# tzrzf - Man Page

tzrzf: RZ factor

## Synopsis

### Functions

subroutine ctzrzf (m, n, a, lda, tau, work, lwork, info)
CTZRZF
subroutine dtzrzf (m, n, a, lda, tau, work, lwork, info)
DTZRZF
subroutine stzrzf (m, n, a, lda, tau, work, lwork, info)
STZRZF
subroutine ztzrzf (m, n, a, lda, tau, work, lwork, info)
ZTZRZF

## Function Documentation

### subroutine ctzrzf (integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CTZRZF

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.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= M.```

A

```          A is 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

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).```

TAU

```          TAU is COMPLEX array, dimension (M)
The scalar factors of the elementary reflectors.```

WORK

```          WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is 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

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

```  The N-by-N matrix Z can be computed by

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

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**H

with v(k) is the kth row vector of the M-by-N matrix

V = ( I   A(:,M+1:N) )

I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from CTZRZF,
and tau(k) is the kth element of the array TAU.```

Definition at line 150 of file ctzrzf.f.

### subroutine dtzrzf (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DTZRZF

Purpose:

``` DTZRZF 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.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= M.```

A

```          A is DOUBLE PRECISION 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
orthogonal matrix Z as a product of M elementary reflectors.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).```

TAU

```          TAU is DOUBLE PRECISION array, dimension (M)
The scalar factors of the elementary reflectors.```

WORK

```          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is 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

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

```  The N-by-N matrix Z can be computed by

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

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**T

with v(k) is the kth row vector of the M-by-N matrix

V = ( I   A(:,M+1:N) )

I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from DTZRZF,
and tau(k) is the kth element of the array TAU.```

Definition at line 150 of file dtzrzf.f.

### subroutine stzrzf (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

STZRZF

Purpose:

``` STZRZF 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.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= M.```

A

```          A is REAL 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
orthogonal matrix Z as a product of M elementary reflectors.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).```

TAU

```          TAU is REAL array, dimension (M)
The scalar factors of the elementary reflectors.```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is 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

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

```  The N-by-N matrix Z can be computed by

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

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**T

with v(k) is the kth row vector of the M-by-N matrix

V = ( I   A(:,M+1:N) )

I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from STZRZF,
and tau(k) is the kth element of the array TAU.```

Definition at line 150 of file stzrzf.f.

### subroutine ztzrzf (integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZTZRZF

Purpose:

``` ZTZRZF 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.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the matrix A.  N >= M.```

A

```          A is 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

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).```

TAU

```          TAU is COMPLEX*16 array, dimension (M)
The scalar factors of the elementary reflectors.```

WORK

```          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is 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

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

```  The N-by-N matrix Z can be computed by

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

where each N-by-N Z(k) is given by

Z(k) = I - tau(k)*v(k)*v(k)**H

with v(k) is the kth row vector of the M-by-N matrix

V = ( I   A(:,M+1:N) )

I is the M-by-M identity matrix, A(:,M+1:N)
is the output stored in A on exit from ZTZRZF,
and tau(k) is the kth element of the array TAU.```

Definition at line 150 of file ztzrzf.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK