hetrf_rk(3)

NAME
hetrf_rk - {he,sy}trf_rk: triangular factor

SYNOPSIS

  Functions

subroutine chetrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
CHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine csytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine dsytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine ssytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
SSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine zhetrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
ZHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine zsytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
ZSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Detailed Description

Function Documentation

  subroutine chetrf_rk (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

    !> CHETRF_RK computes the factorization of a complex Hermitian matrix A
    !> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
    !>
    !>    A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
    !>
    !> where U (or L) is unit upper (or lower) triangular matrix,
    !> U**H (or L**H) is the conjugate of U (or L), P is a permutation
    !> matrix, P**T is the transpose of P, and D is Hermitian and block
    !> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the blocked version of the algorithm, calling Level 3 BLAS.
    !> For more information see Further Details section.
    !>

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, contains:
    !>            a) ONLY diagonal elements of the Hermitian block diagonal
    !>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
    !>               (superdiagonal (or subdiagonal) elements of D
    !>                are stored on exit in array E), and
    !>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': factor L in the subdiagonal part of A.
    !>

    LDA

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

    E

    !>          E is COMPLEX array, dimension (N)
    !>          On exit, contains the superdiagonal (or subdiagonal)
    !>          elements of the Hermitian block diagonal matrix D
    !>          with 1-by-1 or 2-by-2 diagonal blocks, where
    !>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    !>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    !>
    !>          NOTE: For 1-by-1 diagonal block D(k), where
    !>          1 <= k <= N, the element E(k) is set to 0 in both
    !>          UPLO = 'U' or UPLO = 'L' cases.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step. For more info see Further
    !>          Details section.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    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 length of WORK.  LWORK >= 1.  For best performance
    !>          LWORK >= N*NB, where NB is the block size returned
    !>          by ILAENV.
    !>
    !>          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 = -k, the k-th argument had an illegal value
    !>
    !>          > 0: If INFO = k, the matrix A is singular, because:
    !>                 If UPLO = 'U': column k in the upper
    !>                 triangular part of A contains all zeros.
    !>                 If UPLO = 'L': column k in the lower
    !>                 triangular part of A contains all zeros.
    !>
    !>               Therefore D(k,k) is exactly zero, and superdiagonal
    !>               elements of column k of U (or subdiagonal elements of
    !>               column k of L ) are all zeros. 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.
    !>
    !>               NOTE: INFO only stores the first occurrence of
    !>               a singularity, any subsequent occurrence of singularity
    !>               is not stored in INFO even though the factorization
    !>               always completes.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> TODO: put correct description
    !>

Contributors:

    !>
    !>  December 2016,  Igor Kozachenko,
    !>                  Computer Science Division,
    !>                  University of California, Berkeley
    !>
    !>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
    !>                  School of Mathematics,
    !>                  University of Manchester
    !>
    !>

  subroutine csytrf_rk (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( * ) work, integer lwork, integer info)

CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

    !> CSYTRF_RK computes the factorization of a complex symmetric matrix A
    !> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
    !>
    !>    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
    !>
    !> where U (or L) is unit upper (or lower) triangular matrix,
    !> U**T (or L**T) is the transpose of U (or L), P is a permutation
    !> matrix, P**T is the transpose of P, and D is symmetric and block
    !> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the blocked version of the algorithm, calling Level 3 BLAS.
    !> For more information see Further Details section.
    !>

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, contains:
    !>            a) ONLY diagonal elements of the symmetric block diagonal
    !>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
    !>               (superdiagonal (or subdiagonal) elements of D
    !>                are stored on exit in array E), and
    !>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': factor L in the subdiagonal part of A.
    !>

    LDA

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

    E

    !>          E is COMPLEX array, dimension (N)
    !>          On exit, contains the superdiagonal (or subdiagonal)
    !>          elements of the symmetric block diagonal matrix D
    !>          with 1-by-1 or 2-by-2 diagonal blocks, where
    !>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    !>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    !>
    !>          NOTE: For 1-by-1 diagonal block D(k), where
    !>          1 <= k <= N, the element E(k) is set to 0 in both
    !>          UPLO = 'U' or UPLO = 'L' cases.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step. For more info see Further
    !>          Details section.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    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 length of WORK.  LWORK >=1.  For best performance
    !>          LWORK >= N*NB, where NB is the block size returned
    !>          by ILAENV.
    !>
    !>          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 = -k, the k-th argument had an illegal value
    !>
    !>          > 0: If INFO = k, the matrix A is singular, because:
    !>                 If UPLO = 'U': column k in the upper
    !>                 triangular part of A contains all zeros.
    !>                 If UPLO = 'L': column k in the lower
    !>                 triangular part of A contains all zeros.
    !>
    !>               Therefore D(k,k) is exactly zero, and superdiagonal
    !>               elements of column k of U (or subdiagonal elements of
    !>               column k of L ) are all zeros. 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.
    !>
    !>               NOTE: INFO only stores the first occurrence of
    !>               a singularity, any subsequent occurrence of singularity
    !>               is not stored in INFO even though the factorization
    !>               always completes.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> TODO: put correct description
    !>

Contributors:

    !>
    !>  December 2016,  Igor Kozachenko,
    !>                  Computer Science Division,
    !>                  University of California, Berkeley
    !>
    !>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
    !>                  School of Mathematics,
    !>                  University of Manchester
    !>
    !>

  subroutine dsytrf_rk (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

    !> DSYTRF_RK computes the factorization of a real symmetric matrix A
    !> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
    !>
    !>    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
    !>
    !> where U (or L) is unit upper (or lower) triangular matrix,
    !> U**T (or L**T) is the transpose of U (or L), P is a permutation
    !> matrix, P**T is the transpose of P, and D is symmetric and block
    !> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the blocked version of the algorithm, calling Level 3 BLAS.
    !> For more information see Further Details section.
    !>

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, contains:
    !>            a) ONLY diagonal elements of the symmetric block diagonal
    !>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
    !>               (superdiagonal (or subdiagonal) elements of D
    !>                are stored on exit in array E), and
    !>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': factor L in the subdiagonal part of A.
    !>

    LDA

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

    E

    !>          E is DOUBLE PRECISION array, dimension (N)
    !>          On exit, contains the superdiagonal (or subdiagonal)
    !>          elements of the symmetric block diagonal matrix D
    !>          with 1-by-1 or 2-by-2 diagonal blocks, where
    !>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    !>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    !>
    !>          NOTE: For 1-by-1 diagonal block D(k), where
    !>          1 <= k <= N, the element E(k) is set to 0 in both
    !>          UPLO = 'U' or UPLO = 'L' cases.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step. For more info see Further
    !>          Details section.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    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 length of WORK.  LWORK >= 1.  For best performance
    !>          LWORK >= N*NB, where NB is the block size returned
    !>          by ILAENV.
    !>
    !>          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 = -k, the k-th argument had an illegal value
    !>
    !>          > 0: If INFO = k, the matrix A is singular, because:
    !>                 If UPLO = 'U': column k in the upper
    !>                 triangular part of A contains all zeros.
    !>                 If UPLO = 'L': column k in the lower
    !>                 triangular part of A contains all zeros.
    !>
    !>               Therefore D(k,k) is exactly zero, and superdiagonal
    !>               elements of column k of U (or subdiagonal elements of
    !>               column k of L ) are all zeros. 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.
    !>
    !>               NOTE: INFO only stores the first occurrence of
    !>               a singularity, any subsequent occurrence of singularity
    !>               is not stored in INFO even though the factorization
    !>               always completes.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> TODO: put correct description
    !>

Contributors:

    !>
    !>  December 2016,  Igor Kozachenko,
    !>                  Computer Science Division,
    !>                  University of California, Berkeley
    !>
    !>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
    !>                  School of Mathematics,
    !>                  University of Manchester
    !>
    !>

  subroutine ssytrf_rk (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real, dimension( * ) work, integer lwork, integer info)

SSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

    !> SSYTRF_RK computes the factorization of a real symmetric matrix A
    !> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
    !>
    !>    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
    !>
    !> where U (or L) is unit upper (or lower) triangular matrix,
    !> U**T (or L**T) is the transpose of U (or L), P is a permutation
    !> matrix, P**T is the transpose of P, and D is symmetric and block
    !> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the blocked version of the algorithm, calling Level 3 BLAS.
    !> For more information see Further Details section.
    !>

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, contains:
    !>            a) ONLY diagonal elements of the symmetric block diagonal
    !>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
    !>               (superdiagonal (or subdiagonal) elements of D
    !>                are stored on exit in array E), and
    !>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': factor L in the subdiagonal part of A.
    !>

    LDA

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

    E

    !>          E is REAL array, dimension (N)
    !>          On exit, contains the superdiagonal (or subdiagonal)
    !>          elements of the symmetric block diagonal matrix D
    !>          with 1-by-1 or 2-by-2 diagonal blocks, where
    !>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    !>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    !>
    !>          NOTE: For 1-by-1 diagonal block D(k), where
    !>          1 <= k <= N, the element E(k) is set to 0 in both
    !>          UPLO = 'U' or UPLO = 'L' cases.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step. For more info see Further
    !>          Details section.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    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 length of WORK.  LWORK >= 1.  For best performance
    !>          LWORK >= N*NB, where NB is the block size returned
    !>          by ILAENV.
    !>
    !>          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 = -k, the k-th argument had an illegal value
    !>
    !>          > 0: If INFO = k, the matrix A is singular, because:
    !>                 If UPLO = 'U': column k in the upper
    !>                 triangular part of A contains all zeros.
    !>                 If UPLO = 'L': column k in the lower
    !>                 triangular part of A contains all zeros.
    !>
    !>               Therefore D(k,k) is exactly zero, and superdiagonal
    !>               elements of column k of U (or subdiagonal elements of
    !>               column k of L ) are all zeros. 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.
    !>
    !>               NOTE: INFO only stores the first occurrence of
    !>               a singularity, any subsequent occurrence of singularity
    !>               is not stored in INFO even though the factorization
    !>               always completes.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> TODO: put correct description
    !>

Contributors:

    !>
    !>  December 2016,  Igor Kozachenko,
    !>                  Computer Science Division,
    !>                  University of California, Berkeley
    !>
    !>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
    !>                  School of Mathematics,
    !>                  University of Manchester
    !>
    !>

  subroutine zhetrf_rk (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

    !> ZHETRF_RK computes the factorization of a complex Hermitian matrix A
    !> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
    !>
    !>    A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
    !>
    !> where U (or L) is unit upper (or lower) triangular matrix,
    !> U**H (or L**H) is the conjugate of U (or L), P is a permutation
    !> matrix, P**T is the transpose of P, and D is Hermitian and block
    !> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the blocked version of the algorithm, calling Level 3 BLAS.
    !> For more information see Further Details section.
    !>

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, contains:
    !>            a) ONLY diagonal elements of the Hermitian block diagonal
    !>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
    !>               (superdiagonal (or subdiagonal) elements of D
    !>                are stored on exit in array E), and
    !>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': factor L in the subdiagonal part of A.
    !>

    LDA

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

    E

    !>          E is COMPLEX*16 array, dimension (N)
    !>          On exit, contains the superdiagonal (or subdiagonal)
    !>          elements of the Hermitian block diagonal matrix D
    !>          with 1-by-1 or 2-by-2 diagonal blocks, where
    !>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    !>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    !>
    !>          NOTE: For 1-by-1 diagonal block D(k), where
    !>          1 <= k <= N, the element E(k) is set to 0 in both
    !>          UPLO = 'U' or UPLO = 'L' cases.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step. For more info see Further
    !>          Details section.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    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 length of WORK.  LWORK >= 1.  For best performance
    !>          LWORK >= N*NB, where NB is the block size returned
    !>          by ILAENV.
    !>
    !>          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 = -k, the k-th argument had an illegal value
    !>
    !>          > 0: If INFO = k, the matrix A is singular, because:
    !>                 If UPLO = 'U': column k in the upper
    !>                 triangular part of A contains all zeros.
    !>                 If UPLO = 'L': column k in the lower
    !>                 triangular part of A contains all zeros.
    !>
    !>               Therefore D(k,k) is exactly zero, and superdiagonal
    !>               elements of column k of U (or subdiagonal elements of
    !>               column k of L ) are all zeros. 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.
    !>
    !>               NOTE: INFO only stores the first occurrence of
    !>               a singularity, any subsequent occurrence of singularity
    !>               is not stored in INFO even though the factorization
    !>               always completes.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> TODO: put correct description
    !>

Contributors:

    !>
    !>  December 2016,  Igor Kozachenko,
    !>                  Computer Science Division,
    !>                  University of California, Berkeley
    !>
    !>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
    !>                  School of Mathematics,
    !>                  University of Manchester
    !>
    !>

  subroutine zsytrf_rk (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:

    !> ZSYTRF_RK computes the factorization of a complex symmetric matrix A
    !> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
    !>
    !>    A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
    !>
    !> where U (or L) is unit upper (or lower) triangular matrix,
    !> U**T (or L**T) is the transpose of U (or L), P is a permutation
    !> matrix, P**T is the transpose of P, and D is symmetric and block
    !> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
    !>
    !> This is the blocked version of the algorithm, calling Level 3 BLAS.
    !> For more information see Further Details section.
    !>

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, contains:
    !>            a) ONLY diagonal elements of the symmetric block diagonal
    !>               matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
    !>               (superdiagonal (or subdiagonal) elements of D
    !>                are stored on exit in array E), and
    !>            b) If UPLO = 'U': factor U in the superdiagonal part of A.
    !>               If UPLO = 'L': factor L in the subdiagonal part of A.
    !>

    LDA

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

    E

    !>          E is COMPLEX*16 array, dimension (N)
    !>          On exit, contains the superdiagonal (or subdiagonal)
    !>          elements of the symmetric block diagonal matrix D
    !>          with 1-by-1 or 2-by-2 diagonal blocks, where
    !>          If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
    !>          If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
    !>
    !>          NOTE: For 1-by-1 diagonal block D(k), where
    !>          1 <= k <= N, the element E(k) is set to 0 in both
    !>          UPLO = 'U' or UPLO = 'L' cases.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step. For more info see Further
    !>          Details section.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the matrix A(1:N,1:N).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the matrix A(1:N,1:N).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    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 length of WORK.  LWORK >=1.  For best performance
    !>          LWORK >= N*NB, where NB is the block size returned
    !>          by ILAENV.
    !>
    !>          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 = -k, the k-th argument had an illegal value
    !>
    !>          > 0: If INFO = k, the matrix A is singular, because:
    !>                 If UPLO = 'U': column k in the upper
    !>                 triangular part of A contains all zeros.
    !>                 If UPLO = 'L': column k in the lower
    !>                 triangular part of A contains all zeros.
    !>
    !>               Therefore D(k,k) is exactly zero, and superdiagonal
    !>               elements of column k of U (or subdiagonal elements of
    !>               column k of L ) are all zeros. 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.
    !>
    !>               NOTE: INFO only stores the first occurrence of
    !>               a singularity, any subsequent occurrence of singularity
    !>               is not stored in INFO even though the factorization
    !>               always completes.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !> TODO: put correct description
    !>

Contributors:

    !>
    !>  December 2016,  Igor Kozachenko,
    !>                  Computer Science Division,
    !>                  University of California, Berkeley
    !>
    !>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
    !>                  School of Mathematics,
    !>                  University of Manchester
    !>
    !>

Author

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