516 lines
16 KiB
FortranFixed
516 lines
16 KiB
FortranFixed
SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
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$ LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
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$ Q, LDQ, WORK, NCYCLE, INFO )
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*
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* -- LAPACK routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER JOBQ, JOBU, JOBV
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INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
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$ NCYCLE, P
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DOUBLE PRECISION TOLA, TOLB
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ),
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$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
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$ V( LDV, * ), WORK( * )
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* ..
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*
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* Purpose
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* =======
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*
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* DTGSJA computes the generalized singular value decomposition (GSVD)
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* of two real upper triangular (or trapezoidal) matrices A and B.
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*
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* On entry, it is assumed that matrices A and B have the following
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* forms, which may be obtained by the preprocessing subroutine DGGSVP
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* from a general M-by-N matrix A and P-by-N matrix B:
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*
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* N-K-L K L
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* A = K ( 0 A12 A13 ) if M-K-L >= 0;
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* L ( 0 0 A23 )
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* M-K-L ( 0 0 0 )
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*
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* N-K-L K L
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* A = K ( 0 A12 A13 ) if M-K-L < 0;
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* M-K ( 0 0 A23 )
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*
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* N-K-L K L
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* B = L ( 0 0 B13 )
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* P-L ( 0 0 0 )
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*
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* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
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* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
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* otherwise A23 is (M-K)-by-L upper trapezoidal.
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*
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* On exit,
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*
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* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ),
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*
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* where U, V and Q are orthogonal matrices, Z' denotes the transpose
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* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
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* ``diagonal'' matrices, which are of the following structures:
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*
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* If M-K-L >= 0,
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*
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* K L
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* D1 = K ( I 0 )
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* L ( 0 C )
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* M-K-L ( 0 0 )
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*
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* K L
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* D2 = L ( 0 S )
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* P-L ( 0 0 )
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*
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* N-K-L K L
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* ( 0 R ) = K ( 0 R11 R12 ) K
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* L ( 0 0 R22 ) L
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*
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* where
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*
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* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
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* S = diag( BETA(K+1), ... , BETA(K+L) ),
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* C**2 + S**2 = I.
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*
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* R is stored in A(1:K+L,N-K-L+1:N) on exit.
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*
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* If M-K-L < 0,
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*
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* K M-K K+L-M
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* D1 = K ( I 0 0 )
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* M-K ( 0 C 0 )
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*
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* K M-K K+L-M
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* D2 = M-K ( 0 S 0 )
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* K+L-M ( 0 0 I )
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* P-L ( 0 0 0 )
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*
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* N-K-L K M-K K+L-M
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* ( 0 R ) = K ( 0 R11 R12 R13 )
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* M-K ( 0 0 R22 R23 )
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* K+L-M ( 0 0 0 R33 )
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*
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* where
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* C = diag( ALPHA(K+1), ... , ALPHA(M) ),
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* S = diag( BETA(K+1), ... , BETA(M) ),
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* C**2 + S**2 = I.
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*
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* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
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* ( 0 R22 R23 )
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* in B(M-K+1:L,N+M-K-L+1:N) on exit.
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*
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* The computation of the orthogonal transformation matrices U, V or Q
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* is optional. These matrices may either be formed explicitly, or they
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* may be postmultiplied into input matrices U1, V1, or Q1.
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*
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* Arguments
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* =========
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*
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* JOBU (input) CHARACTER*1
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* = 'U': U must contain an orthogonal matrix U1 on entry, and
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* the product U1*U is returned;
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* = 'I': U is initialized to the unit matrix, and the
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* orthogonal matrix U is returned;
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* = 'N': U is not computed.
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*
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* JOBV (input) CHARACTER*1
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* = 'V': V must contain an orthogonal matrix V1 on entry, and
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* the product V1*V is returned;
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* = 'I': V is initialized to the unit matrix, and the
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* orthogonal matrix V is returned;
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* = 'N': V is not computed.
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*
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* JOBQ (input) CHARACTER*1
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* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and
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* the product Q1*Q is returned;
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* = 'I': Q is initialized to the unit matrix, and the
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* orthogonal matrix Q is returned;
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* = 'N': Q is not computed.
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*
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* M (input) INTEGER
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* The number of rows of the matrix A. M >= 0.
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*
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* P (input) INTEGER
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* The number of rows of the matrix B. P >= 0.
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*
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* N (input) INTEGER
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* The number of columns of the matrices A and B. N >= 0.
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*
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* K (input) INTEGER
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* L (input) INTEGER
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* K and L specify the subblocks in the input matrices A and B:
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* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
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* of A and B, whose GSVD is going to be computed by DTGSJA.
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* See Further details.
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*
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
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* On entry, the M-by-N matrix A.
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* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
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* matrix R or part of R. See Purpose for details.
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,M).
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*
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* B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
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* On entry, the P-by-N matrix B.
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* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
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* a part of R. See Purpose for details.
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*
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* LDB (input) INTEGER
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* The leading dimension of the array B. LDB >= max(1,P).
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*
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* TOLA (input) DOUBLE PRECISION
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* TOLB (input) DOUBLE PRECISION
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* TOLA and TOLB are the convergence criteria for the Jacobi-
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* Kogbetliantz iteration procedure. Generally, they are the
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* same as used in the preprocessing step, say
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* TOLA = max(M,N)*norm(A)*MAZHEPS,
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* TOLB = max(P,N)*norm(B)*MAZHEPS.
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*
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* ALPHA (output) DOUBLE PRECISION array, dimension (N)
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* BETA (output) DOUBLE PRECISION array, dimension (N)
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* On exit, ALPHA and BETA contain the generalized singular
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* value pairs of A and B;
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* ALPHA(1:K) = 1,
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* BETA(1:K) = 0,
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* and if M-K-L >= 0,
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* ALPHA(K+1:K+L) = diag(C),
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* BETA(K+1:K+L) = diag(S),
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* or if M-K-L < 0,
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* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
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* BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
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* Furthermore, if K+L < N,
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* ALPHA(K+L+1:N) = 0 and
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* BETA(K+L+1:N) = 0.
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*
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* U (input/output) DOUBLE PRECISION array, dimension (LDU,M)
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* On entry, if JOBU = 'U', U must contain a matrix U1 (usually
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* the orthogonal matrix returned by DGGSVP).
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* On exit,
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* if JOBU = 'I', U contains the orthogonal matrix U;
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* if JOBU = 'U', U contains the product U1*U.
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* If JOBU = 'N', U is not referenced.
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*
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* LDU (input) INTEGER
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* The leading dimension of the array U. LDU >= max(1,M) if
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* JOBU = 'U'; LDU >= 1 otherwise.
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*
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* V (input/output) DOUBLE PRECISION array, dimension (LDV,P)
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* On entry, if JOBV = 'V', V must contain a matrix V1 (usually
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* the orthogonal matrix returned by DGGSVP).
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* On exit,
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* if JOBV = 'I', V contains the orthogonal matrix V;
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* if JOBV = 'V', V contains the product V1*V.
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* If JOBV = 'N', V is not referenced.
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*
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* LDV (input) INTEGER
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* The leading dimension of the array V. LDV >= max(1,P) if
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* JOBV = 'V'; LDV >= 1 otherwise.
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*
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* Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
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* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
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* the orthogonal matrix returned by DGGSVP).
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* On exit,
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* if JOBQ = 'I', Q contains the orthogonal matrix Q;
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* if JOBQ = 'Q', Q contains the product Q1*Q.
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* If JOBQ = 'N', Q is not referenced.
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*
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* LDQ (input) INTEGER
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* The leading dimension of the array Q. LDQ >= max(1,N) if
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* JOBQ = 'Q'; LDQ >= 1 otherwise.
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*
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* WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
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*
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* NCYCLE (output) INTEGER
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* The number of cycles required for convergence.
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -i, the i-th argument had an illegal value.
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* = 1: the procedure does not converge after MAXIT cycles.
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*
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* Internal Parameters
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* ===================
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*
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* MAXIT INTEGER
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* MAXIT specifies the total loops that the iterative procedure
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* may take. If after MAXIT cycles, the routine fails to
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* converge, we return INFO = 1.
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*
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* Further Details
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* ===============
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*
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* DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
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* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
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* matrix B13 to the form:
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*
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* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
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*
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* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
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* of Z. C1 and S1 are diagonal matrices satisfying
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*
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* C1**2 + S1**2 = I,
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*
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* and R1 is an L-by-L nonsingular upper triangular matrix.
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*
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* =====================================================================
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*
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* .. Parameters ..
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INTEGER MAXIT
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PARAMETER ( MAXIT = 40 )
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* ..
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* .. Local Scalars ..
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*
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LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
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INTEGER I, J, KCYCLE
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DOUBLE PRECISION A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
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$ GAMMA, RWK, SNQ, SNU, SNV, SSMIN
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL DCOPY, DLAGS2, DLAPLL, DLARTG, DLASET, DROT,
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$ DSCAL, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Decode and test the input parameters
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*
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INITU = LSAME( JOBU, 'I' )
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WANTU = INITU .OR. LSAME( JOBU, 'U' )
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*
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INITV = LSAME( JOBV, 'I' )
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WANTV = INITV .OR. LSAME( JOBV, 'V' )
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*
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INITQ = LSAME( JOBQ, 'I' )
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WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
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*
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INFO = 0
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IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
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INFO = -2
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ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
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INFO = -3
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ELSE IF( M.LT.0 ) THEN
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INFO = -4
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ELSE IF( P.LT.0 ) THEN
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INFO = -5
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ELSE IF( N.LT.0 ) THEN
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INFO = -6
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -10
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ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
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INFO = -12
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ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
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INFO = -18
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ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
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INFO = -20
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ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
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INFO = -22
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DTGSJA', -INFO )
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RETURN
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END IF
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*
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* Initialize U, V and Q, if necessary
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*
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IF( INITU )
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$ CALL DLASET( 'Full', M, M, ZERO, ONE, U, LDU )
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IF( INITV )
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$ CALL DLASET( 'Full', P, P, ZERO, ONE, V, LDV )
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IF( INITQ )
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$ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
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*
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* Loop until convergence
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*
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UPPER = .FALSE.
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DO 40 KCYCLE = 1, MAXIT
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*
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UPPER = .NOT.UPPER
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*
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DO 20 I = 1, L - 1
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DO 10 J = I + 1, L
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*
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A1 = ZERO
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A2 = ZERO
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A3 = ZERO
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IF( K+I.LE.M )
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$ A1 = A( K+I, N-L+I )
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IF( K+J.LE.M )
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$ A3 = A( K+J, N-L+J )
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*
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B1 = B( I, N-L+I )
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B3 = B( J, N-L+J )
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*
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IF( UPPER ) THEN
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IF( K+I.LE.M )
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$ A2 = A( K+I, N-L+J )
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B2 = B( I, N-L+J )
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ELSE
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IF( K+J.LE.M )
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$ A2 = A( K+J, N-L+I )
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B2 = B( J, N-L+I )
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END IF
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*
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CALL DLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
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$ CSV, SNV, CSQ, SNQ )
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*
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* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A
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*
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IF( K+J.LE.M )
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$ CALL DROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
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$ LDA, CSU, SNU )
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*
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* Update I-th and J-th rows of matrix B: V'*B
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*
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CALL DROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
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$ CSV, SNV )
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*
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* Update (N-L+I)-th and (N-L+J)-th columns of matrices
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* A and B: A*Q and B*Q
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*
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CALL DROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
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$ A( 1, N-L+I ), 1, CSQ, SNQ )
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*
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CALL DROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
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$ SNQ )
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*
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IF( UPPER ) THEN
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IF( K+I.LE.M )
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$ A( K+I, N-L+J ) = ZERO
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B( I, N-L+J ) = ZERO
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ELSE
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IF( K+J.LE.M )
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$ A( K+J, N-L+I ) = ZERO
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B( J, N-L+I ) = ZERO
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END IF
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*
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* Update orthogonal matrices U, V, Q, if desired.
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*
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IF( WANTU .AND. K+J.LE.M )
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$ CALL DROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
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$ SNU )
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*
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IF( WANTV )
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$ CALL DROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
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*
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IF( WANTQ )
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$ CALL DROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
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$ SNQ )
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*
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10 CONTINUE
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20 CONTINUE
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*
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IF( .NOT.UPPER ) THEN
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*
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* The matrices A13 and B13 were lower triangular at the start
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* of the cycle, and are now upper triangular.
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*
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* Convergence test: test the parallelism of the corresponding
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* rows of A and B.
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*
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ERROR = ZERO
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DO 30 I = 1, MIN( L, M-K )
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CALL DCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
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CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
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CALL DLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
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ERROR = MAX( ERROR, SSMIN )
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30 CONTINUE
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*
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IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
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$ GO TO 50
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END IF
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*
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* End of cycle loop
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*
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40 CONTINUE
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*
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* The algorithm has not converged after MAXIT cycles.
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*
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INFO = 1
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GO TO 100
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*
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50 CONTINUE
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*
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* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
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* Compute the generalized singular value pairs (ALPHA, BETA), and
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* set the triangular matrix R to array A.
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*
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DO 60 I = 1, K
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ALPHA( I ) = ONE
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BETA( I ) = ZERO
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60 CONTINUE
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*
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|
DO 70 I = 1, MIN( L, M-K )
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*
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A1 = A( K+I, N-L+I )
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B1 = B( I, N-L+I )
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*
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IF( A1.NE.ZERO ) THEN
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GAMMA = B1 / A1
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*
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|
* change sign if necessary
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|
*
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|
IF( GAMMA.LT.ZERO ) THEN
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|
CALL DSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
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IF( WANTV )
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$ CALL DSCAL( P, -ONE, V( 1, I ), 1 )
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END IF
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*
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|
CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
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$ RWK )
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|
*
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|
IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
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CALL DSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
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|
$ LDA )
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|
ELSE
|
|
CALL DSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
|
|
$ LDB )
|
|
CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
|
|
$ LDA )
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
ALPHA( K+I ) = ZERO
|
|
BETA( K+I ) = ONE
|
|
CALL DCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
|
|
$ LDA )
|
|
*
|
|
END IF
|
|
*
|
|
70 CONTINUE
|
|
*
|
|
* Post-assignment
|
|
*
|
|
DO 80 I = M + 1, K + L
|
|
ALPHA( I ) = ZERO
|
|
BETA( I ) = ONE
|
|
80 CONTINUE
|
|
*
|
|
IF( K+L.LT.N ) THEN
|
|
DO 90 I = K + L + 1, N
|
|
ALPHA( I ) = ZERO
|
|
BETA( I ) = ZERO
|
|
90 CONTINUE
|
|
END IF
|
|
*
|
|
100 CONTINUE
|
|
NCYCLE = KCYCLE
|
|
RETURN
|
|
*
|
|
* End of DTGSJA
|
|
*
|
|
END
|