SUBROUTINE MB04ND( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
$ TAU, DWORK )
C
C PURPOSE
C
C To calculate an RQ factorization of the first block row and
C apply the orthogonal transformations (from the right) also to the
C second block row of a structured matrix, as follows
C _
C [ A R ] [ 0 R ]
C [ ] * Q' = [ _ _ ]
C [ C B ] [ C B ]
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
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 rows of the matrices B and C. M >= 0.
C
C P (input) INTEGER
C The number of columns of the matrices A and C. 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,P)
C On entry, if UPLO = 'F', the leading N-by-P part of this
C array must contain the matrix A. For UPLO = 'U', if
C N <= P, the upper triangle of the subarray A(1:N,P-N+1:P)
C must contain the N-by-N upper triangular matrix A, and if
C N >= P, the elements on and above the (N-P)-th subdiagonal
C must contain the N-by-P upper trapezoidal matrix A.
C On exit, if UPLO = 'F', the leading N-by-P part of this
C array contains the trailing components (the vectors v, see
C METHOD) of the elementary reflectors used in the
C factorization. If UPLO = 'U', the upper triangle of the
C subarray A(1:N,P-N+1:P) (if N <= P), or the elements on
C and above the (N-P)-th subdiagonal (if N >= P), contain
C the trailing components (the vectors v, see METHOD) of the
C elementary reflectors used in the factorization.
C The remaining elements are not referenced.
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,N)
C On entry, the leading M-by-N part of this array must
C contain the matrix B.
C On exit, the leading M-by-N part of this array contains
C _
C the computed matrix B.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,M).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,P)
C On entry, the leading M-by-P part of this array must
C contain the matrix C.
C On exit, the leading M-by-P part of this array contains
C _
C the computed matrix C.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,M).
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 (MAX(N-1,M))
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 a min(N-i+1,P)-vector,
C i
C if UPLO = 'U'. The components of v are stored in the i-th row
C i
C of A, and tau is stored in TAU(i), i = N,N-1,...,1.
C i
C In-line code for applying Householder transformations is used
C whenever possible (see MB04NY routine).
C
C NUMERICAL ASPECTS
C
C The algorithm is backward stable.
C
C CONTRIBUTORS
C
C V. Sima, Katholieke Univ. Leuven, Belgium, Apr. 1998.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Elementary reflector, RQ factorization, orthogonal transformation.
C
C ******************************************************************
C
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, IP
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DLARFG, MB04NY
C .. Intrinsic Functions ..
INTRINSIC MIN
C .. Executable Statements ..
C
C For efficiency reasons, the parameters are not checked.
C
IF( MIN( N, P ).EQ.0 )
$ RETURN
C
LUPLO = LSAME( UPLO, 'U' )
IF ( LUPLO ) THEN
C
DO 10 I = N, 1, -1
C
C Annihilate the I-th row of A and apply the transformations
C to the entire block matrix, exploiting its structure.
C
IM = MIN( N-I+1, P )
IP = MAX( P-N+I, 1 )
CALL DLARFG( IM+1, R(I,I), A(I,IP), LDA, TAU(I) )
C
C Compute
C [ 1 ]
C w := [ R(1:I-1,I) A(1:I-1,IP:P) ] * [ ],
C [ v ]
C
C [ R(1:I-1,I) A(1:I-1,IP:P) ] =
C [ R(1:I-1,I) A(1:I-1,IP:P) ] - tau * w * [ 1 v' ].
C
IF ( I.GT.0 )
C
$ CALL MB04NY( I-1, IM, A(I,IP), LDA, TAU(I), R(1,I), LDR,
$ A(1,IP), LDA, DWORK )
C
C Compute
C [ 1 ]
C w := [ B(:,I) C(:,IP:P) ] * [ ],
C [ v ]
C
C [ B(:,I) C(:,IP:P) ] = [ B(:,I) C(:,IP:P) ] -
C tau * w * [ 1 v' ].
C
IF ( M.GT.0 )
$ CALL MB04NY( M, IM, A(I,IP), LDA, TAU(I), B(1,I), LDB,
$ C(1,IP), LDC, DWORK )
10 CONTINUE
C
ELSE
C
DO 20 I = N, 2 , -1
C
C Annihilate the I-th row of A and apply the transformations
C to the first block row, exploiting its structure.
C
CALL DLARFG( P+1, R(I,I), A(I,1), LDA, TAU(I) )
C
C Compute
C [ 1 ]
C w := [ R(1:I-1,I) A(1:I-1,:) ] * [ ],
C [ v ]
C
C [ R(1:I-1,I) A(1:I-1,:) ] = [ R(1:I-1,I) A(1:I-1,:) ] -
C tau * w * [ 1 v' ].
C
CALL MB04NY( I-1, P, A(I,1), LDA, TAU(I), R(1,I), LDR, A,
$ LDA, DWORK )
20 CONTINUE
C
CALL DLARFG( P+1, R(1,1), A(1,1), LDA, TAU(1) )
IF ( M.GT.0 ) THEN
C
C Apply the transformations to the second block row.
C
DO 30 I = N, 1, -1
C
C Compute
C [ 1 ]
C w := [ B(:,I) C ] * [ ],
C [ v ]
C
C [ B(:,I) C ] = [ B(:,I) C ] - tau * w * [ 1 v' ].
C
CALL MB04NY( M, P, A(I,1), LDA, TAU(I), B(1,I), LDB, C,
$ LDC, DWORK )
30 CONTINUE
C
END IF
END IF
RETURN
C *** Last line of MB04ND ***
END