210 lines
7.5 KiB
Fortran
210 lines
7.5 KiB
Fortran
SUBROUTINE MB04KD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
|
|
$ TAU, DWORK )
|
|
C
|
|
C SLICOT RELEASE 5.0.
|
|
C
|
|
C Copyright (c) 2002-2009 NICONET e.V.
|
|
C
|
|
C This program is free software: you can redistribute it and/or
|
|
C modify it under the terms of the GNU General Public License as
|
|
C published by the Free Software Foundation, either version 2 of
|
|
C the License, or (at your option) any later version.
|
|
C
|
|
C This program is distributed in the hope that it will be useful,
|
|
C but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
|
C GNU General Public License for more details.
|
|
C
|
|
C You should have received a copy of the GNU General Public License
|
|
C along with this program. If not, see
|
|
C <http://www.gnu.org/licenses/>.
|
|
C
|
|
C PURPOSE
|
|
C
|
|
C To calculate a QR factorization of the first block column and
|
|
C apply the orthogonal transformations (from the left) also to the
|
|
C second block column of a structured matrix, as follows
|
|
C _
|
|
C [ R 0 ] [ R C ]
|
|
C Q' * [ ] = [ ]
|
|
C [ A B ] [ 0 D ]
|
|
C _
|
|
C where R and R are upper triangular. The matrix A can be full or
|
|
C upper trapezoidal/triangular. The problem structure is exploited.
|
|
C This computation is useful, for instance, in combined measurement
|
|
C and time update of one iteration of the Kalman filter (square
|
|
C root information filter).
|
|
C
|
|
C ARGUMENTS
|
|
C
|
|
C Mode Parameters
|
|
C
|
|
C UPLO CHARACTER*1
|
|
C Indicates if the matrix A is or not triangular as follows:
|
|
C = 'U': Matrix A is upper trapezoidal/triangular;
|
|
C = 'F': Matrix A is full.
|
|
C
|
|
C Input/Output Parameters
|
|
C
|
|
C N (input) INTEGER _
|
|
C The order of the matrices R and R. N >= 0.
|
|
C
|
|
C M (input) INTEGER
|
|
C The number of columns of the matrices B, C and D. M >= 0.
|
|
C
|
|
C P (input) INTEGER
|
|
C The number of rows of the matrices A, B and D. P >= 0.
|
|
C
|
|
C R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
|
|
C On entry, the leading N-by-N upper triangular part of this
|
|
C array must contain the upper triangular matrix R.
|
|
C On exit, the leading N-by-N upper triangular part of this
|
|
C _
|
|
C array contains the upper triangular matrix R.
|
|
C The strict lower triangular part of this array is not
|
|
C referenced.
|
|
C
|
|
C LDR INTEGER
|
|
C The leading dimension of array R. LDR >= MAX(1,N).
|
|
C
|
|
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
|
|
C On entry, if UPLO = 'F', the leading P-by-N part of this
|
|
C array must contain the matrix A. If UPLO = 'U', the
|
|
C leading MIN(P,N)-by-N part of this array must contain the
|
|
C upper trapezoidal (upper triangular if P >= N) matrix A,
|
|
C and the elements below the diagonal are not referenced.
|
|
C On exit, the leading P-by-N part (upper trapezoidal or
|
|
C triangular, if UPLO = 'U') of this array contains the
|
|
C trailing components (the vectors v, see Method) of the
|
|
C elementary reflectors used in the factorization.
|
|
C
|
|
C LDA INTEGER
|
|
C The leading dimension of array A. LDA >= MAX(1,P).
|
|
C
|
|
C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
|
|
C On entry, the leading P-by-M part of this array must
|
|
C contain the matrix B.
|
|
C On exit, the leading P-by-M part of this array contains
|
|
C the computed matrix D.
|
|
C
|
|
C LDB INTEGER
|
|
C The leading dimension of array B. LDB >= MAX(1,P).
|
|
C
|
|
C C (output) DOUBLE PRECISION array, dimension (LDC,M)
|
|
C The leading N-by-M part of this array contains the
|
|
C computed matrix C.
|
|
C
|
|
C LDC INTEGER
|
|
C The leading dimension of array C. LDC >= MAX(1,N).
|
|
C
|
|
C TAU (output) DOUBLE PRECISION array, dimension (N)
|
|
C The scalar factors of the elementary reflectors used.
|
|
C
|
|
C Workspace
|
|
C
|
|
C DWORK DOUBLE PRECISION array, dimension (N)
|
|
C
|
|
C METHOD
|
|
C
|
|
C The routine uses N Householder transformations exploiting the zero
|
|
C pattern of the block matrix. A Householder matrix has the form
|
|
C
|
|
C ( 1 ),
|
|
C H = I - tau *u *u', u = ( v )
|
|
C i i i i i ( i)
|
|
C
|
|
C where v is a P-vector, if UPLO = 'F', or an min(i,P)-vector, if
|
|
C i
|
|
C UPLO = 'U'. The components of v are stored in the i-th column
|
|
C i
|
|
C of A, and tau is stored in TAU(i).
|
|
C i
|
|
C
|
|
C NUMERICAL ASPECTS
|
|
C
|
|
C The algorithm is backward stable.
|
|
C
|
|
C CONTRIBUTORS
|
|
C
|
|
C V. Sima, Katholieke Univ. Leuven, Belgium, Feb. 1997.
|
|
C
|
|
C REVISIONS
|
|
C
|
|
C -
|
|
C
|
|
C KEYWORDS
|
|
C
|
|
C Elementary reflector, QR factorization, orthogonal transformation.
|
|
C
|
|
C ******************************************************************
|
|
C
|
|
C .. Parameters ..
|
|
DOUBLE PRECISION ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
|
|
C .. Scalar Arguments ..
|
|
CHARACTER UPLO
|
|
INTEGER LDA, LDB, LDC, LDR, M, N, P
|
|
C .. Array Arguments ..
|
|
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
|
|
$ R(LDR,*), TAU(*)
|
|
C .. Local Scalars ..
|
|
LOGICAL LUPLO
|
|
INTEGER I, IM
|
|
C .. External Functions ..
|
|
LOGICAL LSAME
|
|
EXTERNAL LSAME
|
|
C .. External Subroutines ..
|
|
EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, DSCAL
|
|
C .. Intrinsic Functions ..
|
|
INTRINSIC MIN
|
|
C .. Executable Statements ..
|
|
C
|
|
IF( MIN( N, P ).EQ.0 )
|
|
$ RETURN
|
|
C
|
|
LUPLO = LSAME( UPLO, 'U' )
|
|
IM = P
|
|
C
|
|
DO 10 I = 1, N
|
|
C
|
|
C Annihilate the I-th column of A and apply the transformations
|
|
C to the entire block matrix, exploiting its structure.
|
|
C
|
|
IF( LUPLO ) IM = MIN( I, P )
|
|
CALL DLARFG( IM+1, R(I,I), A(1,I), 1, TAU(I) )
|
|
IF( TAU(I).NE.ZERO ) THEN
|
|
C
|
|
C [ R(I,I+1:N) 0 ]
|
|
C [ w C(I,:) ] := [ 1 v' ] * [ ]
|
|
C [ A(1:IM,I+1:N) B(1:IM,:) ]
|
|
C
|
|
IF( I.LT.N ) THEN
|
|
CALL DCOPY( N-I, R(I,I+1), LDR, DWORK, 1 )
|
|
CALL DGEMV( 'Transpose', IM, N-I, ONE, A(1,I+1), LDA,
|
|
$ A(1,I), 1, ONE, DWORK, 1 )
|
|
END IF
|
|
CALL DGEMV( 'Transpose', IM, M, ONE, B, LDB, A(1,I), 1,
|
|
$ ZERO, C(I,1), LDC )
|
|
C
|
|
C [ R(I,I+1:N) C(I,:) ] [ R(I,I+1:N) 0 ]
|
|
C [ ] := [ ]
|
|
C [ A(1:IM,I+1:N) D(1:IM,:) ] [ A(1:IM,I+1:N) B(1:IM,:) ]
|
|
C
|
|
C [ 1 ]
|
|
C - tau * [ ] * [ w C(I,:) ]
|
|
C [ v ]
|
|
C
|
|
IF( I.LT.N ) THEN
|
|
CALL DAXPY( N-I, -TAU(I), DWORK, 1, R(I,I+1), LDR )
|
|
CALL DGER( IM, N-I, -TAU(I), A(1,I), 1, DWORK, 1,
|
|
$ A(1,I+1), LDA )
|
|
END IF
|
|
CALL DSCAL( M, -TAU(I), C(I,1), LDC )
|
|
CALL DGER( IM, M, ONE, A(1,I), 1, C(I,1), LDC, B, LDB )
|
|
END IF
|
|
10 CONTINUE
|
|
C
|
|
RETURN
|
|
C *** Last line of MB04KD ***
|
|
END
|