# 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.

## Detailed Description

## 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

Univ. of Colorado Denver

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

Univ. of Colorado Denver

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

Univ. of Colorado Denver

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**KB**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**W*W is REAL array, dimension (LDW,NB)

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

*INFO***Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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**KB**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**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***Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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

*NB**KB**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**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***Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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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