SUBROUTINE SLATRZ( M, N, L, A, LDA, TAU, WORK ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER L, LDA, M, N * .. * .. Array Arguments .. REAL A( LDA, * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * SLATRZ 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) REAL 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) REAL array, dimension (M) * The scalar factors of the elementary reflectors. * * WORK (workspace) REAL 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 .. REAL ZERO PARAMETER ( ZERO = 0.0E+0 ) * .. * .. Local Scalars .. INTEGER I * .. * .. External Subroutines .. EXTERNAL SLARFG, SLARZ * .. * .. 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 SLARFG( 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 SLARZ( '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 SLATRZ * END