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complex16HEcomputational(3) LAPACK complex16HEcomputational(3)

NAME

complex16HEcomputational

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.

Date:

December 2016

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)

IWORK

          IWORK is INTEGER 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.

Date:

June 2017

Contributors:

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

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

subroutine 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.

Date:

June 2017

Contributors:

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

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

subroutine 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.

Date:

April 2012

References:

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

subroutine 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.

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.

Date:

December 2016

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.

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.

Date:

December 2016

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.

Date:

December 2016

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 symmetric matrix A.  If UPLO = 'U', the leading N-by-N
     upper triangular part of A contains the upper triangular
     part of the matrix A, and the strictly lower triangular
     part of A is not referenced.  If UPLO = 'L', the leading
     N-by-N lower triangular part of A contains the lower
     triangular part of the matrix A, and the strictly upper
     triangular part of A is not referenced.

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**T or A =
     L*D*L**T as computed by DSYTRF.

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 DSYTRF.

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 DGETRS.
     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 .LT. 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 .LE. 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 .LT. 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.

Date:

April 2012

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.

Date:

December 2016

Further Details:

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(n-1) . . . H(2) H(1).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(n-1).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
  and tau in TAU(i).
  The contents of A on exit are illustrated by the following examples
  with n = 5:
  if UPLO = 'U':                       if UPLO = 'L':
    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )
  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i).

subroutine 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.

Date:

November 2013

Further Details:

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

Contributors:

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

subroutine 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.

Date:

December 2016

Further Details:

 TODO: put further details

Contributors:

  December 2016,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester
  01-01-96 - Based on modifications by
    J. Lewis, Boeing Computer Services Company
    A. Petitet, Computer Science Dept.,
                Univ. of Tenn., Knoxville abd , USA

subroutine 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.

Date:

November 2013

Further Details:

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

Contributors:

  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester
  01-01-96 - Based on modifications by
    J. Lewis, Boeing Computer Services Company
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

subroutine 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.

Date:

December 2016

Further Details:

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(n-1) . . . H(2) H(1).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
  A(1:i-1,i+1), and tau in TAU(i).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(n-1).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
  and tau in TAU(i).
  The contents of A on exit are illustrated by the following examples
  with n = 5:
  if UPLO = 'U':                       if UPLO = 'L':
    (  d   e   v2  v3  v4 )              (  d                  )
    (      d   e   v3  v4 )              (  e   d              )
    (          d   e   v4 )              (  v1  e   d          )
    (              d   e  )              (  v1  v2  e   d      )
    (                  d  )              (  v1  v2  v3  e   d  )
  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i).

subroutine 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, that
          store the Householder representation of the stage2
          band to tridiagonal.

LHOUS2

          LHOUS2 is INTEGER
          The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension)
          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.
          LHOUS2 = MAX(1, dimension) where
          dimension = 4*N if VECT='N'
          not available now if VECT='H'

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.

Date:

November 2017

Further Details:

  Implemented by Azzam Haidar.
  All details are available on technical report, SC11, SC13 papers.
  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
  Parallel reduction to condensed forms for symmetric eigenvalue problems
  using aggregated fine-grained and memory-aware kernels. In Proceedings
  of 2011 International Conference for High Performance Computing,
  Networking, Storage and Analysis (SC '11), New York, NY, USA,
  Article 8 , 11 pages.
  http://doi.acm.org/10.1145/2063384.2063394
  A. Haidar, J. Kurzak, P. Luszczek, 2013.
  An improved parallel singular value algorithm and its implementation 
  for multicore hardware, In Proceedings of 2013 International Conference
  for High Performance Computing, Networking, Storage and Analysis (SC '13).
  Denver, Colorado, USA, 2013.
  Article 90, 12 pages.
  http://doi.acm.org/10.1145/2503210.2503292
  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
  A novel hybrid CPU-GPU generalized eigensolver for electronic structure 
  calculations based on fine-grained memory aware tasks.
  International Journal of High Performance Computing Applications.
  Volume 28 Issue 2, Pages 196-209, May 2014.
  http://hpc.sagepub.com/content/28/2/196 

subroutine 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.

Date:

November 2017

Further Details:

  Implemented by Azzam Haidar.
  All details are available on technical report, SC11, SC13 papers.
  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
  Parallel reduction to condensed forms for symmetric eigenvalue problems
  using aggregated fine-grained and memory-aware kernels. In Proceedings
  of 2011 International Conference for High Performance Computing,
  Networking, Storage and Analysis (SC '11), New York, NY, USA,
  Article 8 , 11 pages.
  http://doi.acm.org/10.1145/2063384.2063394
  A. Haidar, J. Kurzak, P. Luszczek, 2013.
  An improved parallel singular value algorithm and its implementation 
  for multicore hardware, In Proceedings of 2013 International Conference
  for High Performance Computing, Networking, Storage and Analysis (SC '13).
  Denver, Colorado, USA, 2013.
  Article 90, 12 pages.
  http://doi.acm.org/10.1145/2503210.2503292
  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
  A novel hybrid CPU-GPU generalized eigensolver for electronic structure 
  calculations based on fine-grained memory aware tasks.
  International Journal of High Performance Computing Applications.
  Volume 28 Issue 2, Pages 196-209, May 2014.
  http://hpc.sagepub.com/content/28/2/196 

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
  A(i,i+kd+1:n), and tau in TAU(i).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(k), where k = n-kd.
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
  A(i+kd+2:n,i), and tau in TAU(i).
  The contents of A on exit are illustrated by the following examples
  with n = 5:
  if UPLO = 'U':                       if UPLO = 'L':
    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
    (                            ab    )              (  v1     v2     v3     ab/v4 ab )
  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i)..fi
 

subroutine 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.

Date:

December 2016

Further Details:

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

subroutine 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*T*U**H  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.

Date:

November 2017

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.

Date:

December 2016

Further Details:

 TODO: put correct description

Contributors:

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

subroutine zhetrf_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.

Date:

June 2016

Further Details:

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

Contributors:

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

subroutine 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.

Date:

December 2016

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 NB 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 NB 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.

Date:

November 2017

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.

Date:

June 2017

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.

Date:

November 2017

Contributors:

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

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.

Date:

June 2017

Contributors:

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

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.

Date:

November 2013

Contributors:

  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester

subroutine 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.

Date:

December 2016

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.

Date:

June 2016

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.

Date:

June 2017

Contributors:

  June 2017,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester

subroutine 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*T*U**H or
 A = L*T*L**T 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*T*U**H;
          = '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 DOUBLE array, dimension (MAX(1,LWORK))

LWORK

          LWORK is INTEGER, LWORK >= MAX(1,3*N-2).
 aram[out]

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.

Date:

November 2017

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.

Date:

November 2013

Contributors:

  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester

subroutine 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.

Date:

June 2017

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

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.

Date:

December 2016

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.

Date:

December 2016

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 resonsible 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', '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 .LT. 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 definte 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.

Date:

June 2017

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.

Date:

June 2016

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.

Date:

December 2016

Contributors:

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

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.

Date:

November 2017

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.

Date:

December 2016

Contributors:

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

subroutine 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.

Date:

November 2013

Contributors:

  November 2013,  Igor Kozachenko,
                  Computer Science Division,
                  University of California, Berkeley
  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
                  School of Mathematics,
                  University of Manchester

Author

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