SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
*
* -- LAPACK routine (version 3.2.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2010
*
* .. Scalar Arguments ..
INTEGER L, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
* [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
* of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
* matrix and, R and A1 are M-by-M upper triangular matrices.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* L (input) INTEGER
* The number of columns of the matrix A containing the
* meaningful part of the Householder vectors. N-M >= L >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the leading M-by-N upper trapezoidal part of the
* array A must contain the matrix to be factorized.
* On exit, the leading M-by-M upper triangular part of A
* contains the upper triangular matrix R, and elements N-L+1 to
* N of the first M rows of A, with the array TAU, represent the
* orthogonal matrix Z as a product of M elementary reflectors.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* TAU (output) DOUBLE PRECISION array, dimension (M)
* The scalar factors of the elementary reflectors.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (M)
*
* Further Details
* ===============
*
* Based on contributions by
* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
* The factorization is obtained by Householder's method. The kth
* transformation matrix, Z( k ), which is used to introduce zeros into
* the ( m - k + 1 )th row of A, is given in the form
*
* Z( k ) = ( I 0 ),
* ( 0 T( k ) )
*
* where
*
* T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
* ( 0 )
* ( z( k ) )
*
* tau is a scalar and z( k ) is an l element vector. tau and z( k )
* are chosen to annihilate the elements of the kth row of A2.
*
* The scalar tau is returned in the kth element of TAU and the vector
* u( k ) in the kth row of A2, such that the elements of z( k ) are
* in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
* the upper triangular part of A1.
*
* Z is given by
*
* Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL DLARFG, DLARZ
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
* Quick return if possible
*
IF( M.EQ.0 ) THEN
RETURN
ELSE IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
RETURN
END IF
*
DO 20 I = M, 1, -1
*
* Generate elementary reflector H(i) to annihilate
* [ A(i,i) A(i,n-l+1:n) ]
*
CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
*
* Apply H(i) to A(1:i-1,i:n) from the right
*
CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
$ TAU( I ), A( 1, I ), LDA, WORK )
*
20 CONTINUE
*
RETURN
*
* End of DLATRZ
*
END