SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C
C PURPOSE
C
C To calculate an LQ factorization of the first block row and apply
C the orthogonal transformations (from the right) also to the second
C block row of a structured matrix, as follows
C _
C [ L A ] [ L 0 ]
C [ ]*Q = [ ]
C [ 0 B ] [ C D ]
C _
C where L and L are lower triangular. The matrix A can be full or
C lower 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 covariance 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 = 'L': Matrix A is lower trapezoidal/triangular;
C = 'F': Matrix A is full.
C
C Input/Output Parameters
C
C N (input) INTEGER _
C The order of the matrices L and L. N >= 0.
C
C M (input) INTEGER
C The number of columns of the matrices A, B and D. M >= 0.
C
C P (input) INTEGER
C The number of rows of the matrices B, C and D. P >= 0.
C
C L (input/output) DOUBLE PRECISION array, dimension (LDL,N)
C On entry, the leading N-by-N lower triangular part of this
C array must contain the lower triangular matrix L.
C On exit, the leading N-by-N lower triangular part of this
C _
C array contains the lower triangular matrix L.
C The strict upper triangular part of this array is not
C referenced.
C
C LDL INTEGER
C The leading dimension of array L. LDL >= MAX(1,N).
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,M)
C On entry, if UPLO = 'F', the leading N-by-M part of this
C array must contain the matrix A. If UPLO = 'L', the
C leading N-by-MIN(N,M) part of this array must contain the
C lower trapezoidal (lower triangular if N <= M) matrix A,
C and the elements above the diagonal are not referenced.
C On exit, the leading N-by-M part (lower trapezoidal or
C triangular, if UPLO = 'L') 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,N).
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,N)
C The leading P-by-N part of this array contains the
C computed matrix C.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,P).
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 an M-vector, if UPLO = 'F', or an min(i,M)-vector, if
C i
C UPLO = 'L'. The components of v are stored in the i-th row of A,
C i
C 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, LQ 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, LDL, M, N, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
$ L(LDL,*), 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( M, N ).EQ.0 )
$ RETURN
C
LUPLO = LSAME( UPLO, 'L' )
IM = M
C
DO 10 I = 1, N
C
C Annihilate the I-th row of A and apply the transformations to
C the entire block matrix, exploiting its structure.
C
IF( LUPLO ) IM = MIN( I, M )
CALL DLARFG( IM+1, L(I,I), A(I,1), LDA, TAU(I) )
IF( TAU(I).NE.ZERO ) THEN
C
C [ w ] [ L(I+1:N,I) A(I+1:N,1:IM) ] [ 1 ]
C [ ] := [ ] * [ ]
C [ C(:,I) ] [ 0 B(:,1:IM) ] [ v ]
C
IF( I.LT.N ) THEN
CALL DCOPY( N-I, L(I+1,I), 1, DWORK, 1 )
CALL DGEMV( 'No transpose', N-I, IM, ONE, A(I+1,1), LDA,
$ A(I,1), LDA, ONE, DWORK, 1 )
END IF
CALL DGEMV( 'No transpose', P, IM, ONE, B, LDB, A(I,1),
$ LDA, ZERO, C(1,I), 1 )
C
C [ L(I+1:N,I) A(I+1:N,1:IM) ] [ L(I+1:N,I) A(I+1:N,1:IM) ]
C [ ] := [ ]
C [ C(:,I) D(:,1:IM) ] [ 0 B(:,1:IM) ]
C
C [ w ]
C - tau * [ ] * [ 1 , v']
C [ C(:,I) ]
C
IF( I.LT.N ) THEN
CALL DAXPY( N-I, -TAU(I), DWORK, 1, L(I+1,I), 1 )
CALL DGER( N-I, IM, -TAU(I), DWORK, 1, A(I,1), LDA,
$ A(I+1,1), LDA )
END IF
CALL DSCAL( P, -TAU(I), C(1,I), 1 )
CALL DGER( P, IM, ONE, C(1,I), 1, A(I,1), LDA, B, LDB )
END IF
10 CONTINUE
C
RETURN
C *** Last line of MB04LD ***
END