# lahef - Man Page

la{he,sy}f: step in hetrf

## Synopsis

### Functions

subroutine clahef (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
CLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
subroutine clasyf (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine dlasyf (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine slasyf (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine zlahef (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
subroutine zlasyf (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

## Function Documentation

### subroutine clahef (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)

CLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).

Purpose:

``` CLAHEF computes a partial factorization of a complex Hermitian
matrix A using the Bunch-Kaufman diagonal pivoting method. The
partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I      0     )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**H U22**H )

A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L'
( L21  I ) (  0  A22 ) (  0      I     )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**H denotes the conjugate transpose of U.

CLAHEF is an auxiliary routine called by CHETRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.```

LDA

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

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = 'U':
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = 'L':
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.```

W

`          W is COMPLEX array, dimension (LDW,NB)`

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).```

INFO

```          INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
has been completed, but the block diagonal matrix D is
exactly singular.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  November 2013,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 176 of file clahef.f.

### subroutine clasyf (character uplo, integer n, integer nb, integer kb, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex, dimension( ldw, * ) w, integer ldw, integer info)

CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:

``` CLASYF computes a partial factorization of a complex symmetric matrix
A using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) ( D    0  ) ( L11**T L21**T )  if UPLO = 'L'
( L21  I ) ( 0   A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**T denotes the transpose of U.

CLASYF is an auxiliary routine called by CSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.```

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the symmetric matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.```

LDA

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

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = 'U':
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = 'L':
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.```

W

`          W is COMPLEX array, dimension (LDW,NB)`

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).```

INFO

```          INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
has been completed, but the block diagonal matrix D is
exactly singular.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  November 2013,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 176 of file clasyf.f.

### subroutine dlasyf (character uplo, integer n, integer nb, integer kb, double precision, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, double precision, dimension( ldw, * ) w, integer ldw, integer info)

DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:

``` DLASYF computes a partial factorization of a real symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.

DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.```

A

```          A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.```

LDA

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

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = 'U':
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = 'L':
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.```

W

`          W is DOUBLE PRECISION array, dimension (LDW,NB)`

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).```

INFO

```          INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
has been completed, but the block diagonal matrix D is
exactly singular.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  November 2013,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 175 of file dlasyf.f.

### subroutine slasyf (character uplo, integer n, integer nb, integer kb, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, real, dimension( ldw, * ) w, integer ldw, integer info)

SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:

``` SLASYF computes a partial factorization of a real symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.

SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.```

A

```          A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.```

LDA

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

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = 'U':
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = 'L':
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.```

W

`          W is REAL array, dimension (LDW,NB)`

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).```

INFO

```          INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
has been completed, but the block diagonal matrix D is
exactly singular.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  November 2013,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 175 of file slasyf.f.

### subroutine zlahef (character uplo, integer n, integer nb, integer kb, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w, integer ldw, integer info)

ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).

Purpose:

``` ZLAHEF computes a partial factorization of a complex Hermitian
matrix A using the Bunch-Kaufman diagonal pivoting method. The
partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I      0     )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**H U22**H )

A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L'
( L21  I ) (  0  A22 ) (  0      I     )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**H denotes the conjugate transpose of U.

ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.```

LDA

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

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = 'U':
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = 'L':
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.```

W

`          W is COMPLEX*16 array, dimension (LDW,NB)`

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).```

INFO

```          INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
has been completed, but the block diagonal matrix D is
exactly singular.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  December 2016,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 176 of file zlahef.f.

### subroutine zlasyf (character uplo, integer n, integer nb, integer kb, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w, integer ldw, integer info)

ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:

``` ZLASYF computes a partial factorization of a complex symmetric matrix
A using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) ( D    0  ) ( L11**T L21**T )  if UPLO = 'L'
( L21  I ) ( 0   A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**T denotes the transpose of U.

ZLASYF is an auxiliary routine called by ZSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U':  Upper triangular
= 'L':  Lower triangular```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

NB

```          NB is INTEGER
The maximum number of columns of the matrix A that should be
factored.  NB should be at least 2 to allow for 2-by-2 pivot
blocks.```

KB

```          KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
On entry, the symmetric matrix A.  If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced.  If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.```

LDA

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

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.

If UPLO = 'U':
Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.

If UPLO = 'L':
Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.```

W

`          W is COMPLEX*16 array, dimension (LDW,NB)`

LDW

```          LDW is INTEGER
The leading dimension of the array W.  LDW >= max(1,N).```

INFO

```          INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero.  The factorization
has been completed, but the block diagonal matrix D is
exactly singular.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  November 2013,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 176 of file zlasyf.f.

## Author

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## Info

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