Provided by: liblapack-doc_3.11.0-2build1_all bug

NAME

       complexHEcomputational - complex

SYNOPSIS

   Functions
       subroutine checon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
           CHECON
       subroutine checon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO)
           CHECON_3
       subroutine checon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO)
            CHECON_ROOK estimates the reciprocal of the condition number fort HE matrices using
           factorization obtained with one of the bounded diagonal pivoting methods (max 2
           interchanges)
       subroutine cheequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
           CHEEQUB
       subroutine chegs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
           CHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using
           the factorization results obtained from cpotrf (unblocked algorithm).
       subroutine chegst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
           CHEGST
       subroutine cherfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR,
           WORK, RWORK, INFO)
           CHERFS
       subroutine cherfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX,
           RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK,
           INFO)
           CHERFSX
       subroutine chetd2 (UPLO, N, A, LDA, D, E, TAU, INFO)
           CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary
           similarity transformation (unblocked algorithm).
       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 chetf2_rk (UPLO, N, A, LDA, E, IPIV, INFO)
           CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using
           the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
       subroutine chetf2_rook (UPLO, N, A, LDA, IPIV, INFO)
           CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using
           the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
       subroutine chetrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
           CHETRD
       subroutine chetrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK,
           INFO)
           CHETRD_2STAGE
       subroutine chetrd_he2hb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
           CHETRD_HE2HB
       subroutine chetrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
           CHETRF
       subroutine chetrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
           CHETRF_AA
       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 chetrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
           CHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using
           the bounded Bunch-Kaufman ('rook') diagonal pivoting method (blocked algorithm,
           calling Level 3 BLAS).
       subroutine chetri (UPLO, N, A, LDA, IPIV, WORK, INFO)
           CHETRI
       subroutine chetri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
           CHETRI2
       subroutine chetri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO)
           CHETRI2X
       subroutine chetri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
           CHETRI_3
       subroutine chetri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO)
           CHETRI_3X
       subroutine chetri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO)
           CHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with
           the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
       subroutine chetrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
           CHETRS
       subroutine chetrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)
           CHETRS2
       subroutine chetrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
           CHETRS_3
       subroutine chetrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)
           CHETRS_AA
       subroutine chetrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
           CHETRS_ROOK computes the solution to a system of linear equations A * X = B for HE
           matrices using factorization obtained with one of the bounded diagonal pivoting
           methods (max 2 interchanges)
       subroutine cla_heamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
           CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to
           calculate error bounds.
       real function cla_hercond_c (UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK,
           RWORK)
           CLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for
           Hermitian indefinite matrices.
       real function cla_hercond_x (UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK)
           CLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for
           Hermitian indefinite matrices.
       subroutine cla_herfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU,
           C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY,
           Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
           CLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for
           Hermitian indefinite matrices by performing extra-precise iterative refinement and
           provides error bounds and backward error estimates for the solution.
       real function cla_herpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
           CLA_HERPVGRW
       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 clahef_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)
           CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix
           using bounded Bunch-Kaufman (rook) diagonal pivoting method.
       subroutine clahef_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)

Detailed Description

       This is the group of complex computational functions for HE matrices

Function Documentation

   subroutine checon (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK,
       integer INFO)
       CHECON

       Purpose:

            CHECON estimates the reciprocal of the condition number of a complex
            Hermitian matrix A using the factorization A = U*D*U**H or
            A = L*D*L**H computed by CHETRF.

            An estimate is obtained for norm(inv(A)), and the reciprocal of the
            condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The block diagonal matrix D and the multipliers used to
                     obtain the factor U or L as computed by CHETRF.

           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
                     as determined by CHETRF.

           ANORM

                     ANORM is REAL
                     The 1-norm of the original matrix A.

           RCOND

                     RCOND is REAL
                     The reciprocal of the condition number of the matrix A,
                     computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
                     estimate of the 1-norm of inv(A) computed in this routine.

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine checon_3 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       complex, dimension( * ) E, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex,
       dimension( * ) WORK, integer INFO)
       CHECON_3

       Purpose:

            CHECON_3 estimates the reciprocal of the condition number (in the
            1-norm) of a complex Hermitian matrix A using the factorization
            computed by CHETRF_RK or CHETRF_BK:

               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.

            An estimate is obtained for norm(inv(A)), and the reciprocal of the
            condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
            This routine uses BLAS3 solver CHETRS_3.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are
                     stored as an upper or lower triangular matrix:
                     = 'U':  Upper triangular, form is A = P*U*D*(U**H)*(P**T);
                     = 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     Diagonal of the block diagonal matrix D and factors U or L
                     as computed by CHETRF_RK and CHETRF_BK:
                       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
                           should be provided on entry 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 entry, 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) not referenced;
                     If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

                     NOTE: For 1-by-1 diagonal block D(k), where
                     1 <= k <= N, the element E(k) is not referenced in both
                     UPLO = 'U' or UPLO = 'L' cases.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     Details of the interchanges and the block structure of D
                     as determined by CHETRF_RK or CHETRF_BK.

           ANORM

                     ANORM is REAL
                     The 1-norm of the original matrix A.

           RCOND

                     RCOND is REAL
                     The reciprocal of the condition number of the matrix A,
                     computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
                     estimate of the 1-norm of inv(A) computed in this routine.

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:

             June 2017,  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 checon_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, complex, dimension( * ) WORK,
       integer INFO)
        CHECON_ROOK estimates the reciprocal of the condition number fort HE matrices using
       factorization obtained with one of the bounded diagonal pivoting methods (max 2
       interchanges)

       Purpose:

            CHECON_ROOK estimates the reciprocal of the condition number of a complex
            Hermitian matrix A using the factorization A = U*D*U**H or
            A = L*D*L**H computed by CHETRF_ROOK.

            An estimate is obtained for norm(inv(A)), and the reciprocal of the
            condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The block diagonal matrix D and the multipliers used to
                     obtain the factor U or L as computed by CHETRF_ROOK.

           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
                     as determined by CHETRF_ROOK.

           ANORM

                     ANORM is REAL
                     The 1-norm of the original matrix A.

           RCOND

                     RCOND is REAL
                     The reciprocal of the condition number of the matrix A,
                     computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
                     estimate of the 1-norm of inv(A) computed in this routine.

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       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

             September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                             School of Mathematics,
                             University of Manchester

   subroutine cheequb (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) S, real SCOND, real AMAX, complex, dimension( * ) WORK, integer INFO)
       CHEEQUB

       Purpose:

            CHEEQUB computes row and column scalings intended to equilibrate a
            Hermitian matrix A (with respect to the Euclidean norm) and reduce
            its condition number. The scale factors S are computed by the BIN
            algorithm (see references) so that the scaled matrix B with elements
            B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
            the smallest possible condition number over all possible diagonal
            scalings.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The N-by-N Hermitian matrix whose scaling factors are to be
                     computed.

           LDA

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

           S

                     S is REAL array, dimension (N)
                     If INFO = 0, S contains the scale factors for A.

           SCOND

                     SCOND is REAL
                     If INFO = 0, S contains the ratio of the smallest S(i) to
                     the largest S(i). If SCOND >= 0.1 and AMAX is neither too
                     large nor too small, it is not worth scaling by S.

           AMAX

                     AMAX is REAL
                     Largest absolute value of any matrix element. If AMAX is
                     very close to overflow or very close to underflow, the
                     matrix should be scaled.

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, the i-th diagonal element is nonpositive.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       References:
           Livne, O.E. and Golub, G.H., 'Scaling by Binormalization',
            Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
            DOI 10.1023/B:NUMA.0000016606.32820.69
            Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

   subroutine chegs2 (integer ITYPE, character UPLO, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer INFO)
       CHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the
       factorization results obtained from cpotrf (unblocked algorithm).

       Purpose:

            CHEGS2 reduces a complex Hermitian-definite generalized
            eigenproblem to standard form.

            If ITYPE = 1, the problem is A*x = lambda*B*x,
            and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)

            If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
            B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H *A*L.

            B must have been previously factorized as U**H *U or L*L**H by ZPOTRF.

       Parameters
           ITYPE

                     ITYPE is INTEGER
                     = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
                     = 2 or 3: compute U*A*U**H or L**H *A*L.

           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the upper or lower triangular part of the
                     Hermitian matrix A is stored, and how B has been factorized.
                     = 'U':  Upper triangular
                     = 'L':  Lower triangular

           N

                     N is INTEGER
                     The order of the matrices A and B.  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, if INFO = 0, the transformed matrix, stored in the
                     same format as A.

           LDA

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

           B

                     B is COMPLEX array, dimension (LDB,N)
                     The triangular factor from the Cholesky factorization of B,
                     as returned by CPOTRF.
                     B is modified by the routine but restored on exit.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chegst (integer ITYPE, character UPLO, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldb, * ) B, integer LDB, integer INFO)
       CHEGST

       Purpose:

            CHEGST reduces a complex Hermitian-definite generalized
            eigenproblem to standard form.

            If ITYPE = 1, the problem is A*x = lambda*B*x,
            and A is overwritten by inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H)

            If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
            B*A*x = lambda*x, and A is overwritten by U*A*U**H or L**H*A*L.

            B must have been previously factorized as U**H*U or L*L**H by CPOTRF.

       Parameters
           ITYPE

                     ITYPE is INTEGER
                     = 1: compute inv(U**H)*A*inv(U) or inv(L)*A*inv(L**H);
                     = 2 or 3: compute U*A*U**H or L**H*A*L.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored and B is factored as
                             U**H*U;
                     = 'L':  Lower triangle of A is stored and B is factored as
                             L*L**H.

           N

                     N is INTEGER
                     The order of the matrices A and B.  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, if INFO = 0, the transformed matrix, stored in the
                     same format as A.

           LDA

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

           B

                     B is COMPLEX array, dimension (LDB,N)
                     The triangular factor from the Cholesky factorization of B,
                     as returned by CPOTRF.
                     B is modified by the routine but restored on exit.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cherfs (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
       complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX,
       real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real,
       dimension( * ) RWORK, integer INFO)
       CHERFS

       Purpose:

            CHERFS improves the computed solution to a system of linear
            equations when the coefficient matrix is Hermitian indefinite, and
            provides error bounds and backward error estimates for the solution.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N

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

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrices B and X.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     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.

           LDA

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

           AF

                     AF is COMPLEX array, dimension (LDAF,N)
                     The factored form of the matrix A.  AF contains the block
                     diagonal matrix D and the multipliers used to obtain the
                     factor U or L from the factorization A = U*D*U**H or
                     A = L*D*L**H as computed by CHETRF.

           LDAF

                     LDAF is INTEGER
                     The leading dimension of the array AF.  LDAF >= max(1,N).

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     Details of the interchanges and the block structure of D
                     as determined by CHETRF.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     The right hand side matrix B.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           X

                     X is COMPLEX array, dimension (LDX,NRHS)
                     On entry, the solution matrix X, as computed by CHETRS.
                     On exit, the improved solution matrix X.

           LDX

                     LDX is INTEGER
                     The leading dimension of the array X.  LDX >= max(1,N).

           FERR

                     FERR is REAL array, dimension (NRHS)
                     The estimated forward error bound for each solution vector
                     X(j) (the j-th column of the solution matrix X).
                     If XTRUE is the true solution corresponding to X(j), FERR(j)
                     is an estimated upper bound for the magnitude of the largest
                     element in (X(j) - XTRUE) divided by the magnitude of the
                     largest element in X(j).  The estimate is as reliable as
                     the estimate for RCOND, and is almost always a slight
                     overestimate of the true error.

           BERR

                     BERR is REAL array, dimension (NRHS)
                     The componentwise relative backward error of each solution
                     vector X(j) (i.e., the smallest relative change in
                     any element of A or B that makes X(j) an exact solution).

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           RWORK

                     RWORK is REAL array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Internal Parameters:

             ITMAX is the maximum number of steps of iterative refinement.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cherfsx (character UPLO, character EQUED, integer N, integer NRHS, complex,
       dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
       integer, dimension( * ) IPIV, real, dimension( * ) S, complex, dimension( ldb, * ) B,
       integer LDB, complex, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * )
       BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs,
       * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, complex, dimension( * )
       WORK, real, dimension( * ) RWORK, integer INFO)
       CHERFSX

       Purpose:

               CHERFSX improves the computed solution to a system of linear
               equations when the coefficient matrix is Hermitian indefinite, and
               provides error bounds and backward error estimates for the
               solution.  In addition to normwise error bound, the code provides
               maximum componentwise error bound if possible.  See comments for
               ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.

               The original system of linear equations may have been equilibrated
               before calling this routine, as described by arguments EQUED and S
               below. In this case, the solution and error bounds returned are
               for the original unequilibrated system.

                Some optional parameters are bundled in the PARAMS array.  These
                settings determine how refinement is performed, but often the
                defaults are acceptable.  If the defaults are acceptable, users
                can pass NPARAMS = 0 which prevents the source code from accessing
                the PARAMS argument.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                  = 'U':  Upper triangle of A is stored;
                  = 'L':  Lower triangle of A is stored.

           EQUED

                     EQUED is CHARACTER*1
                Specifies the form of equilibration that was done to A
                before calling this routine. This is needed to compute
                the solution and error bounds correctly.
                  = 'N':  No equilibration
                  = 'Y':  Both row and column equilibration, i.e., A has been
                          replaced by diag(S) * A * diag(S).
                          The right hand side B has been changed accordingly.

           N

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

           NRHS

                     NRHS is INTEGER
                The number of right hand sides, i.e., the number of columns
                of the matrices B and X.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                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.

           LDA

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

           AF

                     AF is COMPLEX array, dimension (LDAF,N)
                The factored form of the matrix A.  AF contains the block
                diagonal matrix D and the multipliers used to obtain the
                factor U or L from the factorization A = U*D*U**H or A =
                L*D*L**H as computed by CHETRF.

           LDAF

                     LDAF is INTEGER
                The leading dimension of the array AF.  LDAF >= max(1,N).

           IPIV

                     IPIV is INTEGER array, dimension (N)
                Details of the interchanges and the block structure of D
                as determined by CHETRF.

           S

                     S is REAL array, dimension (N)
                The scale factors for A.  If EQUED = 'Y', A is multiplied on
                the left and right by diag(S).  S is an input argument if FACT =
                'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
                = 'Y', each element of S must be positive.  If S is output, each
                element of S is a power of the radix. If S is input, each element
                of S should be a power of the radix to ensure a reliable solution
                and error estimates. Scaling by powers of the radix does not cause
                rounding errors unless the result underflows or overflows.
                Rounding errors during scaling lead to refining with a matrix that
                is not equivalent to the input matrix, producing error estimates
                that may not be reliable.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                The right hand side matrix B.

           LDB

                     LDB is INTEGER
                The leading dimension of the array B.  LDB >= max(1,N).

           X

                     X is COMPLEX array, dimension (LDX,NRHS)
                On entry, the solution matrix X, as computed by CHETRS.
                On exit, the improved solution matrix X.

           LDX

                     LDX is INTEGER
                The leading dimension of the array X.  LDX >= max(1,N).

           RCOND

                     RCOND is REAL
                Reciprocal scaled condition number.  This is an estimate of the
                reciprocal Skeel condition number of the matrix A after
                equilibration (if done).  If this is less than the machine
                precision (in particular, if it is zero), the matrix is singular
                to working precision.  Note that the error may still be small even
                if this number is very small and the matrix appears ill-
                conditioned.

           BERR

                     BERR is REAL array, dimension (NRHS)
                Componentwise relative backward error.  This is the
                componentwise relative backward error of each solution vector X(j)
                (i.e., the smallest relative change in any element of A or B that
                makes X(j) an exact solution).

           N_ERR_BNDS

                     N_ERR_BNDS is INTEGER
                Number of error bounds to return for each right hand side
                and each type (normwise or componentwise).  See ERR_BNDS_NORM and
                ERR_BNDS_COMP below.

           ERR_BNDS_NORM

                     ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

                The array is indexed by the type of error information as described
                below. There currently are up to three pieces of information
                returned.

                The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_NORM(:,err) contains the following
                three fields:
                err = 1 'Trust/don't trust' boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('Epsilon').

                err = 2 'Guaranteed' error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * slamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated normwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * slamch('Epsilon') to determine if the error
                         estimate is 'guaranteed'. These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*A, where S scales each row by a power of the
                         radix so all absolute row sums of Z are approximately 1.

                See Lapack Working Note 165 for further details and extra
                cautions.

           ERR_BNDS_COMP

                     ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

                The array is indexed by the right-hand side i (on which the
                componentwise relative error depends), and the type of error
                information as described below. There currently are up to three
                pieces of information returned for each right-hand side. If
                componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
                the first (:,N_ERR_BNDS) entries are returned.

                The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_COMP(:,err) contains the following
                three fields:
                err = 1 'Trust/don't trust' boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('Epsilon').

                err = 2 'Guaranteed' error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * slamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated componentwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * slamch('Epsilon') to determine if the error
                         estimate is 'guaranteed'. These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*(A*diag(x)), where x is the solution for the
                         current right-hand side and S scales each row of
                         A*diag(x) by a power of the radix so all absolute row
                         sums of Z are approximately 1.

                See Lapack Working Note 165 for further details and extra
                cautions.

           NPARAMS

                     NPARAMS is INTEGER
                Specifies the number of parameters set in PARAMS.  If <= 0, the
                PARAMS array is never referenced and default values are used.

           PARAMS

                     PARAMS is REAL array, dimension NPARAMS
                Specifies algorithm parameters.  If an entry is < 0.0, then
                that entry will be filled with default value used for that
                parameter.  Only positions up to NPARAMS are accessed; defaults
                are used for higher-numbered parameters.

                  PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                       refinement or not.
                    Default: 1.0
                       = 0.0:  No refinement is performed, and no error bounds are
                               computed.
                       = 1.0:  Use the double-precision refinement algorithm,
                               possibly with doubled-single computations if the
                               compilation environment does not support DOUBLE
                               PRECISION.
                         (other values are reserved for future use)

                  PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                       computations allowed for refinement.
                    Default: 10
                    Aggressive: Set to 100 to permit convergence using approximate
                                factorizations or factorizations other than LU. If
                                the factorization uses a technique other than
                                Gaussian elimination, the guarantees in
                                err_bnds_norm and err_bnds_comp may no longer be
                                trustworthy.

                  PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                       will attempt to find a solution with small componentwise
                       relative error in the double-precision algorithm.  Positive
                       is true, 0.0 is false.
                    Default: 1.0 (attempt componentwise convergence)

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           RWORK

                     RWORK is REAL array, dimension (2*N)

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit. The solution to every right-hand side is
                    guaranteed.
                  < 0:  If INFO = -i, the i-th argument had an illegal value
                  > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
                    has been completed, but the factor U is exactly singular, so
                    the solution and error bounds could not be computed. RCOND = 0
                    is returned.
                  = N+J: The solution corresponding to the Jth right-hand side is
                    not guaranteed. The solutions corresponding to other right-
                    hand sides K with K > J may not be guaranteed as well, but
                    only the first such right-hand side is reported. If a small
                    componentwise error is not requested (PARAMS(3) = 0.0) then
                    the Jth right-hand side is the first with a normwise error
                    bound that is not guaranteed (the smallest J such
                    that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
                    the Jth right-hand side is the first with either a normwise or
                    componentwise error bound that is not guaranteed (the smallest
                    J such that either ERR_BNDS_NORM(J,1) = 0.0 or
                    ERR_BNDS_COMP(J,1) = 0.0). See the definition of
                    ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
                    about all of the right-hand sides check ERR_BNDS_NORM or
                    ERR_BNDS_COMP.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetd2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, integer INFO)
       CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary
       similarity transformation (unblocked algorithm).

       Purpose:

            CHETD2 reduces a complex Hermitian matrix A to real symmetric
            tridiagonal form T by a unitary similarity transformation:
            Q**H * A * Q = T.

       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, if UPLO = 'U', the diagonal and first superdiagonal
                     of A are overwritten by the corresponding elements of the
                     tridiagonal matrix T, and the elements above the first
                     superdiagonal, with the array TAU, represent the unitary
                     matrix Q as a product of elementary reflectors; if UPLO
                     = 'L', the diagonal and first subdiagonal of A are over-
                     written by the corresponding elements of the tridiagonal
                     matrix T, and the elements below the first subdiagonal, with
                     the array TAU, represent the unitary matrix Q as a product
                     of elementary reflectors. See Further Details.

           LDA

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

           D

                     D is REAL array, dimension (N)
                     The diagonal elements of the tridiagonal matrix T:
                     D(i) = A(i,i).

           E

                     E is REAL array, dimension (N-1)
                     The off-diagonal elements of the tridiagonal matrix T:
                     E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

           TAU

                     TAU is COMPLEX array, dimension (N-1)
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             If UPLO = 'U', the matrix Q is represented as a product of elementary
             reflectors

                Q = H(n-1) . . . H(2) H(1).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
             A(1:i-1,i+1), and tau in TAU(i).

             If UPLO = 'L', the matrix Q is represented as a product of elementary
             reflectors

                Q = H(1) H(2) . . . H(n-1).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
             and tau in TAU(i).

             The contents of A on exit are illustrated by the following examples
             with n = 5:

             if UPLO = 'U':                       if UPLO = 'L':

               (  d   e   v2  v3  v4 )              (  d                  )
               (      d   e   v3  v4 )              (  e   d              )
               (          d   e   v4 )              (  v1  e   d          )
               (              d   e  )              (  v1  v2  e   d      )
               (                  d  )              (  v1  v2  v3  e   d  )

             where d and e denote diagonal and off-diagonal elements of T, and vi
             denotes an element of the vector defining H(i).

   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 chetf2_rk (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       complex, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)
       CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the
       bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

       Purpose:

            CHETF2_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 unblocked version of the algorithm, calling Level 2 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.

           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 further details

       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

             01-01-96 - Based on modifications by
               J. Lewis, Boeing Computer Services Company
               A. Petitet, Computer Science Dept.,
                           Univ. of Tenn., Knoxville abd , USA

   subroutine chetf2_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) IPIV, integer INFO)
       CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the
       bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).

       Purpose:

            CHETF2_ROOK computes the factorization of a complex Hermitian matrix A
            using the bounded Bunch-Kaufman ('rook') 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) < 0 and IPIV(k-1) < 0, then rows and
                        columns k and -IPIV(k) were interchanged and rows and
                        columns k-1 and -IPIV(k-1) were inerchaged,
                        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) < 0 and IPIV(k+1) < 0, then rows and
                        columns k and -IPIV(k) were interchanged and rows and
                        columns k+1 and -IPIV(k+1) were inerchaged,
                        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:

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

             September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                             School of Mathematics,
                             University of Manchester

             01-01-96 - Based on modifications by
               J. Lewis, Boeing Computer Services Company
               A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

   subroutine chetrd (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, complex,
       dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRD

       Purpose:

            CHETRD reduces a complex Hermitian matrix A to real symmetric
            tridiagonal form T by a unitary similarity transformation:
            Q**H * A * Q = T.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           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, if UPLO = 'U', the diagonal and first superdiagonal
                     of A are overwritten by the corresponding elements of the
                     tridiagonal matrix T, and the elements above the first
                     superdiagonal, with the array TAU, represent the unitary
                     matrix Q as a product of elementary reflectors; if UPLO
                     = 'L', the diagonal and first subdiagonal of A are over-
                     written by the corresponding elements of the tridiagonal
                     matrix T, and the elements below the first subdiagonal, with
                     the array TAU, represent the unitary matrix Q as a product
                     of elementary reflectors. See Further Details.

           LDA

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

           D

                     D is REAL array, dimension (N)
                     The diagonal elements of the tridiagonal matrix T:
                     D(i) = A(i,i).

           E

                     E is REAL array, dimension (N-1)
                     The off-diagonal elements of the tridiagonal matrix T:
                     E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

           TAU

                     TAU is COMPLEX array, dimension (N-1)
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           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 dimension of the array WORK.  LWORK >= 1.
                     For optimum performance LWORK >= N*NB, where NB is the
                     optimal blocksize.

                     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 = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             If UPLO = 'U', the matrix Q is represented as a product of elementary
             reflectors

                Q = H(n-1) . . . H(2) H(1).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
             A(1:i-1,i+1), and tau in TAU(i).

             If UPLO = 'L', the matrix Q is represented as a product of elementary
             reflectors

                Q = H(1) H(2) . . . H(n-1).

             Each H(i) has the form

                H(i) = I - tau * v * v**H

             where tau is a complex scalar, and v is a complex vector with
             v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
             and tau in TAU(i).

             The contents of A on exit are illustrated by the following examples
             with n = 5:

             if UPLO = 'U':                       if UPLO = 'L':

               (  d   e   v2  v3  v4 )              (  d                  )
               (      d   e   v3  v4 )              (  e   d              )
               (          d   e   v4 )              (  v1  e   d          )
               (              d   e  )              (  v1  v2  e   d      )
               (                  d  )              (  v1  v2  v3  e   d  )

             where d and e denote diagonal and off-diagonal elements of T, and vi
             denotes an element of the vector defining H(i).

   subroutine chetrd_2stage (character VECT, character UPLO, integer N, complex, dimension( lda,
       * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( *
       ) TAU, complex, dimension( * ) HOUS2, integer LHOUS2, complex, dimension( * ) WORK,
       integer LWORK, integer INFO)
       CHETRD_2STAGE

       Purpose:

            CHETRD_2STAGE reduces a complex Hermitian matrix A to real symmetric
            tridiagonal form T by a unitary similarity transformation:
            Q1**H Q2**H* A * Q2 * Q1 = T.

       Parameters
           VECT

                     VECT is CHARACTER*1
                     = 'N':  No need for the Housholder representation,
                             in particular for the second stage (Band to
                             tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
                     = 'V':  the Householder representation is needed to
                             either generate Q1 Q2 or to apply Q1 Q2,
                             then LHOUS2 is to be queried and computed.
                             (NOT AVAILABLE IN THIS RELEASE).

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           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, if UPLO = 'U', the band superdiagonal
                     of A are overwritten by the corresponding elements of the
                     internal band-diagonal matrix AB, and the elements above
                     the KD superdiagonal, with the array TAU, represent the unitary
                     matrix Q1 as a product of elementary reflectors; if UPLO
                     = 'L', the diagonal and band subdiagonal of A are over-
                     written by the corresponding elements of the internal band-diagonal
                     matrix AB, and the elements below the KD subdiagonal, with
                     the array TAU, represent the unitary matrix Q1 as a product
                     of elementary reflectors. See Further Details.

           LDA

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

           D

                     D is REAL array, dimension (N)
                     The diagonal elements of the tridiagonal matrix T.

           E

                     E is REAL array, dimension (N-1)
                     The off-diagonal elements of the tridiagonal matrix T.

           TAU

                     TAU is COMPLEX array, dimension (N-KD)
                     The scalar factors of the elementary reflectors of
                     the first stage (see Further Details).

           HOUS2

                     HOUS2 is COMPLEX array, dimension (LHOUS2)
                     Stores the Householder representation of the stage2
                     band to tridiagonal.

           LHOUS2

                     LHOUS2 is INTEGER
                     The dimension of the array HOUS2.
                     If LWORK = -1, or LHOUS2=-1,
                     then a query is assumed; the routine
                     only calculates the optimal size of the HOUS2 array, returns
                     this value as the first entry of the HOUS2 array, and no error
                     message related to LHOUS2 is issued by XERBLA.
                     If VECT='N', LHOUS2 = max(1, 4*n);
                     if VECT='V', option not yet available.

           WORK

                     WORK is COMPLEX array, dimension (LWORK)

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK = MAX(1, dimension)
                     If LWORK = -1, or LHOUS2 = -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.
                     LWORK = MAX(1, dimension) where
                     dimension   = max(stage1,stage2) + (KD+1)*N
                                 = N*KD + N*max(KD+1,FACTOPTNB)
                                   + max(2*KD*KD, KD*NTHREADS)
                                   + (KD+1)*N
                     where KD is the blocking size of the reduction,
                     FACTOPTNB is the blocking used by the QR or LQ
                     algorithm, usually FACTOPTNB=128 is a good choice
                     NTHREADS is the number of threads used when
                     openMP compilation is enabled, otherwise =1.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Implemented by Azzam Haidar.

             All details are available on technical report, SC11, SC13 papers.

             Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
             Parallel reduction to condensed forms for symmetric eigenvalue problems
             using aggregated fine-grained and memory-aware kernels. In Proceedings
             of 2011 International Conference for High Performance Computing,
             Networking, Storage and Analysis (SC '11), New York, NY, USA,
             Article 8 , 11 pages.
             http://doi.acm.org/10.1145/2063384.2063394

             A. Haidar, J. Kurzak, P. Luszczek, 2013.
             An improved parallel singular value algorithm and its implementation
             for multicore hardware, In Proceedings of 2013 International Conference
             for High Performance Computing, Networking, Storage and Analysis (SC '13).
             Denver, Colorado, USA, 2013.
             Article 90, 12 pages.
             http://doi.acm.org/10.1145/2503210.2503292

             A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
             A novel hybrid CPU-GPU generalized eigensolver for electronic structure
             calculations based on fine-grained memory aware tasks.
             International Journal of High Performance Computing Applications.
             Volume 28 Issue 2, Pages 196-209, May 2014.
             http://hpc.sagepub.com/content/28/2/196

   subroutine chetrd_he2hb (character UPLO, integer N, integer KD, complex, dimension( lda, * )
       A, integer LDA, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( * )
       TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRD_HE2HB

       Purpose:

            CHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
            band-diagonal form AB by a unitary similarity transformation:
            Q**H * A * Q = AB.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N

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

           KD

                     KD is INTEGER
                     The number of superdiagonals of the reduced matrix if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
                     The reduced matrix is stored in the array AB.

           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, if UPLO = 'U', the diagonal and first superdiagonal
                     of A are overwritten by the corresponding elements of the
                     tridiagonal matrix T, and the elements above the first
                     superdiagonal, with the array TAU, represent the unitary
                     matrix Q as a product of elementary reflectors; if UPLO
                     = 'L', the diagonal and first subdiagonal of A are over-
                     written by the corresponding elements of the tridiagonal
                     matrix T, and the elements below the first subdiagonal, with
                     the array TAU, represent the unitary matrix Q as a product
                     of elementary reflectors. See Further Details.

           LDA

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

           AB

                     AB is COMPLEX array, dimension (LDAB,N)
                     On exit, the upper or lower triangle of the Hermitian band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

           LDAB

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KD+1.

           TAU

                     TAU is COMPLEX array, dimension (N-KD)
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is COMPLEX array, dimension (LWORK)
                     On exit, if INFO = 0, or if LWORK=-1,
                     WORK(1) returns the size of LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK which should be calculated
                     by a workspace query. LWORK = MAX(1, LWORK_QUERY)
                     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.
                     LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
                     where FACTOPTNB is the blocking used by the QR or LQ
                     algorithm, usually FACTOPTNB=128 is a good choice otherwise
                     putting LWORK=-1 will provide the size of WORK.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Implemented by Azzam Haidar.

             All details are available on technical report, SC11, SC13 papers.

             Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
             Parallel reduction to condensed forms for symmetric eigenvalue problems
             using aggregated fine-grained and memory-aware kernels. In Proceedings
             of 2011 International Conference for High Performance Computing,
             Networking, Storage and Analysis (SC '11), New York, NY, USA,
             Article 8 , 11 pages.
             http://doi.acm.org/10.1145/2063384.2063394

             A. Haidar, J. Kurzak, P. Luszczek, 2013.
             An improved parallel singular value algorithm and its implementation
             for multicore hardware, In Proceedings of 2013 International Conference
             for High Performance Computing, Networking, Storage and Analysis (SC '13).
             Denver, Colorado, USA, 2013.
             Article 90, 12 pages.
             http://doi.acm.org/10.1145/2503210.2503292

             A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
             A novel hybrid CPU-GPU generalized eigensolver for electronic structure
             calculations based on fine-grained memory aware tasks.
             International Journal of High Performance Computing Applications.
             Volume 28 Issue 2, Pages 196-209, May 2014.
             http://hpc.sagepub.com/content/28/2/196

         If UPLO = 'U', the matrix Q is represented as a product of elementary
         reflectors

            Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.

         Each H(i) has the form

            H(i) = I - tau * v * v**H

         where tau is a complex scalar, and v is a complex vector with
         v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
         A(i,i+kd+1:n), and tau in TAU(i).

         If UPLO = 'L', the matrix Q is represented as a product of elementary
         reflectors

            Q = H(1) H(2) . . . H(k), where k = n-kd.

         Each H(i) has the form

            H(i) = I - tau * v * v**H

         where tau is a complex scalar, and v is a complex vector with
         v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
         A(i+kd+2:n,i), and tau in TAU(i).

         The contents of A on exit are illustrated by the following examples
         with n = 5:

         if UPLO = 'U':                       if UPLO = 'L':

           (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
           (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
           (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
           (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
           (                            ab    )              (  v1     v2     v3     ab/v4 ab )

         where d and e denote diagonal and off-diagonal elements of T, and vi
         denotes an element of the vector defining H(i)..fi

   subroutine chetrf (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRF

       Purpose:

            CHETRF computes the factorization of a complex Hermitian matrix A
            using the Bunch-Kaufman diagonal pivoting method.  The form of the
            factorization is

               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, 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.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           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 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 UPLO = 'U' and 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' and 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.

           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.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, D(i,i) 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).

   subroutine chetrf_aa (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRF_AA

       Purpose:

            CHETRF_AA computes the factorization of a complex hermitian matrix A
            using the Aasen's algorithm.  The form of the factorization is

               A = U**H*T*U  or  A = L*T*L**H

            where U (or L) is a product of permutation and unit upper (lower)
            triangular matrices, and T is a hermitian tridiagonal matrix.

            This is the blocked version of the algorithm, calling Level 3 BLAS.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           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 tridiagonal matrix is stored in the diagonals
                     and the subdiagonals of A just below (or above) the diagonals,
                     and L is stored below (or above) the subdiaonals, when UPLO
                     is 'L' (or 'U').

           LDA

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

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     On exit, it contains the details of the interchanges, i.e.,
                     the row and column k of A were interchanged with the
                     row and column IPIV(k).

           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 >= 2*N. For optimum performance
                     LWORK >= N*(1+NB), where NB is the optimal blocksize.

                     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 = -i, the i-th argument had an illegal value.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   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 chetrf_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer
       INFO)
       CHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using the
       bounded Bunch-Kaufman ('rook') diagonal pivoting method (blocked algorithm, calling Level
       3 BLAS).

       Purpose:

            CHETRF_ROOK computes the factorization of a complex Hermitian matrix A
            using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
            The form of the factorization is

               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, 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.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

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

           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 = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, D(i,i) 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:

             June 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 chetri (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer INFO)
       CHETRI

       Purpose:

            CHETRI computes the inverse of a complex Hermitian indefinite matrix
            A using the factorization A = U*D*U**H or A = L*D*L**H computed by
            CHETRF.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the block diagonal matrix D and the multipliers
                     used to obtain the factor U or L as computed by CHETRF.

                     On exit, if INFO = 0, the (Hermitian) inverse of the original
                     matrix.  If UPLO = 'U', the upper triangular part of the
                     inverse is formed and the part of A below the diagonal is not
                     referenced; if UPLO = 'L' the lower triangular part of the
                     inverse is formed and the part of A above the diagonal is
                     not referenced.

           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
                     as determined by CHETRF.

           WORK

                     WORK is COMPLEX array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value
                     > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                          inverse could not be computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetri2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRI2

       Purpose:

            CHETRI2 computes the inverse of a COMPLEX hermitian indefinite matrix
            A using the factorization A = U*D*U**T or A = L*D*L**T computed by
            CHETRF. CHETRI2 set the LEADING DIMENSION of the workspace
            before calling CHETRI2X that actually computes the inverse.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**T;
                     = 'L':  Lower triangular, form is A = L*D*L**T.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the block diagonal matrix D and the multipliers
                     used to obtain the factor U or L as computed by CHETRF.

                     On exit, if INFO = 0, the (symmetric) inverse of the original
                     matrix.  If UPLO = 'U', the upper triangular part of the
                     inverse is formed and the part of A below the diagonal is not
                     referenced; if UPLO = 'L' the lower triangular part of the
                     inverse is formed and the part of A above the diagonal is
                     not referenced.

           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
                     as determined by CHETRF.

           WORK

                     WORK is COMPLEX array, dimension (N+NB+1)*(NB+3)

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     WORK is size >= (N+NB+1)*(NB+3)
                     If LWORK = -1, then a workspace query is assumed; the routine
                      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 = -i, the i-th argument had an illegal value
                     > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                          inverse could not be computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetri2x (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       integer, dimension( * ) IPIV, complex, dimension( n+nb+1,* ) WORK, integer NB, integer
       INFO)
       CHETRI2X

       Purpose:

            CHETRI2X computes the inverse of a complex Hermitian indefinite matrix
            A using the factorization A = U*D*U**H or A = L*D*L**H computed by
            CHETRF.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the NNB diagonal matrix D and the multipliers
                     used to obtain the factor U or L as computed by CHETRF.

                     On exit, if INFO = 0, the (symmetric) inverse of the original
                     matrix.  If UPLO = 'U', the upper triangular part of the
                     inverse is formed and the part of A below the diagonal is not
                     referenced; if UPLO = 'L' the lower triangular part of the
                     inverse is formed and the part of A above the diagonal is
                     not referenced.

           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 NNB structure of D
                     as determined by CHETRF.

           WORK

                     WORK is COMPLEX array, dimension (N+NB+1,NB+3)

           NB

                     NB is INTEGER
                     Block size

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value
                     > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                          inverse could not be computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetri_3 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( * ) WORK,
       integer LWORK, integer INFO)
       CHETRI_3

       Purpose:

            CHETRI_3 computes the inverse of a complex Hermitian indefinite
            matrix A using the factorization computed by CHETRF_RK or CHETRF_BK:

                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.

            CHETRI_3 sets the leading dimension of the workspace  before calling
            CHETRI_3X that actually computes the inverse.  This is the blocked
            version of the algorithm, calling Level 3 BLAS.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are
                     stored as an upper or lower triangular matrix.
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, diagonal of the block diagonal matrix D and
                     factors U or L as computed by CHETRF_RK and CHETRF_BK:
                       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
                           should be provided on entry 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.

                     On exit, if INFO = 0, the Hermitian inverse of the original
                     matrix.
                        If UPLO = 'U': the upper triangular part of the inverse
                        is formed and the part of A below the diagonal is not
                        referenced;
                        If UPLO = 'L': the lower triangular part of the inverse
                        is formed and the part of A above the diagonal is not
                        referenced.

           LDA

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

           E

                     E is COMPLEX array, dimension (N)
                     On entry, 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) not referenced;
                     If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

                     NOTE: For 1-by-1 diagonal block D(k), where
                     1 <= k <= N, the element E(k) is not referenced in both
                     UPLO = 'U' or UPLO = 'L' cases.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     Details of the interchanges and the block structure of D
                     as determined by CHETRF_RK or CHETRF_BK.

           WORK

                     WORK is COMPLEX array, dimension (N+NB+1)*(NB+3).
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The length of WORK. LWORK >= (N+NB+1)*(NB+3).

                     If LDWORK = -1, then a workspace query is assumed;
                     the routine only calculates the optimal size of 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 = -i, the i-th argument had an illegal value
                     > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                          inverse could not be computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:

             November 2017,  Igor Kozachenko,
                             Computer Science Division,
                             University of California, Berkeley

   subroutine chetri_3x (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA,
       complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension( n+nb+1, * )
       WORK, integer NB, integer INFO)
       CHETRI_3X

       Purpose:

            CHETRI_3X computes the inverse of a complex Hermitian indefinite
            matrix A using the factorization computed by CHETRF_RK or CHETRF_BK:

                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.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are
                     stored as an upper or lower triangular matrix.
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, diagonal of the block diagonal matrix D and
                     factors U or L as computed by CHETRF_RK and CHETRF_BK:
                       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
                           should be provided on entry 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.

                     On exit, if INFO = 0, the Hermitian inverse of the original
                     matrix.
                        If UPLO = 'U': the upper triangular part of the inverse
                        is formed and the part of A below the diagonal is not
                        referenced;
                        If UPLO = 'L': the lower triangular part of the inverse
                        is formed and the part of A above the diagonal is not
                        referenced.

           LDA

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

           E

                     E is COMPLEX array, dimension (N)
                     On entry, 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) not referenced;
                     If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.

                     NOTE: For 1-by-1 diagonal block D(k), where
                     1 <= k <= N, the element E(k) is not referenced in both
                     UPLO = 'U' or UPLO = 'L' cases.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     Details of the interchanges and the block structure of D
                     as determined by CHETRF_RK or CHETRF_BK.

           WORK

                     WORK is COMPLEX array, dimension (N+NB+1,NB+3).

           NB

                     NB is INTEGER
                     Block size.

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value
                     > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                          inverse could not be computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:

             June 2017,  Igor Kozachenko,
                             Computer Science Division,
                             University of California, Berkeley

   subroutine chetri_rook (character UPLO, integer N, complex, dimension( lda, * ) A, integer
       LDA, integer, dimension( * ) IPIV, complex, dimension( * ) WORK, integer INFO)
       CHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the
       bounded Bunch-Kaufman ('rook') diagonal pivoting method.

       Purpose:

            CHETRI_ROOK computes the inverse of a complex Hermitian indefinite matrix
            A using the factorization A = U*D*U**H or A = L*D*L**H computed by
            CHETRF_ROOK.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           A

                     A is COMPLEX array, dimension (LDA,N)
                     On entry, the block diagonal matrix D and the multipliers
                     used to obtain the factor U or L as computed by CHETRF_ROOK.

                     On exit, if INFO = 0, the (Hermitian) inverse of the original
                     matrix.  If UPLO = 'U', the upper triangular part of the
                     inverse is formed and the part of A below the diagonal is not
                     referenced; if UPLO = 'L' the lower triangular part of the
                     inverse is formed and the part of A above the diagonal is
                     not referenced.

           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
                     as determined by CHETRF_ROOK.

           WORK

                     WORK is COMPLEX array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value
                     > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
                          inverse could not be computed.

       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

             September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                             School of Mathematics,
                             University of Manchester

   subroutine chetrs (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A,
       integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB,
       integer INFO)
       CHETRS

       Purpose:

            CHETRS solves a system of linear equations A*X = B with a complex
            Hermitian matrix A using the factorization A = U*D*U**H or
            A = L*D*L**H computed by CHETRF.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The block diagonal matrix D and the multipliers used to
                     obtain the factor U or L as computed by CHETRF.

           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
                     as determined by CHETRF.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetrs2 (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A,
       integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB,
       complex, dimension( * ) WORK, integer INFO)
       CHETRS2

       Purpose:

            CHETRS2 solves a system of linear equations A*X = B with a complex
            Hermitian matrix A using the factorization A = U*D*U**H or
            A = L*D*L**H computed by CHETRF and converted by CSYCONV.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The block diagonal matrix D and the multipliers used to
                     obtain the factor U or L as computed by CHETRF.

           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
                     as determined by CHETRF.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           WORK

                     WORK is COMPLEX array, dimension (N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetrs_3 (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex, dimension(
       ldb, * ) B, integer LDB, integer INFO)
       CHETRS_3

       Purpose:

            CHETRS_3 solves a system of linear equations A * X = B with a complex
            Hermitian matrix A using the factorization computed
            by CHETRF_RK or CHETRF_BK:

               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 algorithm is using Level 3 BLAS.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are
                     stored as an upper or lower triangular matrix:
                     = 'U':  Upper triangular, form is A = P*U*D*(U**H)*(P**T);
                     = 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).

           N

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

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     Diagonal of the block diagonal matrix D and factors U or L
                     as computed by CHETRF_RK and CHETRF_BK:
                       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
                           should be provided on entry 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 entry, 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) not referenced;
                     If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

                     NOTE: For 1-by-1 diagonal block D(k), where
                     1 <= k <= N, the element E(k) is not referenced in both
                     UPLO = 'U' or UPLO = 'L' cases.

           IPIV

                     IPIV is INTEGER array, dimension (N)
                     Details of the interchanges and the block structure of D
                     as determined by CHETRF_RK or CHETRF_BK.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:

             June 2017,  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 chetrs_aa (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A,
       integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB,
       complex, dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRS_AA

       Purpose:

            CHETRS_AA solves a system of linear equations A*X = B with a complex
            hermitian matrix A using the factorization A = U**H*T*U or
            A = L*T*L**H computed by CHETRF_AA.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U**H*T*U;
                     = 'L':  Lower triangular, form is A = L*T*L**H.

           N

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

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     Details of factors computed by CHETRF_AA.

           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 as computed by CHETRF_AA.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           WORK

                     WORK is COMPLEX array, dimension (MAX(1,LWORK))

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= max(1,3*N-2).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine chetrs_rook (character UPLO, integer N, integer NRHS, complex, dimension( lda, * )
       A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB,
       integer INFO)
       CHETRS_ROOK computes the solution to a system of linear equations A * X = B for HE
       matrices using factorization obtained with one of the bounded diagonal pivoting methods
       (max 2 interchanges)

       Purpose:

            CHETRS_ROOK solves a system of linear equations A*X = B with a complex
            Hermitian matrix A using the factorization A = U*D*U**H or
            A = L*D*L**H computed by CHETRF_ROOK.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     Specifies whether the details of the factorization are stored
                     as an upper or lower triangular matrix.
                     = 'U':  Upper triangular, form is A = U*D*U**H;
                     = 'L':  Lower triangular, form is A = L*D*L**H.

           N

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

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrix B.  NRHS >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The block diagonal matrix D and the multipliers used to
                     obtain the factor U or L as computed by CHETRF_ROOK.

           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
                     as determined by CHETRF_ROOK.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                     On entry, the right hand side matrix B.
                     On exit, the solution matrix X.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,N).

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       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

             September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                             School of Mathematics,
                             University of Manchester

   subroutine cla_heamv (integer UPLO, integer N, real ALPHA, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y,
       integer INCY)
       CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to
       calculate error bounds.

       Purpose:

            CLA_SYAMV  performs the matrix-vector operation

                    y := alpha*abs(A)*abs(x) + beta*abs(y),

            where alpha and beta are scalars, x and y are vectors and A is an
            n by n symmetric matrix.

            This function is primarily used in calculating error bounds.
            To protect against underflow during evaluation, components in
            the resulting vector are perturbed away from zero by (N+1)
            times the underflow threshold.  To prevent unnecessarily large
            errors for block-structure embedded in general matrices,
            'symbolically' zero components are not perturbed.  A zero
            entry is considered 'symbolic' if all multiplications involved
            in computing that entry have at least one zero multiplicand.

       Parameters
           UPLO

                     UPLO is INTEGER
                      On entry, UPLO specifies whether the upper or lower
                      triangular part of the array A is to be referenced as
                      follows:

                         UPLO = BLAS_UPPER   Only the upper triangular part of A
                                             is to be referenced.

                         UPLO = BLAS_LOWER   Only the lower triangular part of A
                                             is to be referenced.

                      Unchanged on exit.

           N

                     N is INTEGER
                      On entry, N specifies the number of columns of the matrix A.
                      N must be at least zero.
                      Unchanged on exit.

           ALPHA

                     ALPHA is REAL .
                      On entry, ALPHA specifies the scalar alpha.
                      Unchanged on exit.

           A

                     A is COMPLEX array, dimension ( LDA, n ).
                      Before entry, the leading m by n part of the array A must
                      contain the matrix of coefficients.
                      Unchanged on exit.

           LDA

                     LDA is INTEGER
                      On entry, LDA specifies the first dimension of A as declared
                      in the calling (sub) program. LDA must be at least
                      max( 1, n ).
                      Unchanged on exit.

           X

                     X is COMPLEX array, dimension
                      ( 1 + ( n - 1 )*abs( INCX ) )
                      Before entry, the incremented array X must contain the
                      vector x.
                      Unchanged on exit.

           INCX

                     INCX is INTEGER
                      On entry, INCX specifies the increment for the elements of
                      X. INCX must not be zero.
                      Unchanged on exit.

           BETA

                     BETA is REAL .
                      On entry, BETA specifies the scalar beta. When BETA is
                      supplied as zero then Y need not be set on input.
                      Unchanged on exit.

           Y

                     Y is REAL array, dimension
                      ( 1 + ( n - 1 )*abs( INCY ) )
                      Before entry with BETA non-zero, the incremented array Y
                      must contain the vector y. On exit, Y is overwritten by the
                      updated vector y.

           INCY

                     INCY is INTEGER
                      On entry, INCY specifies the increment for the elements of
                      Y. INCY must not be zero.
                      Unchanged on exit.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             Level 2 Blas routine.

             -- Written on 22-October-1986.
                Jack Dongarra, Argonne National Lab.
                Jeremy Du Croz, Nag Central Office.
                Sven Hammarling, Nag Central Office.
                Richard Hanson, Sandia National Labs.
             -- Modified for the absolute-value product, April 2006
                Jason Riedy, UC Berkeley

   real function cla_hercond_c (character UPLO, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
       real, dimension ( * ) C, logical CAPPLY, integer INFO, complex, dimension( * ) WORK, real,
       dimension( * ) RWORK)
       CLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for
       Hermitian indefinite matrices.

       Purpose:

               CLA_HERCOND_C computes the infinity norm condition number of
               op(A) * inv(diag(C)) where C is a REAL vector.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                  = 'U':  Upper triangle of A is stored;
                  = 'L':  Lower triangle of A is stored.

           N

                     N is INTEGER
                The number of linear equations, i.e., the order of the
                matrix A.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                On entry, the N-by-N matrix A

           LDA

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

           AF

                     AF is COMPLEX array, dimension (LDAF,N)
                The block diagonal matrix D and the multipliers used to
                obtain the factor U or L as computed by CHETRF.

           LDAF

                     LDAF is INTEGER
                The leading dimension of the array AF.  LDAF >= max(1,N).

           IPIV

                     IPIV is INTEGER array, dimension (N)
                Details of the interchanges and the block structure of D
                as determined by CHETRF.

           C

                     C is REAL array, dimension (N)
                The vector C in the formula op(A) * inv(diag(C)).

           CAPPLY

                     CAPPLY is LOGICAL
                If .TRUE. then access the vector C in the formula above.

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                i > 0:  The ith argument is invalid.

           WORK

                     WORK is COMPLEX array, dimension (2*N).
                Workspace.

           RWORK

                     RWORK is REAL array, dimension (N).
                Workspace.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function cla_hercond_x (character UPLO, integer N, complex, dimension( lda, * ) A,
       integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
       complex, dimension( * ) X, integer INFO, complex, dimension( * ) WORK, real, dimension( *
       ) RWORK)
       CLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian
       indefinite matrices.

       Purpose:

               CLA_HERCOND_X computes the infinity norm condition number of
               op(A) * diag(X) where X is a COMPLEX vector.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                  = 'U':  Upper triangle of A is stored;
                  = 'L':  Lower triangle of A is stored.

           N

                     N is INTEGER
                The number of linear equations, i.e., the order of the
                matrix A.  N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is COMPLEX array, dimension (LDAF,N)
                The block diagonal matrix D and the multipliers used to
                obtain the factor U or L as computed by CHETRF.

           LDAF

                     LDAF is INTEGER
                The leading dimension of the array AF.  LDAF >= max(1,N).

           IPIV

                     IPIV is INTEGER array, dimension (N)
                Details of the interchanges and the block structure of D
                as determined by CHETRF.

           X

                     X is COMPLEX array, dimension (N)
                The vector X in the formula op(A) * diag(X).

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                i > 0:  The ith argument is invalid.

           WORK

                     WORK is COMPLEX array, dimension (2*N).
                Workspace.

           RWORK

                     RWORK is REAL array, dimension (N).
                Workspace.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine cla_herfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS,
       complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer
       LDAF, integer, dimension( * ) IPIV, logical COLEQU, real, dimension( * ) C, complex,
       dimension( ldb, * ) B, integer LDB, complex, dimension( ldy, * ) Y, integer LDY, real,
       dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real,
       dimension( nrhs, * ) ERR_BNDS_COMP, complex, dimension( * ) RES, real, dimension( * ) AYB,
       complex, dimension( * ) DY, complex, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH,
       real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)
       CLA_HERFSX_EXTENDED improves the computed solution to a system of linear equations for
       Hermitian indefinite matrices by performing extra-precise iterative refinement and
       provides error bounds and backward error estimates for the solution.

       Purpose:

            CLA_HERFSX_EXTENDED improves the computed solution to a system of
            linear equations by performing extra-precise iterative refinement
            and provides error bounds and backward error estimates for the solution.
            This subroutine is called by CHERFSX to perform iterative refinement.
            In addition to normwise error bound, the code provides maximum
            componentwise error bound if possible. See comments for ERR_BNDS_NORM
            and ERR_BNDS_COMP for details of the error bounds. Note that this
            subroutine is only responsible for setting the second fields of
            ERR_BNDS_NORM and ERR_BNDS_COMP.

       Parameters
           PREC_TYPE

                     PREC_TYPE is INTEGER
                Specifies the intermediate precision to be used in refinement.
                The value is defined by ILAPREC(P) where P is a CHARACTER and P
                     = 'S':  Single
                     = 'D':  Double
                     = 'I':  Indigenous
                     = 'X' or 'E':  Extra

           UPLO

                     UPLO is CHARACTER*1
                  = 'U':  Upper triangle of A is stored;
                  = 'L':  Lower triangle of A is stored.

           N

                     N is INTEGER
                The number of linear equations, i.e., the order of the
                matrix A.  N >= 0.

           NRHS

                     NRHS is INTEGER
                The number of right-hand-sides, i.e., the number of columns of the
                matrix B.

           A

                     A is COMPLEX array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is COMPLEX array, dimension (LDAF,N)
                The block diagonal matrix D and the multipliers used to
                obtain the factor U or L as computed by CHETRF.

           LDAF

                     LDAF is INTEGER
                The leading dimension of the array AF.  LDAF >= max(1,N).

           IPIV

                     IPIV is INTEGER array, dimension (N)
                Details of the interchanges and the block structure of D
                as determined by CHETRF.

           COLEQU

                     COLEQU is LOGICAL
                If .TRUE. then column equilibration was done to A before calling
                this routine. This is needed to compute the solution and error
                bounds correctly.

           C

                     C is REAL array, dimension (N)
                The column scale factors for A. If COLEQU = .FALSE., C
                is not accessed. If C is input, each element of C should be a power
                of the radix to ensure a reliable solution and error estimates.
                Scaling by powers of the radix does not cause rounding errors unless
                the result underflows or overflows. Rounding errors during scaling
                lead to refining with a matrix that is not equivalent to the
                input matrix, producing error estimates that may not be
                reliable.

           B

                     B is COMPLEX array, dimension (LDB,NRHS)
                The right-hand-side matrix B.

           LDB

                     LDB is INTEGER
                The leading dimension of the array B.  LDB >= max(1,N).

           Y

                     Y is COMPLEX array, dimension (LDY,NRHS)
                On entry, the solution matrix X, as computed by CHETRS.
                On exit, the improved solution matrix Y.

           LDY

                     LDY is INTEGER
                The leading dimension of the array Y.  LDY >= max(1,N).

           BERR_OUT

                     BERR_OUT is REAL array, dimension (NRHS)
                On exit, BERR_OUT(j) contains the componentwise relative backward
                error for right-hand-side j from the formula
                    max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
                where abs(Z) is the componentwise absolute value of the matrix
                or vector Z. This is computed by CLA_LIN_BERR.

           N_NORMS

                     N_NORMS is INTEGER
                Determines which error bounds to return (see ERR_BNDS_NORM
                and ERR_BNDS_COMP).
                If N_NORMS >= 1 return normwise error bounds.
                If N_NORMS >= 2 return componentwise error bounds.

           ERR_BNDS_NORM

                     ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                normwise relative error, which is defined as follows:

                Normwise relative error in the ith solution vector:
                        max_j (abs(XTRUE(j,i) - X(j,i)))
                       ------------------------------
                             max_j abs(X(j,i))

                The array is indexed by the type of error information as described
                below. There currently are up to three pieces of information
                returned.

                The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_NORM(:,err) contains the following
                three fields:
                err = 1 'Trust/don't trust' boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('Epsilon').

                err = 2 'Guaranteed' error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * slamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated normwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * slamch('Epsilon') to determine if the error
                         estimate is 'guaranteed'. These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*A, where S scales each row by a power of the
                         radix so all absolute row sums of Z are approximately 1.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           ERR_BNDS_COMP

                     ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
                For each right-hand side, this array contains information about
                various error bounds and condition numbers corresponding to the
                componentwise relative error, which is defined as follows:

                Componentwise relative error in the ith solution vector:
                               abs(XTRUE(j,i) - X(j,i))
                        max_j ----------------------
                                    abs(X(j,i))

                The array is indexed by the right-hand side i (on which the
                componentwise relative error depends), and the type of error
                information as described below. There currently are up to three
                pieces of information returned for each right-hand side. If
                componentwise accuracy is not requested (PARAMS(3) = 0.0), then
                ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
                the first (:,N_ERR_BNDS) entries are returned.

                The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
                right-hand side.

                The second index in ERR_BNDS_COMP(:,err) contains the following
                three fields:
                err = 1 'Trust/don't trust' boolean. Trust the answer if the
                         reciprocal condition number is less than the threshold
                         sqrt(n) * slamch('Epsilon').

                err = 2 'Guaranteed' error bound: The estimated forward error,
                         almost certainly within a factor of 10 of the true error
                         so long as the next entry is greater than the threshold
                         sqrt(n) * slamch('Epsilon'). This error bound should only
                         be trusted if the previous boolean is true.

                err = 3  Reciprocal condition number: Estimated componentwise
                         reciprocal condition number.  Compared with the threshold
                         sqrt(n) * slamch('Epsilon') to determine if the error
                         estimate is 'guaranteed'. These reciprocal condition
                         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                         appropriately scaled matrix Z.
                         Let Z = S*(A*diag(x)), where x is the solution for the
                         current right-hand side and S scales each row of
                         A*diag(x) by a power of the radix so all absolute row
                         sums of Z are approximately 1.

                This subroutine is only responsible for setting the second field
                above.
                See Lapack Working Note 165 for further details and extra
                cautions.

           RES

                     RES is COMPLEX array, dimension (N)
                Workspace to hold the intermediate residual.

           AYB

                     AYB is REAL array, dimension (N)
                Workspace.

           DY

                     DY is COMPLEX array, dimension (N)
                Workspace to hold the intermediate solution.

           Y_TAIL

                     Y_TAIL is COMPLEX array, dimension (N)
                Workspace to hold the trailing bits of the intermediate solution.

           RCOND

                     RCOND is REAL
                Reciprocal scaled condition number.  This is an estimate of the
                reciprocal Skeel condition number of the matrix A after
                equilibration (if done).  If this is less than the machine
                precision (in particular, if it is zero), the matrix is singular
                to working precision.  Note that the error may still be small even
                if this number is very small and the matrix appears ill-
                conditioned.

           ITHRESH

                     ITHRESH is INTEGER
                The maximum number of residual computations allowed for
                refinement. The default is 10. For 'aggressive' set to 100 to
                permit convergence using approximate factorizations or
                factorizations other than LU. If the factorization uses a
                technique other than Gaussian elimination, the guarantees in
                ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

           RTHRESH

                     RTHRESH is REAL
                Determines when to stop refinement if the error estimate stops
                decreasing. Refinement will stop when the next solution no longer
                satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
                the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
                default value is 0.5. For 'aggressive' set to 0.9 to permit
                convergence on extremely ill-conditioned matrices. See LAWN 165
                for more details.

           DZ_UB

                     DZ_UB is REAL
                Determines when to start considering componentwise convergence.
                Componentwise convergence is only considered after each component
                of the solution Y is stable, which we define as the relative
                change in each component being less than DZ_UB. The default value
                is 0.25, requiring the first bit to be stable. See LAWN 165 for
                more details.

           IGNORE_CWISE

                     IGNORE_CWISE is LOGICAL
                If .TRUE. then ignore componentwise convergence. Default value
                is .FALSE..

           INFO

                     INFO is INTEGER
                  = 0:  Successful exit.
                  < 0:  if INFO = -i, the ith argument to CLA_HERFSX_EXTENDED had an illegal
                        value

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   real function cla_herpvgrw (character*1 UPLO, integer N, integer INFO, complex, dimension(
       lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF, integer,
       dimension( * ) IPIV, real, dimension( * ) WORK)
       CLA_HERPVGRW

       Purpose:

            CLA_HERPVGRW computes the reciprocal pivot growth factor
            norm(A)/norm(U). The 'max absolute element' norm is used. If this is
            much less than 1, the stability of the LU factorization of the
            (equilibrated) matrix A could be poor. This also means that the
            solution X, estimated condition numbers, and error bounds could be
            unreliable.

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                  = 'U':  Upper triangle of A is stored;
                  = 'L':  Lower triangle of A is stored.

           N

                     N is INTEGER
                The number of linear equations, i.e., the order of the
                matrix A.  N >= 0.

           INFO

                     INFO is INTEGER
                The value of INFO returned from SSYTRF, .i.e., the pivot in
                column INFO is exactly 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                On entry, the N-by-N matrix A.

           LDA

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

           AF

                     AF is COMPLEX array, dimension (LDAF,N)
                The block diagonal matrix D and the multipliers used to
                obtain the factor U or L as computed by CHETRF.

           LDAF

                     LDAF is INTEGER
                The leading dimension of the array AF.  LDAF >= max(1,N).

           IPIV

                     IPIV is INTEGER array, dimension (N)
                Details of the interchanges and the block structure of D
                as determined by CHETRF.

           WORK

                     WORK is REAL array, dimension (2*N)

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   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

   subroutine clahef_rk (character UPLO, integer N, integer NB, integer KB, complex, dimension(
       lda, * ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV, complex,
       dimension( ldw, * ) W, integer LDW, integer INFO)
       CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using
       bounded Bunch-Kaufman (rook) diagonal pivoting method.

       Purpose:

            CLAHEF_RK computes a partial factorization of a complex Hermitian
            matrix A using the bounded Bunch-Kaufman (rook) 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.

            CLAHEF_RK is an auxiliary routine called by CHETRF_RK. 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, 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.

                     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 submatrix A(1:N,N-KB+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,N-KB+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 submatrix A(1:N,N-KB+1:N).
                             If -IPIV(k-1) = k-1, no interchange occurred.

                       c) In both cases a) and b) is 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 submatrix A(1:N,1:KB).
                          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 submatrix A(1:N,1:KB).
                             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 submatrix A(1:N,1:KB).
                             If -IPIV(k+1) = k+1, no interchange occurred.

                       c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.

                       d) NOTE: Any entry IPIV(k) is always NONZERO on output.

           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, 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.

       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 clahef_rook (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)
       Purpose:

            CLAHEF_ROOK computes a partial factorization of a complex Hermitian
            matrix A using the bounded Bunch-Kaufman ('rook') 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_ROOK is an auxiliary routine called by CHETRF_ROOK. 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) < 0 and IPIV(k-1) < 0, then rows and
                        columns k and -IPIV(k) were interchanged and rows and
                        columns k-1 and -IPIV(k-1) were inerchaged,
                        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) < 0 and IPIV(k+1) < 0, then rows and
                        columns k and -IPIV(k) were interchanged and rows and
                        columns k+1 and -IPIV(k+1) were inerchaged,
                        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

             September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                             School of Mathematics,
                             University of Manchester

Author

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