><span class="comment">*</span><span class="comment"> described in Purpose.
</span><span class="comment">*</span><span class="comment"> K + L = effective numerical rank of (A',B')'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> U (output) DOUBLE PRECISION array, dimension (LDU,M)
</span><span class="comment">*</span><span class="comment"> If JOBU = 'U', U contains the orthogonal matrix U.
</span><span class="comment">*</span><span class="comment"> If JOBU = 'N', U is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDU (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array U. LDU >= max(1,M) if
</span><span class="comment">*</span><span class="comment"> JOBU = 'U'; LDU >= 1 otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> V (output) DOUBLE PRECISION array, dimension (LDV,M)
</span><span class="comment">*</span><span class="comment"> If JOBV = 'V', V contains the orthogonal matrix V.
</span><span class="comment">*</span><span class="comment"> If JOBV = 'N', V is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDV (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array V. LDV >= max(1,P) if
</span><span class="comment">*</span><span class="comment"> JOBV = 'V'; LDV >= 1 otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
</span><span class="comment">*</span><span class="comment"> If JOBQ = 'Q', Q contains the orthogonal matrix Q.
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