SUBROUTINE MB04KD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
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