---
name: "latrz"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/latrz"
---

# latrz(3)

latrz: RZ factor step

## NAME

latrz - latrz: RZ factor step

## SYNOPSIS

### Functions

subroutine **clatrz** (m, n, l, a, lda, tau, work)\
 **CLATRZ** factors an upper trapezoidal matrix by means of unitary transformations.  subroutine **dlatrz** (m, n, l, a, lda, tau, work)\
 **DLATRZ** factors an upper trapezoidal matrix by means of orthogonal transformations.  subroutine **slatrz** (m, n, l, a, lda, tau, work)\
 **SLATRZ** factors an upper trapezoidal matrix by means of orthogonal transformations.  subroutine **zlatrz** (m, n, l, a, lda, tau, work)\
 **ZLATRZ** factors an upper trapezoidal matrix by means of unitary transformations.

## Detailed Description

## Function Documentation

### subroutine clatrz (integer m, integer n, integer l, complex, dimension( lda, \* ) a, integer lda, complex, dimension( \* ) tau, complex, dimension( \* ) work)

**CLATRZ** factors an upper trapezoidal matrix by means of unitary transformations.

**Purpose:**

```
!>
!> CLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
!> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
!> of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
!> matrix and, R and A1 are M-by-M upper triangular matrices.
!>
```

**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 >= 0.
!>
```

*L*

```
!>          L is INTEGER
!>          The number of columns of the matrix A containing the
!>          meaningful part of the Householder vectors. N-M >= L >= 0.
!>
```

*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 N-L+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 (M)
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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

**Further Details:**

```
!>
!>  The factorization is obtained by Householder's method.  The kth
!>  transformation matrix, Z( k ), which 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 )**H,   u( k ) = (   1    ),
!>                                                 (   0    )
!>                                                 ( z( k ) )
!>
!>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
!>  are chosen to annihilate the elements of the kth row of A2.
!>
!>  The scalar tau is returned in the kth element of TAU and the vector
!>  u( k ) in the kth row of A2, such that the elements of z( k ) are
!>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
!>  the upper triangular part of A1.
!>
!>  Z is given by
!>
!>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
!>
```

Definition at line **139** of file **clatrz.f**.

### subroutine dlatrz (integer m, integer n, integer l, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( \* ) tau, double precision, dimension( \* ) work)

**DLATRZ** factors an upper trapezoidal matrix by means of orthogonal transformations.

**Purpose:**

```
!>
!> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
!> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
!> of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
!> matrix and, R and A1 are M-by-M upper triangular matrices.
!>
```

**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 >= 0.
!>
```

*L*

```
!>          L is INTEGER
!>          The number of columns of the matrix A containing the
!>          meaningful part of the Householder vectors. N-M >= L >= 0.
!>
```

*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 N-L+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 (M)
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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

**Further Details:**

```
!>
!>  The factorization is obtained by Householder's method.  The kth
!>  transformation matrix, Z( k ), which 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 )**T,   u( k ) = (   1    ),
!>                                                 (   0    )
!>                                                 ( z( k ) )
!>
!>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
!>  are chosen to annihilate the elements of the kth row of A2.
!>
!>  The scalar tau is returned in the kth element of TAU and the vector
!>  u( k ) in the kth row of A2, such that the elements of z( k ) are
!>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
!>  the upper triangular part of A1.
!>
!>  Z is given by
!>
!>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
!>
```

Definition at line **139** of file **dlatrz.f**.

### subroutine slatrz (integer m, integer n, integer l, real, dimension( lda, \* ) a, integer lda, real, dimension( \* ) tau, real, dimension( \* ) work)

**SLATRZ** factors an upper trapezoidal matrix by means of orthogonal transformations.

**Purpose:**

```
!>
!> SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
!> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
!> of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
!> matrix and, R and A1 are M-by-M upper triangular matrices.
!>
```

**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 >= 0.
!>
```

*L*

```
!>          L is INTEGER
!>          The number of columns of the matrix A containing the
!>          meaningful part of the Householder vectors. N-M >= L >= 0.
!>
```

*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 N-L+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 (M)
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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

**Further Details:**

```
!>
!>  The factorization is obtained by Householder's method.  The kth
!>  transformation matrix, Z( k ), which 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 )**T,   u( k ) = (   1    ),
!>                                                 (   0    )
!>                                                 ( z( k ) )
!>
!>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
!>  are chosen to annihilate the elements of the kth row of A2.
!>
!>  The scalar tau is returned in the kth element of TAU and the vector
!>  u( k ) in the kth row of A2, such that the elements of z( k ) are
!>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
!>  the upper triangular part of A1.
!>
!>  Z is given by
!>
!>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
!>
```

Definition at line **139** of file **slatrz.f**.

### subroutine zlatrz (integer m, integer n, integer l, complex\*16, dimension( lda, \* ) a, integer lda, complex\*16, dimension( \* ) tau, complex\*16, dimension( \* ) work)

**ZLATRZ** factors an upper trapezoidal matrix by means of unitary transformations.

**Purpose:**

```
!>
!> ZLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
!> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
!> of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
!> matrix and, R and A1 are M-by-M upper triangular matrices.
!>
```

**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 >= 0.
!>
```

*L*

```
!>          L is INTEGER
!>          The number of columns of the matrix A containing the
!>          meaningful part of the Householder vectors. N-M >= L >= 0.
!>
```

*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 N-L+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 (M)
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

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

**Further Details:**

```
!>
!>  The factorization is obtained by Householder's method.  The kth
!>  transformation matrix, Z( k ), which 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 )**H,   u( k ) = (   1    ),
!>                                                 (   0    )
!>                                                 ( z( k ) )
!>
!>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
!>  are chosen to annihilate the elements of the kth row of A2.
!>
!>  The scalar tau is returned in the kth element of TAU and the vector
!>  u( k ) in the kth row of A2, such that the elements of z( k ) are
!>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
!>  the upper triangular part of A1.
!>
!>  Z is given by
!>
!>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
!>
```

Definition at line **139** of file **zlatrz.f**.

## Author

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