ctzrqf(l) — SunSoft Performance Library

NAME

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

SYNOPSIS

SUBROUTINE CTZRQF(
M, N, A, LDA, TAU, INFO )

INTEGER INFO, LDA, M, N

COMPLEX A( LDA, ∗ ), TAU( ∗ )

PURPOSE

CTZRQF 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.

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

The factorization is obtained by Householder’s method. The kth transformation matrix, Z( k ), whose conjugate transpose 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 ( 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 ).