403 lines
12 KiB
FortranFixed
403 lines
12 KiB
FortranFixed
SUBROUTINE ZGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB,
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$ TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ,
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$ IWORK, RWORK, TAU, WORK, 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, P
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DOUBLE PRECISION TOLA, TOLB
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* ..
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* .. Array Arguments ..
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INTEGER IWORK( * )
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DOUBLE PRECISION RWORK( * )
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COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
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$ TAU( * ), U( LDU, * ), 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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* ZGGSVP computes unitary matrices U, V and Q such that
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*
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* N-K-L K L
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* U'*A*Q = 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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* = 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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* V'*B*Q = 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. K+L = the effective
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* numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the
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* conjugate transpose of Z.
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*
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* This decomposition is the preprocessing step for computing the
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* Generalized Singular Value Decomposition (GSVD), see subroutine
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* ZGGSVD.
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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': Unitary matrix U is computed;
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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': Unitary matrix V is computed;
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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': Unitary matrix Q is computed;
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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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* A (input/output) COMPLEX*16 array, dimension (LDA,N)
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* On entry, the M-by-N matrix A.
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* On exit, A contains the triangular (or trapezoidal) matrix
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* described in the Purpose section.
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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) COMPLEX*16 array, dimension (LDB,N)
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* On entry, the P-by-N matrix B.
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* On exit, B contains the triangular matrix described in
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* the Purpose section.
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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 thresholds to determine the effective
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* numerical rank of matrix B and a subblock of A. Generally,
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* they are set to
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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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* The size of TOLA and TOLB may affect the size of backward
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* errors of the decomposition.
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*
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* K (output) INTEGER
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* L (output) INTEGER
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* On exit, K and L specify the dimension of the subblocks
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* described in Purpose section.
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* K + L = effective numerical rank of (A',B')'.
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*
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* U (output) COMPLEX*16 array, dimension (LDU,M)
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* If JOBU = 'U', U contains the unitary matrix 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 (output) COMPLEX*16 array, dimension (LDV,M)
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* If JOBV = 'V', V contains the unitary matrix 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 (output) COMPLEX*16 array, dimension (LDQ,N)
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* If JOBQ = 'Q', Q contains the unitary matrix 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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* IWORK (workspace) INTEGER array, dimension (N)
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*
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* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
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*
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* TAU (workspace) COMPLEX*16 array, dimension (N)
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*
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* WORK (workspace) COMPLEX*16 array, dimension (max(3*N,M,P))
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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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*
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* Further Details
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* ===============
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*
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* The subroutine uses LAPACK subroutine ZGEQPF for the QR factorization
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* with column pivoting to detect the effective numerical rank of the
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* a matrix. It may be replaced by a better rank determination strategy.
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*
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* =====================================================================
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*
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* .. Parameters ..
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COMPLEX*16 CZERO, CONE
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PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
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$ CONE = ( 1.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL FORWRD, WANTQ, WANTU, WANTV
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INTEGER I, J
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COMPLEX*16 T
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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 XERBLA, ZGEQPF, ZGEQR2, ZGERQ2, ZLACPY, ZLAPMT,
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$ ZLASET, ZUNG2R, ZUNM2R, ZUNMR2
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
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* ..
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* .. Statement Functions ..
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DOUBLE PRECISION CABS1
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* ..
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* .. Statement Function definitions ..
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CABS1( T ) = ABS( DBLE( T ) ) + ABS( DIMAG( T ) )
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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WANTU = LSAME( JOBU, 'U' )
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WANTV = LSAME( JOBV, 'V' )
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WANTQ = LSAME( JOBQ, 'Q' )
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FORWRD = .TRUE.
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*
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INFO = 0
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IF( .NOT.( WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
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INFO = -2
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ELSE IF( .NOT.( 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 = -8
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ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
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INFO = -10
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ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
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INFO = -16
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ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
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INFO = -18
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ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
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INFO = -20
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZGGSVP', -INFO )
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RETURN
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END IF
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*
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* QR with column pivoting of B: B*P = V*( S11 S12 )
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* ( 0 0 )
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*
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DO 10 I = 1, N
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IWORK( I ) = 0
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10 CONTINUE
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CALL ZGEQPF( P, N, B, LDB, IWORK, TAU, WORK, RWORK, INFO )
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*
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* Update A := A*P
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*
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CALL ZLAPMT( FORWRD, M, N, A, LDA, IWORK )
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*
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* Determine the effective rank of matrix B.
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*
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L = 0
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DO 20 I = 1, MIN( P, N )
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IF( CABS1( B( I, I ) ).GT.TOLB )
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$ L = L + 1
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20 CONTINUE
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*
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IF( WANTV ) THEN
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*
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* Copy the details of V, and form V.
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*
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CALL ZLASET( 'Full', P, P, CZERO, CZERO, V, LDV )
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IF( P.GT.1 )
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$ CALL ZLACPY( 'Lower', P-1, N, B( 2, 1 ), LDB, V( 2, 1 ),
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$ LDV )
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CALL ZUNG2R( P, P, MIN( P, N ), V, LDV, TAU, WORK, INFO )
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END IF
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*
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* Clean up B
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*
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DO 40 J = 1, L - 1
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DO 30 I = J + 1, L
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B( I, J ) = CZERO
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30 CONTINUE
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40 CONTINUE
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IF( P.GT.L )
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$ CALL ZLASET( 'Full', P-L, N, CZERO, CZERO, B( L+1, 1 ), LDB )
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*
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IF( WANTQ ) THEN
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*
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* Set Q = I and Update Q := Q*P
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*
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CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
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CALL ZLAPMT( FORWRD, N, N, Q, LDQ, IWORK )
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END IF
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*
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IF( P.GE.L .AND. N.NE.L ) THEN
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*
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* RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z
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*
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CALL ZGERQ2( L, N, B, LDB, TAU, WORK, INFO )
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*
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* Update A := A*Z'
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*
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CALL ZUNMR2( 'Right', 'Conjugate transpose', M, N, L, B, LDB,
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$ TAU, A, LDA, WORK, INFO )
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IF( WANTQ ) THEN
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*
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* Update Q := Q*Z'
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*
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CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N, L, B,
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$ LDB, TAU, Q, LDQ, WORK, INFO )
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END IF
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*
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* Clean up B
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*
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CALL ZLASET( 'Full', L, N-L, CZERO, CZERO, B, LDB )
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DO 60 J = N - L + 1, N
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DO 50 I = J - N + L + 1, L
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B( I, J ) = CZERO
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50 CONTINUE
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60 CONTINUE
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*
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END IF
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*
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* Let N-L L
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* A = ( A11 A12 ) M,
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*
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* then the following does the complete QR decomposition of A11:
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*
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* A11 = U*( 0 T12 )*P1'
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* ( 0 0 )
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*
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DO 70 I = 1, N - L
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IWORK( I ) = 0
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70 CONTINUE
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CALL ZGEQPF( M, N-L, A, LDA, IWORK, TAU, WORK, RWORK, INFO )
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*
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* Determine the effective rank of A11
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*
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K = 0
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DO 80 I = 1, MIN( M, N-L )
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IF( CABS1( A( I, I ) ).GT.TOLA )
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$ K = K + 1
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80 CONTINUE
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*
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* Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N )
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*
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CALL ZUNM2R( 'Left', 'Conjugate transpose', M, L, MIN( M, N-L ),
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$ A, LDA, TAU, A( 1, N-L+1 ), LDA, WORK, INFO )
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*
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IF( WANTU ) THEN
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*
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* Copy the details of U, and form U
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*
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CALL ZLASET( 'Full', M, M, CZERO, CZERO, U, LDU )
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IF( M.GT.1 )
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$ CALL ZLACPY( 'Lower', M-1, N-L, A( 2, 1 ), LDA, U( 2, 1 ),
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$ LDU )
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CALL ZUNG2R( M, M, MIN( M, N-L ), U, LDU, TAU, WORK, INFO )
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END IF
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*
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IF( WANTQ ) THEN
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*
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* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1
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*
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CALL ZLAPMT( FORWRD, N, N-L, Q, LDQ, IWORK )
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END IF
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*
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* Clean up A: set the strictly lower triangular part of
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* A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0.
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*
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DO 100 J = 1, K - 1
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DO 90 I = J + 1, K
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A( I, J ) = CZERO
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90 CONTINUE
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100 CONTINUE
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IF( M.GT.K )
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$ CALL ZLASET( 'Full', M-K, N-L, CZERO, CZERO, A( K+1, 1 ), LDA )
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*
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IF( N-L.GT.K ) THEN
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*
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* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1
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*
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CALL ZGERQ2( K, N-L, A, LDA, TAU, WORK, INFO )
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*
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IF( WANTQ ) THEN
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*
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* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1'
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*
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CALL ZUNMR2( 'Right', 'Conjugate transpose', N, N-L, K, A,
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$ LDA, TAU, Q, LDQ, WORK, INFO )
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END IF
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*
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* Clean up A
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*
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CALL ZLASET( 'Full', K, N-L-K, CZERO, CZERO, A, LDA )
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DO 120 J = N - L - K + 1, N - L
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DO 110 I = J - N + L + K + 1, K
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A( I, J ) = CZERO
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110 CONTINUE
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120 CONTINUE
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*
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END IF
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*
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IF( M.GT.K ) THEN
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*
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* QR factorization of A( K+1:M,N-L+1:N )
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*
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CALL ZGEQR2( M-K, L, A( K+1, N-L+1 ), LDA, TAU, WORK, INFO )
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*
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IF( WANTU ) THEN
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*
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* Update U(:,K+1:M) := U(:,K+1:M)*U1
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*
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CALL ZUNM2R( 'Right', 'No transpose', M, M-K, MIN( M-K, L ),
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$ A( K+1, N-L+1 ), LDA, TAU, U( 1, K+1 ), LDU,
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$ WORK, INFO )
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END IF
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*
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* Clean up
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*
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DO 140 J = N - L + 1, N
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DO 130 I = J - N + K + L + 1, M
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A( I, J ) = CZERO
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130 CONTINUE
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140 CONTINUE
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*
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END IF
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*
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RETURN
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*
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* End of ZGGSVP
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*
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END
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