complex16HEcomputational(3)

NAME
complex16HEcomputational - complex16

SYNOPSIS

  Functions
subroutine zhecon (uplo, n, a, lda, ipiv, anorm, rcond, work, info)
ZHECON
subroutine zhecon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
ZHECON_3
subroutine zhecon_rook (uplo, n, a, lda, ipiv, anorm, rcond, work, info)
ZHECON_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 zheequb (uplo, n, a, lda, s, scond, amax, work, info)
ZHEEQUB
subroutine zhegs2 (itype, uplo, n, a, lda, b, ldb, info)
ZHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the factorization results obtained from cpotrf (unblocked algorithm).
subroutine zhegst (itype, uplo, n, a, lda, b, ldb, info)
ZHEGST
subroutine zherfs (uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZHERFS
subroutine zherfsx (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)
ZHERFSX
subroutine zhetd2 (uplo, n, a, lda, d, e, tau, info)
ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
subroutine zhetf2 (uplo, n, a, lda, ipiv, info)
ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS).
subroutine zhetf2_rk (uplo, n, a, lda, e, ipiv, info)
ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
subroutine zhetf2_rook (uplo, n, a, lda, ipiv, info)
ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
subroutine zhetrd (uplo, n, a, lda, d, e, tau, work, lwork, info)
ZHETRD
subroutine zhetrd_2stage (vect, uplo, n, a, lda, d, e, tau, hous2, lhous2, work, lwork, info)
ZHETRD_2STAGE
subroutine zhetrd_he2hb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info)
ZHETRD_HE2HB
subroutine zhetrf (uplo, n, a, lda, ipiv, work, lwork, info)
ZHETRF
subroutine zhetrf_aa (uplo, n, a, lda, ipiv, work, lwork, info)
ZHETRF_AA
subroutine zhetrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info)
ZHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine zhetrf_rook (uplo, n, a, lda, ipiv, work, lwork, info)
ZHETRF_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 zhetri (uplo, n, a, lda, ipiv, work, info)
ZHETRI
subroutine zhetri2 (uplo, n, a, lda, ipiv, work, lwork, info)
ZHETRI2
subroutine zhetri2x (uplo, n, a, lda, ipiv, work, nb, info)
ZHETRI2X
subroutine zhetri_3 (uplo, n, a, lda, e, ipiv, work, lwork, info)
ZHETRI_3
subroutine zhetri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
ZHETRI_3X
subroutine zhetri_rook (uplo, n, a, lda, ipiv, work, info)
ZHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine zhetrs (uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
ZHETRS
subroutine zhetrs2 (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, info)
ZHETRS2
subroutine zhetrs_3 (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, info)
ZHETRS_3
subroutine zhetrs_aa (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info)
ZHETRS_AA
subroutine zhetrs_rook (uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
ZHETRS_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 zla_heamv (uplo, n, alpha, a, lda, x, incx, beta, y, incy)
ZLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.
double precision function zla_hercond_c (uplo, n, a, lda, af, ldaf, ipiv, c, capply, info, work, rwork)
ZLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.
double precision function zla_hercond_x (uplo, n, a, lda, af, ldaf, ipiv, x, info, work, rwork)
ZLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.
subroutine zla_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)
ZLA_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.
double precision function zla_herpvgrw (uplo, n, info, a, lda, af, ldaf, ipiv, work)
ZLA_HERPVGRW
subroutine zlahef (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)
ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
subroutine zlahef_aa (uplo, j1, m, nb, a, lda, ipiv, h, ldh, work)
ZLAHEF_AA
subroutine zlahef_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)
ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
subroutine zlahef_rook (uplo, n, nb, kb, a, lda, ipiv, w, ldw, info)

Detailed Description

This is the group of complex16 computational functions for HE matrices

Function Documentation

  subroutine zhecon (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZHECON

Purpose:

     ZHECON 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 ZHETRF.
     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*16 array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by ZHETRF.

    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 ZHETRF.

    ANORM

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

    RCOND

              RCOND is DOUBLE PRECISION
              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*16 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.

Definition at line 123 of file zhecon.f.

  subroutine zhecon_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZHECON_3

Purpose:

     ZHECON_3 estimates the reciprocal of the condition number (in the
     1-norm) of a complex Hermitian matrix A using the factorization
     computed by ZHETRF_RK or ZHETRF_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 ZHETRS_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*16 array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by ZHETRF_RK and ZHETRF_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*16 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 ZHETRF_RK or ZHETRF_BK.

    ANORM

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

    RCOND

              RCOND is DOUBLE PRECISION
              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*16 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

Definition at line 164 of file zhecon_3.f.

  subroutine zhecon_rook (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZHECON_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:

     ZHECON_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*16 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 DOUBLE PRECISION
              The 1-norm of the original matrix A.

    RCOND

              RCOND is DOUBLE PRECISION
              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*16 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

Definition at line 137 of file zhecon_rook.f.

  subroutine zheequb (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, double precision scond, double precision amax, complex*16, dimension( * ) work, integer info)

ZHEEQUB

Purpose:

     ZHEEQUB 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*16 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 DOUBLE PRECISION array, dimension (N)
              If INFO = 0, S contains the scale factors for A.

    SCOND

              SCOND is DOUBLE PRECISION
              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 DOUBLE PRECISION
              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*16 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

Definition at line 131 of file zheequb.f.

  subroutine zhegs2 (integer itype, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHEGS2 reduces a Hermitian definite generalized eigenproblem to standard form, using the factorization results obtained from cpotrf (unblocked algorithm).

Purpose:

     ZHEGS2 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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              n by n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n by n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, 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*16 array, dimension (LDB,N)
              The triangular factor from the Cholesky factorization of B,
              as returned by ZPOTRF.
              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.

Definition at line 127 of file zhegs2.f.

  subroutine zhegst (integer itype, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHEGST

Purpose:

     ZHEGST 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
              = '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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, 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*16 array, dimension (LDB,N)
              The triangular factor from the Cholesky factorization of B,
              as returned by ZPOTRF.
              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.

Definition at line 127 of file zhegst.f.

  subroutine zherfs (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHERFS

Purpose:

     ZHERFS 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*16 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*16 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 ZHETRF.

    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 ZHETRF.

    B

              B is COMPLEX*16 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*16 array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by ZHETRS.
              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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION 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.

Definition at line 190 of file zherfs.f.

  subroutine zherfsx (character uplo, character equed, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, double precision, dimension( * ) s, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) berr, integer n_err_bnds, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, integer nparams, double precision, dimension( * ) params, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHERFSX

Purpose:

        ZHERFSX 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*16 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*16 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 ZHETRF.

    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 ZHETRF.

    S

              S is DOUBLE PRECISION 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*16 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*16 array, dimension (LDX,NRHS)
         On entry, the solution matrix X, as computed by ZHETRS.
         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 DOUBLE PRECISION
         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 DOUBLE PRECISION 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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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: 1.0D+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*16 array, dimension (2*N)

    RWORK

              RWORK is DOUBLE PRECISION 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.

Definition at line 397 of file zherfsx.f.

  subroutine zhetd2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, integer info)

ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).

Purpose:

     ZHETD2 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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              n-by-n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n-by-n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, 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 DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION 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*16 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).

Definition at line 174 of file zhetd2.f.

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

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

Purpose:

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

Parameters
    UPLO

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

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              n-by-n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n-by-n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k-1) < 0, then rows and columns
                 k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
                 is a 2-by-2 diagonal block.
              If UPLO = 'L':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k+1) < 0, then rows and columns
                 k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
                 is a 2-by-2 diagonal block.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

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

Contributors:

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

Definition at line 190 of file zhetf2.f.

  subroutine zhetf2_rk (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, integer info)

ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

Purpose:

     ZHETF2_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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.
                If UPLO = 'U': the leading N-by-N upper triangular part
                of A contains the upper triangular part of the matrix A,
                and the strictly lower triangular part of A is not
                referenced.
                If UPLO = 'L': the leading N-by-N lower triangular part
                of A contains the lower triangular part of the matrix A,
                and the strictly upper triangular part of A is not
                referenced.
              On exit, contains:
                a) ONLY diagonal elements of the Hermitian block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

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

    IPIV

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

    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

Definition at line 240 of file zhetf2_rk.f.

  subroutine zhetf2_rook (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).

Purpose:

     ZHETF2_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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              n-by-n upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading n-by-n lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) < 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

Definition at line 193 of file zhetf2_rook.f.

  subroutine zhetrd (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD

Purpose:

     ZHETRD 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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, 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 DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION 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*16 array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The 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).

Definition at line 191 of file zhetrd.f.

  subroutine zhetrd_2stage (character vect, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( * ) hous2, integer lhous2, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_2STAGE

Purpose:

     ZHETRD_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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, 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 DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T.

    E

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

    TAU

              TAU is COMPLEX*16 array, dimension (N-KD)
              The scalar factors of the elementary reflectors of
              the first stage (see Further Details).

    HOUS2

              HOUS2 is COMPLEX*16 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*16 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

Definition at line 222 of file zhetrd_2stage.f.

  subroutine zhetrd_he2hb (character uplo, integer n, integer kd, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_HE2HB

Purpose:

     ZHETRD_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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, 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*16 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*16 array, dimension (N-KD)
              The scalar factors of the elementary reflectors (see Further
              Details).

    WORK

              WORK is COMPLEX*16 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

Definition at line 241 of file zhetrd_he2hb.f.

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

ZHETRF

Purpose:

     ZHETRF 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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If 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*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >=1.  For best performance
              LWORK >= N*NB, where NB is the block size returned by ILAENV.

    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).

Definition at line 176 of file zhetrf.f.

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

ZHETRF_AA

Purpose:

     ZHETRF_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*16 array, dimension (LDA,N)
              On entry, the hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, the 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*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK. LWORK >= MAX(1,2*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.

Definition at line 131 of file zhetrf_aa.f.

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

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

Purpose:

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

Parameters
    UPLO

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

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.
                If UPLO = 'U': the leading N-by-N upper triangular part
                of A contains the upper triangular part of the matrix A,
                and the strictly lower triangular part of A is not
                referenced.
                If UPLO = 'L': the leading N-by-N lower triangular part
                of A contains the lower triangular part of the matrix A,
                and the strictly upper triangular part of A is not
                referenced.
              On exit, contains:
                a) ONLY diagonal elements of the Hermitian block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

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

    IPIV

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

    WORK

              WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >=1.  For best performance
              LWORK >= N*NB, where NB is the block size returned
              by ILAENV.
              If LWORK = -1, then a workspace query is assumed;
              the routine only calculates the optimal size of the WORK
              array, returns this value as the first entry of the WORK
              array, and no error message related to LWORK is issued
              by XERBLA.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: If INFO = -k, the k-th argument had an illegal value
              > 0: If INFO = k, the matrix A is singular, because:
                     If UPLO = 'U': column k in the upper
                     triangular part of A contains all zeros.
                     If UPLO = 'L': column k in the lower
                     triangular part of A contains all zeros.
                   Therefore D(k,k) is exactly zero, and superdiagonal
                   elements of column k of U (or subdiagonal elements of
                   column k of L ) are all zeros. The factorization has
                   been completed, but the block diagonal matrix D is
                   exactly singular, and division by zero will occur if
                   it is used to solve a system of equations.
                   NOTE: INFO only stores the first occurrence of
                   a singularity, any subsequent occurrence of singularity
                   is not stored in INFO even though the factorization
                   always completes.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

     TODO: put correct description

Contributors:

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

Definition at line 257 of file zhetrf_rk.f.

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

ZHETRF_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:

     ZHETRF_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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
              N-by-N upper triangular part of A contains the upper
              triangular part of the matrix A, and the strictly lower
              triangular part of A is not referenced.  If UPLO = 'L', the
              leading N-by-N lower triangular part of A contains the lower
              triangular part of the matrix A, and the strictly upper
              triangular part of A is not referenced.
              On exit, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L (see below for further details).

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 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*16 array, dimension (MAX(1,LWORK)).
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of WORK.  LWORK >=1.  For best performance
              LWORK >= N*NB, where NB is the block size returned by ILAENV.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -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

Definition at line 211 of file zhetrf_rook.f.

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

ZHETRI

Purpose:

     ZHETRI 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
     ZHETRF.

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*16 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 ZHETRF.
              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 ZHETRF.

    WORK

              WORK is COMPLEX*16 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.

Definition at line 113 of file zhetri.f.

  subroutine zhetri2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRI2

Purpose:

     ZHETRI2 computes the inverse of a COMPLEX*16 hermitian indefinite matrix
     A using the factorization A = U*D*U**T or A = L*D*L**T computed by
     ZHETRF. ZHETRI2 set the LEADING DIMENSION of the workspace
     before calling ZHETRI2X 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*16 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 ZHETRF.
              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 ZHETRF.

    WORK

              WORK is COMPLEX*16 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.

Definition at line 126 of file zhetri2.f.

  subroutine zhetri2x (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( n+nb+1,* ) work, integer nb, integer info)

ZHETRI2X

Purpose:

     ZHETRI2X computes the inverse of a COMPLEX*16 Hermitian indefinite matrix
     A using the factorization A = U*D*U**H or A = L*D*L**H computed by
     ZHETRF.

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*16 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 ZHETRF.
              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 ZHETRF.

    WORK

              WORK is COMPLEX*16 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.

Definition at line 119 of file zhetri2x.f.

  subroutine zhetri_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRI_3

Purpose:

     ZHETRI_3 computes the inverse of a complex Hermitian indefinite
     matrix A using the factorization computed by ZHETRF_RK or ZHETRF_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.
     ZHETRI_3 sets the leading dimension of the workspace  before calling
     ZHETRI_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*16 array, dimension (LDA,N)
              On entry, diagonal of the block diagonal matrix D and
              factors U or L as computed by ZHETRF_RK and ZHETRF_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*16 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 ZHETRF_RK or ZHETRF_BK.

    WORK

              WORK is COMPLEX*16 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

Definition at line 168 of file zhetri_3.f.

  subroutine zhetri_3x (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( n+nb+1, * ) work, integer nb, integer info)

ZHETRI_3X

Purpose:

     ZHETRI_3X computes the inverse of a complex Hermitian indefinite
     matrix A using the factorization computed by ZHETRF_RK or ZHETRF_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*16 array, dimension (LDA,N)
              On entry, diagonal of the block diagonal matrix D and
              factors U or L as computed by ZHETRF_RK and ZHETRF_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*16 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 ZHETRF_RK or ZHETRF_BK.

    WORK

              WORK is COMPLEX*16 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

Definition at line 158 of file zhetri_3x.f.

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

ZHETRI_ROOK computes the inverse of HE matrix using the factorization obtained with the bounded Bunch-Kaufman ('rook') diagonal pivoting method.

Purpose:

     ZHETRI_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
     ZHETRF_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*16 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 ZHETRF_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 ZHETRF_ROOK.

    WORK

              WORK is COMPLEX*16 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

Definition at line 127 of file zhetri_rook.f.

  subroutine zhetrs (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHETRS

Purpose:

     ZHETRS 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 ZHETRF.

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*16 array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by ZHETRF.

    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 ZHETRF.

    B

              B is COMPLEX*16 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.

Definition at line 119 of file zhetrs.f.

  subroutine zhetrs2 (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer info)

ZHETRS2

Purpose:

     ZHETRS2 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 ZHETRF and converted by ZSYCONV.

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*16 array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by ZHETRF.

    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 ZHETRF.

    B

              B is COMPLEX*16 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*16 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.

Definition at line 125 of file zhetrs2.f.

  subroutine zhetrs_3 (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHETRS_3

Purpose:

     ZHETRS_3 solves a system of linear equations A * X = B with a complex
     Hermitian matrix A using the factorization computed
     by ZHETRF_RK or ZHETRF_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*16 array, dimension (LDA,N)
              Diagonal of the block diagonal matrix D and factors U or L
              as computed by ZHETRF_RK and ZHETRF_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*16 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 ZHETRF_RK or ZHETRF_BK.

    B

              B is COMPLEX*16 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

Definition at line 163 of file zhetrs_3.f.

  subroutine zhetrs_aa (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRS_AA

Purpose:

     ZHETRS_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 ZHETRF_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*16 array, dimension (LDA,N)
              Details of factors computed by ZHETRF_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 ZHETRF_AA.

    B

              B is COMPLEX*16 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*16 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.

Definition at line 130 of file zhetrs_aa.f.

  subroutine zhetrs_rook (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHETRS_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:

     ZHETRS_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 ZHETRF_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*16 array, dimension (LDA,N)
              The block diagonal matrix D and the multipliers used to
              obtain the factor U or L as computed by ZHETRF_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 ZHETRF_ROOK.

    B

              B is COMPLEX*16 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

Definition at line 134 of file zhetrs_rook.f.

  subroutine zla_heamv (integer uplo, integer n, double precision alpha, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) x, integer incx, double precision beta, double precision, dimension( * ) y, integer incy)

ZLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.

Purpose:

     ZLA_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 DOUBLE PRECISION .
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    A

              A is COMPLEX*16 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*16 array, dimension at least
               ( 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 DOUBLE PRECISION .
               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 DOUBLE PRECISION 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

Definition at line 176 of file zla_heamv.f.

  double precision function zla_hercond_c (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, double precision, dimension ( * ) c, logical capply, integer info, complex*16, dimension( * ) work, double precision, dimension( * ) rwork)

ZLA_HERCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.

Purpose:

        ZLA_HERCOND_C computes the infinity norm condition number of
        op(A) * inv(diag(C)) where C is a DOUBLE PRECISION 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*16 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*16 array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by ZHETRF.

    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 DOUBLE PRECISION 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*16 array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 137 of file zla_hercond_c.f.

  double precision function zla_hercond_x (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, complex*16, dimension( * ) x, integer info, complex*16, dimension( * ) work, double precision, dimension( * ) rwork)

ZLA_HERCOND_X computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.

Purpose:

        ZLA_HERCOND_X computes the infinity norm condition number of
        op(A) * diag(X) where X is a COMPLEX*16 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*16 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*16 array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by ZHETRF.

    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*16 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*16 array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 130 of file zla_hercond_x.f.

  subroutine zla_herfsx_extended (integer prec_type, character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, logical colequ, double precision, dimension( * ) c, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldy, * ) y, integer ldy, double precision, dimension( * ) berr_out, integer n_norms, double precision, dimension( nrhs, * ) err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp, complex*16, dimension( * ) res, double precision, dimension( * ) ayb, complex*16, dimension( * ) dy, complex*16, dimension( * ) y_tail, double precision rcond, integer ithresh, double precision rthresh, double precision dz_ub, logical ignore_cwise, integer info)

ZLA_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:

     ZLA_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 ZHERFSX 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*16 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*16 array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by ZHETRF.

    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 ZHETRF.

    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 DOUBLE PRECISION 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*16 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*16 array, dimension (LDY,NRHS)
         On entry, the solution matrix X, as computed by ZHETRS.
         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 DOUBLE PRECISION 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 ZLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (N)
         Workspace to hold the intermediate residual.

    AYB

              AYB is DOUBLE PRECISION array, dimension (N)
         Workspace.

    DY

              DY is COMPLEX*16 array, dimension (N)
         Workspace to hold the intermediate solution.

    Y_TAIL

              Y_TAIL is COMPLEX*16 array, dimension (N)
         Workspace to hold the trailing bits of the intermediate solution.

    RCOND

              RCOND is DOUBLE PRECISION
         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 DOUBLE PRECISION
         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 DOUBLE PRECISION
         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 ZLA_HERFSX_EXTENDED had an illegal
                 value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 388 of file zla_herfsx_extended.f.

  double precision function zla_herpvgrw (character*1 uplo, integer n, integer info, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldaf, * ) af, integer ldaf, integer, dimension( * ) ipiv, double precision, dimension( * ) work)

ZLA_HERPVGRW

Purpose:

     ZLA_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 ZHETRF, .i.e., the pivot in
         column INFO is exactly 0.

    A

              A is COMPLEX*16 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*16 array, dimension (LDAF,N)
         The block diagonal matrix D and the multipliers used to
         obtain the factor U or L as computed by ZHETRF.

    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 ZHETRF.

    WORK

              WORK is DOUBLE PRECISION array, dimension (2*N)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 121 of file zla_herpvgrw.f.

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

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

Purpose:

     ZLAHEF computes a partial factorization of a complex Hermitian
     matrix A using the Bunch-Kaufman diagonal pivoting method. The
     partial factorization has the form:
     A  =  ( I  U12 ) ( A11  0  ) (  I      0     )  if UPLO = 'U', or:
           ( 0  U22 ) (  0   D  ) ( U12**H U22**H )
     A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L'
           ( L21  I ) (  0  A22 ) (  0      I     )
     where the order of D is at most NB. The actual order is returned in
     the argument KB, and is either NB or NB-1, or N if N <= NB.
     Note that U**H denotes the conjugate transpose of U.
     ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code
     (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
     A22 (if UPLO = 'L').

Parameters
    UPLO

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

    N

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

    NB

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

    KB

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

    A

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

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 Only the last KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k-1) < 0, then rows and columns
                 k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
                 is a 2-by-2 diagonal block.
              If UPLO = 'L':
                 Only the first KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) = IPIV(k+1) < 0, then rows and columns
                 k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
                 is a 2-by-2 diagonal block.

    W

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

    LDW

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

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

Definition at line 176 of file zlahef.f.

  subroutine zlahef_aa (character uplo, integer j1, integer m, integer nb, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) work)

ZLAHEF_AA

Purpose:

     DLAHEF_AA factorizes a panel of a complex hermitian matrix A using
     the Aasen's algorithm. The panel consists of a set of NB rows of A
     when UPLO is U, or a set of NB columns when UPLO is L.
     In order to factorize the panel, the Aasen's algorithm requires the
     last row, or column, of the previous panel. The first row, or column,
     of A is set to be the first row, or column, of an identity matrix,
     which is used to factorize the first panel.
     The resulting J-th row of U, or J-th column of L, is stored in the
     (J-1)-th row, or column, of A (without the unit diagonals), while
     the diagonal and subdiagonal of A are overwritten by those of T.

Parameters
    UPLO

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

    J1

              J1 is INTEGER
              The location of the first row, or column, of the panel
              within the submatrix of A, passed to this routine, e.g.,
              when called by ZHETRF_AA, for the first panel, J1 is 1,
              while for the remaining panels, J1 is 2.

    M

              M is INTEGER
              The dimension of the submatrix. M >= 0.

    NB

              NB is INTEGER
              The dimension of the panel to be facotorized.

    A

              A is COMPLEX*16 array, dimension (LDA,M) for
              the first panel, while dimension (LDA,M+1) for the
              remaining panels.
              On entry, A contains the last row, or column, of
              the previous panel, and the trailing submatrix of A
              to be factorized, except for the first panel, only
              the panel is passed.
              On exit, the leading panel is factorized.

    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 row and column interchanges,
              the row and column k were interchanged with the row and
              column IPIV(k).

    H

              H is COMPLEX*16 workspace, dimension (LDH,NB).

    LDH

              LDH is INTEGER
              The leading dimension of the workspace H. LDH >= max(1,M).

    WORK

              WORK is COMPLEX*16 workspace, dimension (M).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 142 of file zlahef_aa.f.

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

ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Purpose:

     ZLAHEF_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.
     ZLAHEF_RK is an auxiliary routine called by ZHETRF_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*16 array, dimension (LDA,N)
              On entry, the Hermitian matrix A.
                If UPLO = 'U': the leading N-by-N upper triangular part
                of A contains the upper triangular part of the matrix A,
                and the strictly lower triangular part of A is not
                referenced.
                If UPLO = 'L': the leading N-by-N lower triangular part
                of A contains the lower triangular part of the matrix A,
                and the strictly upper triangular part of A is not
                referenced.
              On exit, contains:
                a) ONLY diagonal elements of the Hermitian block diagonal
                   matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
                   (superdiagonal (or subdiagonal) elements of D
                    are stored on exit in array E), and
                b) If UPLO = 'U': factor U in the superdiagonal part of A.
                   If UPLO = 'L': factor L in the subdiagonal part of A.

    LDA

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

    E

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              IPIV describes the permutation matrix P in the factorization
              of matrix A as follows. The absolute value of IPIV(k)
              represents the index of row and column that were
              interchanged with the k-th row and column. The value of UPLO
              describes the order in which the interchanges were applied.
              Also, the sign of IPIV represents the block structure of
              the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
              diagonal blocks which correspond to 1 or 2 interchanges
              at each factorization step.
              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*16 array, dimension (LDW,NB)

    LDW

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

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: If INFO = -k, 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

Definition at line 260 of file zlahef_rk.f.

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

Purpose:

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

    LDA

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

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D.
              If UPLO = 'U':
                 Only the last KB elements of IPIV are set.
                 If IPIV(k) > 0, then rows and columns k and IPIV(k) were
                 interchanged and D(k,k) is a 1-by-1 diagonal block.
                 If IPIV(k) < 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*16 array, dimension (LDW,NB)

    LDW

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    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

Definition at line 182 of file zlahef_rook.f.

Author

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