---
name: "ztzrqf"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/ztzrqf/3lapack"
---

# ztzrqf(3lapack)

routine i deprecated and has been replaced by routine ZTZRZF

## NAME

ZTZRQF - routine i deprecated and has been replaced by routine ZTZRZF

## SYNOPSIS

- SUBROUTINE ZTZRQF(

  M, N, A, LDA, TAU, INFO )

- INTEGER

  INFO, LDA, M, N

- COMPLEX\*16

  A( LDA, \* ), TAU( \* )

## PURPOSE

This routine is deprecated and has been replaced by routine ZTZRZF. ZTZRQF 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\*16 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\*16 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 ).\