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man pages->IRIX man pages -> complib/ztzrqf (3)
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### Contents

```
ZTZRQF(3F)							    ZTZRQF(3F)

```

### NAME[Toc][Back]

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

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	ZTZRQF(	M, N, A, LDA, TAU, INFO	)

INTEGER	INFO, LDA, M, N

COMPLEX*16	A( LDA,	* ), TAU( * )
```

### PURPOSE[Toc][Back]

```     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[Toc][Back]

```     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

Page 1

ZTZRQF(3F)							    ZTZRQF(3F)

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 ).
ZTZRQF(3F)							    ZTZRQF(3F)

```

### NAME[Toc][Back]

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

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	ZTZRQF(	M, N, A, LDA, TAU, INFO	)

INTEGER	INFO, LDA, M, N

COMPLEX*16	A( LDA,	* ), TAU( * )
```

### PURPOSE[Toc][Back]

```     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[Toc][Back]

```     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

Page 1

ZTZRQF(3F)							    ZTZRQF(3F)

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 ).

PPPPaaaaggggeeee 2222```
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