hetf2(3)

NAME
hetf2 - {he,sy}tf2: triangular factor, level 2

SYNOPSIS

  Functions

subroutine chetf2 (uplo, n, a, lda, ipiv, info)
CHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS).
subroutine csytf2 (uplo, n, a, lda, ipiv, info)
CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine dsytf2 (uplo, n, a, lda, ipiv, info)
DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine ssytf2 (uplo, n, a, lda, ipiv, info)
SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine zhetf2 (uplo, n, a, lda, ipiv, info)
ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS).
subroutine zsytf2 (uplo, n, a, lda, ipiv, info)
ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Detailed Description

Function Documentation

  subroutine chetf2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

CHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS).

Purpose:

    !>
    !> CHETF2 computes the factorization of a complex Hermitian matrix A
    !> using the Bunch-Kaufman diagonal pivoting method:
    !>
    !>    A = U*D*U**H  or  A = L*D*L**H
    !>
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, U**H is the conjugate transpose of U, and D is
    !> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the unblocked version of the algorithm, calling Level 2 BLAS.
    !>

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.
    !>

    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, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L (see below for further details).
    !>

    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':
    !>             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':
    !>             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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -k, the k-th argument had an illegal value
    !>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
    !>               has been completed, but the block diagonal matrix D is
    !>               exactly singular, and division by zero will occur if it
    !>               is used to solve a system of equations.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  09-29-06 - patch from
    !>    Bobby Cheng, MathWorks
    !>
    !>    Replace l.210 and l.392
    !>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    !>    by
    !>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
    !>
    !>  01-01-96 - Based on modifications by
    !>    J. Lewis, Boeing Computer Services Company
    !>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
    !>
    !>  If UPLO = 'U', then A = U*D*U**H, where
    !>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
    !>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
    !>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    v    0   )   k-s
    !>     U(k) =  (   0    I    0   )   s
    !>             (   0    0    I   )   n-k
    !>                k-s   s   n-k
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
    !>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
    !>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
    !>
    !>  If UPLO = 'L', then A = L*D*L**H, where
    !>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
    !>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
    !>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    0     0   )  k-1
    !>     L(k) =  (   0    I     0   )  s
    !>             (   0    v     I   )  n-k-s+1
    !>                k-1   s  n-k-s+1
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
    !>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
    !>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
    !>

  subroutine csytf2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

    !>
    !> CSYTF2 computes the factorization of a complex symmetric matrix A
    !> using the Bunch-Kaufman diagonal pivoting method:
    !>
    !>    A = U*D*U**T  or  A = L*D*L**T
    !>
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, U**T is the transpose of U, and D is symmetric and
    !> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the unblocked version of the algorithm, calling Level 2 BLAS.
    !>

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.
    !>

    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, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L (see below for further details).
    !>

    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':
    !>             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':
    !>             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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -k, the k-th argument had an illegal value
    !>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
    !>               has been completed, but the block diagonal matrix D is
    !>               exactly singular, and division by zero will occur if it
    !>               is used to solve a system of equations.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', then A = U*D*U**T, where
    !>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
    !>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
    !>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    v    0   )   k-s
    !>     U(k) =  (   0    I    0   )   s
    !>             (   0    0    I   )   n-k
    !>                k-s   s   n-k
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
    !>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
    !>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
    !>
    !>  If UPLO = 'L', then A = L*D*L**T, where
    !>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
    !>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
    !>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    0     0   )  k-1
    !>     L(k) =  (   0    I     0   )  s
    !>             (   0    v     I   )  n-k-s+1
    !>                k-1   s  n-k-s+1
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
    !>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
    !>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
    !>

Contributors:

    !>
    !>  09-29-06 - patch from
    !>    Bobby Cheng, MathWorks
    !>
    !>    Replace l.209 and l.377
    !>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    !>    by
    !>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
    !>
    !>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
    !>         Company
    !>

  subroutine dsytf2 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

    !>
    !> DSYTF2 computes the factorization of a real symmetric matrix A using
    !> the Bunch-Kaufman diagonal pivoting method:
    !>
    !>    A = U*D*U**T  or  A = L*D*L**T
    !>
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, U**T is the transpose of U, and D is symmetric and
    !> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the unblocked version of the algorithm, calling Level 2 BLAS.
    !>

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.
    !>

    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, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L (see below for further details).
    !>

    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':
    !>             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':
    !>             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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -k, the k-th argument had an illegal value
    !>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
    !>               has been completed, but the block diagonal matrix D is
    !>               exactly singular, and division by zero will occur if it
    !>               is used to solve a system of equations.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', then A = U*D*U**T, where
    !>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
    !>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
    !>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    v    0   )   k-s
    !>     U(k) =  (   0    I    0   )   s
    !>             (   0    0    I   )   n-k
    !>                k-s   s   n-k
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
    !>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
    !>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
    !>
    !>  If UPLO = 'L', then A = L*D*L**T, where
    !>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
    !>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
    !>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    0     0   )  k-1
    !>     L(k) =  (   0    I     0   )  s
    !>             (   0    v     I   )  n-k-s+1
    !>                k-1   s  n-k-s+1
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
    !>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
    !>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
    !>

Contributors:

    !>
    !>  09-29-06 - patch from
    !>    Bobby Cheng, MathWorks
    !>
    !>    Replace l.204 and l.372
    !>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    !>    by
    !>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
    !>
    !>  01-01-96 - Based on modifications by
    !>    J. Lewis, Boeing Computer Services Company
    !>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
    !>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
    !>         Company
    !>

  subroutine ssytf2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

SSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

    !>
    !> SSYTF2 computes the factorization of a real symmetric matrix A using
    !> the Bunch-Kaufman diagonal pivoting method:
    !>
    !>    A = U*D*U**T  or  A = L*D*L**T
    !>
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, U**T is the transpose of U, and D is symmetric and
    !> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the unblocked version of the algorithm, calling Level 2 BLAS.
    !>

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.
    !>

    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, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L (see below for further details).
    !>

    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':
    !>             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':
    !>             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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -k, the k-th argument had an illegal value
    !>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
    !>               has been completed, but the block diagonal matrix D is
    !>               exactly singular, and division by zero will occur if it
    !>               is used to solve a system of equations.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', then A = U*D*U**T, where
    !>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
    !>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
    !>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    v    0   )   k-s
    !>     U(k) =  (   0    I    0   )   s
    !>             (   0    0    I   )   n-k
    !>                k-s   s   n-k
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
    !>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
    !>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
    !>
    !>  If UPLO = 'L', then A = L*D*L**T, where
    !>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
    !>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
    !>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    0     0   )  k-1
    !>     L(k) =  (   0    I     0   )  s
    !>             (   0    v     I   )  n-k-s+1
    !>                k-1   s  n-k-s+1
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
    !>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
    !>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
    !>

Contributors:

    !>
    !>  09-29-06 - patch from
    !>    Bobby Cheng, MathWorks
    !>
    !>    Replace l.204 and l.372
    !>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    !>    by
    !>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN
    !>
    !>  01-01-96 - Based on modifications by
    !>    J. Lewis, Boeing Computer Services Company
    !>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
    !>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
    !>         Company
    !>
    !>

  subroutine zhetf2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS).

Purpose:

    !>
    !> ZHETF2 computes the factorization of a complex Hermitian matrix A
    !> using the Bunch-Kaufman diagonal pivoting method:
    !>
    !>    A = U*D*U**H  or  A = L*D*L**H
    !>
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, U**H is the conjugate transpose of U, and D is
    !> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the unblocked version of the algorithm, calling Level 2 BLAS.
    !>

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.
    !>

    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, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L (see below for further details).
    !>

    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':
    !>             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':
    !>             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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -k, the k-th argument had an illegal value
    !>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
    !>               has been completed, but the block diagonal matrix D is
    !>               exactly singular, and division by zero will occur if it
    !>               is used to solve a system of equations.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', then A = U*D*U**H, where
    !>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
    !>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
    !>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    v    0   )   k-s
    !>     U(k) =  (   0    I    0   )   s
    !>             (   0    0    I   )   n-k
    !>                k-s   s   n-k
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
    !>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
    !>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
    !>
    !>  If UPLO = 'L', then A = L*D*L**H, where
    !>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
    !>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
    !>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    0     0   )  k-1
    !>     L(k) =  (   0    I     0   )  s
    !>             (   0    v     I   )  n-k-s+1
    !>                k-1   s  n-k-s+1
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
    !>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
    !>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
    !>

Contributors:

    !>  09-29-06 - patch from
    !>    Bobby Cheng, MathWorks
    !>
    !>    Replace l.210 and l.393
    !>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    !>    by
    !>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
    !>
    !>  01-01-96 - Based on modifications by
    !>    J. Lewis, Boeing Computer Services Company
    !>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
    !>

  subroutine zsytf2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:

    !>
    !> ZSYTF2 computes the factorization of a complex symmetric matrix A
    !> using the Bunch-Kaufman diagonal pivoting method:
    !>
    !>    A = U*D*U**T  or  A = L*D*L**T
    !>
    !> where U (or L) is a product of permutation and unit upper (lower)
    !> triangular matrices, U**T is the transpose of U, and D is symmetric and
    !> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the unblocked version of the algorithm, calling Level 2 BLAS.
    !>

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.
    !>

    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, the block diagonal matrix D and the multipliers used
    !>          to obtain the factor U or L (see below for further details).
    !>

    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':
    !>             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':
    !>             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.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0: successful exit
    !>          < 0: if INFO = -k, the k-th argument had an illegal value
    !>          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
    !>               has been completed, but the block diagonal matrix D is
    !>               exactly singular, and division by zero will occur if it
    !>               is used to solve a system of equations.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', then A = U*D*U**T, where
    !>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
    !>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
    !>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    v    0   )   k-s
    !>     U(k) =  (   0    I    0   )   s
    !>             (   0    0    I   )   n-k
    !>                k-s   s   n-k
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
    !>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
    !>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
    !>
    !>  If UPLO = 'L', then A = L*D*L**T, where
    !>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
    !>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
    !>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
    !>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
    !>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
    !>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
    !>
    !>             (   I    0     0   )  k-1
    !>     L(k) =  (   0    I     0   )  s
    !>             (   0    v     I   )  n-k-s+1
    !>                k-1   s  n-k-s+1
    !>
    !>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
    !>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
    !>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
    !>

Contributors:

    !>
    !>  09-29-06 - patch from
    !>    Bobby Cheng, MathWorks
    !>
    !>    Replace l.209 and l.377
    !>         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
    !>    by
    !>         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
    !>
    !>  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
    !>         Company
    !>

Author

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