ctzrzf (l) - Linux Manuals

ctzrzf: reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations

Command to display ctzrzf manual in Linux: $ man l ctzrzf

NAME

CTZRZF - reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations

SYNOPSIS

SUBROUTINE CTZRZF(: M, N, A, LDA, TAU, WORK, LWORK, INFO )
: INTEGER INFO, LDA, LWORK, M, N
: COMPLEX A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

CTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations. The upper trapezoidal matrix A is factored as

A = ( R 0 ) * Z,
where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix.

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 >= M.
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 M+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.
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/output) COMPLEX array, dimension (MAX(1,LWORK)): On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER: The dimension of the array WORK. LWORK >= max(1,M). For optimum performance LWORK >= M*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER: = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

Based on contributions by

  A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA The factorization is obtained by Householderaqs 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 )aq,   u( k ) = (   1    ),
                                         (   0    )
                                         ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of X.
The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of A, such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A.
Z is given by

Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).