258 lines
8.8 KiB
Fortran
258 lines
8.8 KiB
Fortran
SUBROUTINE MB04OD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
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$ TAU, DWORK )
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C
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C SLICOT RELEASE 5.0.
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C
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C Copyright (c) 2002-2009 NICONET e.V.
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C
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C This program is free software: you can redistribute it and/or
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C modify it under the terms of the GNU General Public License as
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C published by the Free Software Foundation, either version 2 of
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C the License, or (at your option) any later version.
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C
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C This program is distributed in the hope that it will be useful,
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C but WITHOUT ANY WARRANTY; without even the implied warranty of
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C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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C GNU General Public License for more details.
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C
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C You should have received a copy of the GNU General Public License
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C along with this program. If not, see
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C <http://www.gnu.org/licenses/>.
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C
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C PURPOSE
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C
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C To calculate a QR factorization of the first block column and
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C apply the orthogonal transformations (from the left) also to the
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C second block column of a structured matrix, as follows
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C _ _
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C [ R B ] [ R B ]
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C Q' * [ ] = [ _ ]
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C [ A C ] [ 0 C ]
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C _
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C where R and R are upper triangular. The matrix A can be full or
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C upper trapezoidal/triangular. The problem structure is exploited.
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C
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C ARGUMENTS
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C
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C Mode Parameters
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C
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C UPLO CHARACTER*1
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C Indicates if the matrix A is or not triangular as follows:
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C = 'U': Matrix A is upper trapezoidal/triangular;
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C = 'F': Matrix A is full.
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C
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C Input/Output Parameters
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C
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C N (input) INTEGER _
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C The order of the matrices R and R. N >= 0.
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C
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C M (input) INTEGER
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C The number of columns of the matrices B and C. M >= 0.
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C
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C P (input) INTEGER
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C The number of rows of the matrices A and C. P >= 0.
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C
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C R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
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C On entry, the leading N-by-N upper triangular part of this
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C array must contain the upper triangular matrix R.
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C On exit, the leading N-by-N upper triangular part of this
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C _
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C array contains the upper triangular matrix R.
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C The strict lower triangular part of this array is not
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C referenced.
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C
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C LDR INTEGER
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C The leading dimension of array R. LDR >= MAX(1,N).
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C
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C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
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C On entry, if UPLO = 'F', the leading P-by-N part of this
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C array must contain the matrix A. If UPLO = 'U', the
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C leading MIN(P,N)-by-N part of this array must contain the
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C upper trapezoidal (upper triangular if P >= N) matrix A,
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C and the elements below the diagonal are not referenced.
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C On exit, the leading P-by-N part (upper trapezoidal or
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C triangular, if UPLO = 'U') of this array contains the
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C trailing components (the vectors v, see Method) of the
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C elementary reflectors used in the factorization.
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C
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C LDA INTEGER
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C The leading dimension of array A. LDA >= MAX(1,P).
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C
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C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
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C On entry, the leading N-by-M part of this array must
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C contain the matrix B.
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C On exit, the leading N-by-M part of this array contains
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C _
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C the computed matrix B.
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C
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C LDB INTEGER
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C The leading dimension of array B. LDB >= MAX(1,N).
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C
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C C (input/output) DOUBLE PRECISION array, dimension (LDC,M)
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C On entry, the leading P-by-M part of this array must
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C contain the matrix C.
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C On exit, the leading P-by-M part of this array contains
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C _
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C the computed matrix C.
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C
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C LDC INTEGER
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C The leading dimension of array C. LDC >= MAX(1,P).
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C
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C TAU (output) DOUBLE PRECISION array, dimension (N)
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C The scalar factors of the elementary reflectors used.
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C
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C Workspace
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C
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C DWORK DOUBLE PRECISION array, dimension (MAX(N-1,M))
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C
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C METHOD
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C
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C The routine uses N Householder transformations exploiting the zero
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C pattern of the block matrix. A Householder matrix has the form
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C
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C ( 1 )
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C H = I - tau *u *u', u = ( v ),
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C i i i i i ( i)
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C
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C where v is a P-vector, if UPLO = 'F', or a min(i,P)-vector, if
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C i
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C UPLO = 'U'. The components of v are stored in the i-th column
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C i
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C of A, and tau is stored in TAU(i).
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C i
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C In-line code for applying Householder transformations is used
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C whenever possible (see MB04OY routine).
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C
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C NUMERICAL ASPECTS
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C
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C The algorithm is backward stable.
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C
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C CONTRIBUTORS
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C
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C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1997.
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C
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C REVISIONS
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C
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C Dec. 1997.
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C
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C KEYWORDS
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C
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C Elementary reflector, QR factorization, orthogonal transformation.
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C
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C ******************************************************************
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C
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C .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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C .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER LDA, LDB, LDC, LDR, M, N, P
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C .. Array Arguments ..
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DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
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$ R(LDR,*), TAU(*)
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C .. Local Scalars ..
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LOGICAL LUPLO
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INTEGER I, IM
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C .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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C .. External Subroutines ..
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EXTERNAL DLARFG, MB04OY
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C .. Intrinsic Functions ..
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INTRINSIC MIN
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C .. Executable Statements ..
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C
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C For efficiency reasons, the parameters are not checked.
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C
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IF( MIN( N, P ).EQ.0 )
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$ RETURN
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C
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LUPLO = LSAME( UPLO, 'U' )
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IF ( LUPLO ) THEN
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C
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DO 10 I = 1, N
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C
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C Annihilate the I-th column of A and apply the
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C transformations to the entire block matrix, exploiting
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C its structure.
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C
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IM = MIN( I, P )
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CALL DLARFG( IM+1, R(I,I), A(1,I), 1, TAU(I) )
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C
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C Compute
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C [ R(I,I+1:N) ]
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C w := [ 1 v' ] * [ ],
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C [ A(1:IM,I+1:N) ]
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C
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C [ R(I,I+1:N) ] [ R(I,I+1:N) ] [ 1 ]
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C [ ] := [ ] - tau * [ ] * w .
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C [ A(1:IM,I+1:N) ] [ A(1:IM,I+1:N) ] [ v ]
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C
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IF ( N-I.GT.0 )
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$ CALL MB04OY( IM, N-I, A(1,I), TAU(I), R(I,I+1), LDR,
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$ A(1,I+1), LDA, DWORK )
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C
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C Compute
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C [ B(I,:) ]
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C w := [ 1 v' ] * [ ],
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C [ C(1:IM,:) ]
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C
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C [ B(I,:) ] [ B(I,:) ] [ 1 ]
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C [ ] := [ ] - tau * [ ] * w.
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C [ C(1:IM,:) ] [ C(1:IM,:) ] [ v ]
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C
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C
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IF ( M.GT.0 )
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$ CALL MB04OY( IM, M, A(1,I), TAU(I), B(I,1), LDB, C, LDC,
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$ DWORK )
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10 CONTINUE
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C
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ELSE
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C
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DO 20 I = 1, N - 1
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C
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C Annihilate the I-th column of A and apply the
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C transformations to the first block column, exploiting its
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C structure.
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C
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CALL DLARFG( P+1, R(I,I), A(1,I), 1, TAU(I) )
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C
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C Compute
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C [ R(I,I+1:N) ]
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C w := [ 1 v' ] * [ ],
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C [ A(:,I+1:N) ]
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C
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C [ R(I,I+1:N) ] [ R(I,I+1:N) ] [ 1 ]
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C [ ] := [ ] - tau * [ ] * w .
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C [ A(:,I+1:N) ] [ A(:,I+1:N) ] [ v ]
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C
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CALL MB04OY( P, N-I, A(1,I), TAU(I), R(I,I+1), LDR,
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$ A(1,I+1), LDA, DWORK )
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20 CONTINUE
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C
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CALL DLARFG( P+1, R(N,N), A(1,N), 1, TAU(N) )
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IF ( M.GT.0 ) THEN
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C
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C Apply the transformations to the second block column.
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C
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DO 30 I = 1, N
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C
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C Compute
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C [ B(I,:) ]
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C w := [ 1 v' ] * [ ],
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C [ C ]
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C
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C [ B(I,:) ] [ B(I,:) ] [ 1 ]
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C [ ] := [ ] - tau * [ ] * w.
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C [ C ] [ C ] [ v ]
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C
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CALL MB04OY( P, M, A(1,I), TAU(I), B(I,1), LDB, C, LDC,
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$ DWORK )
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30 CONTINUE
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C
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END IF
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END IF
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RETURN
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C *** Last line of MB04OD ***
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END
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