*> \brief \b SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLATRZ + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLATRZ( M, N, L, A, LDA, TAU, WORK ) * * .. Scalar Arguments .. * INTEGER L, LDA, M, N * .. * .. Array Arguments .. * REAL A( LDA, * ), TAU( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] L *> \verbatim *> L is INTEGER *> The number of columns of the matrix A containing the *> meaningful part of the Householder vectors. N-M >= L >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is REAL array, dimension (M) *> The scalar factors of the elementary reflectors. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (M) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHERcomputational * *> \par Contributors: * ================== *> *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA * *> \par Further Details: * ===================== *> *> \verbatim *> *> 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 )**T, 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 ). *> \endverbatim *> * ===================================================================== SUBROUTINE SLATRZ( M, N, L, A, LDA, TAU, WORK ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER L, LDA, M, N * .. * .. Array Arguments .. REAL A( LDA, * ), TAU( * ), WORK( * ) * .. * * ===================================================================== * * .. 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