CLATRZ(3S)CLATRZ(3S)NAME
CLATRZ - factor the M-by-(M+L) complex upper trapezoidal matrix [ A1 A2 ]
= [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z by means of unitary
transformations, where Z is an (M+L)-by-(M+L) unitary matrix and, R and
A1 are M-by-M upper triangular matrices
SYNOPSIS
SUBROUTINE CLATRZ( M, N, L, A, LDA, TAU, WORK )
INTEGER L, LDA, M, N
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSE
CLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix [ A1 A2 ]
= [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z by means of unitary
transformations, where Z is an (M+L)-by-(M+L) unitary 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) COMPLEX 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 unitary matrix Z as a product of M
elementary reflectors.
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CLATRZ(3S)CLATRZ(3S)
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU (output) COMPLEX array, dimension (M)
The scalar factors of the elementary reflectors.
WORK (workspace) COMPLEX 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 ).
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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