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# dtzrqf

```
NAME
DTZRQF - reduce the M-by-N ( M<=N ) real upper trapezoidal
matrix A to upper triangular form by means of orthogonal
transformations

SYNOPSIS
SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )

INTEGER        INFO, LDA, M, N

DOUBLE         PRECISION A( LDA, * ), TAU( * )

PURPOSE
DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal
matrix A to upper triangular form by means of orthogonal
transformations.

The upper trapezoidal matrix A is factored as

A = ( R  0 ) * Z,

where Z is an N-by-N orthogonal 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) DOUBLE PRECISION array, dimension
(LDA,max(1,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
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) DOUBLE PRECISION array, dimension (max(1,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 ), which is used to intro-
duce 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 ).
```