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

NAME

       complex16OTHEReigen - complex16 Other Eigenvalue routines

SYNOPSIS

   Functions
       subroutine zggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)
           ZGGGLM
       subroutine zhbev (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, RWORK, INFO)
            ZHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine zhbev_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, INFO)
            ZHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine zhbevd (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK,
           IWORK, LIWORK, INFO)
            ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine zhbevd_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK,
           LRWORK, IWORK, LIWORK, INFO)
            ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine zhbevx (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M,
           W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
            ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine zhbevx_2stage (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU,
           ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL, INFO)
            ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine zhbgv (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, RWORK, INFO)
           ZHBGV
       subroutine zhbgvd (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK,
           RWORK, LRWORK, IWORK, LIWORK, INFO)
           ZHBGVD
       subroutine zhbgvx (JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL,
           IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
           ZHBGVX
       subroutine zhpev (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, RWORK, INFO)
            ZHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine zhpevd (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK,
           LIWORK, INFO)
            ZHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine zhpevx (JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
           RWORK, IWORK, IFAIL, INFO)
            ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine zhpgv (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, RWORK, INFO)
           ZHPGV
       subroutine zhpgvd (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK,
           IWORK, LIWORK, INFO)
           ZHPGVD
       subroutine zhpgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z,
           LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
           ZHPGVX

Detailed Description

       This is the group of complex16 Other Eigenvalue routines

Function Documentation

   subroutine zggglm (integer N, integer M, integer P, complex*16, dimension( lda, * ) A, integer
       LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) D,
       complex*16, dimension( * ) X, complex*16, dimension( * ) Y, complex*16, dimension( * )
       WORK, integer LWORK, integer INFO)
       ZGGGLM

       Purpose:

            ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:

                    minimize || y ||_2   subject to   d = A*x + B*y
                        x

            where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
            given N-vector. It is assumed that M <= N <= M+P, and

                       rank(A) = M    and    rank( A B ) = N.

            Under these assumptions, the constrained equation is always
            consistent, and there is a unique solution x and a minimal 2-norm
            solution y, which is obtained using a generalized QR factorization
            of the matrices (A, B) given by

               A = Q*(R),   B = Q*T*Z.
                     (0)

            In particular, if matrix B is square nonsingular, then the problem
            GLM is equivalent to the following weighted linear least squares
            problem

                         minimize || inv(B)*(d-A*x) ||_2
                             x

            where inv(B) denotes the inverse of B.

       Parameters
           N

                     N is INTEGER
                     The number of rows of the matrices A and B.  N >= 0.

           M

                     M is INTEGER
                     The number of columns of the matrix A.  0 <= M <= N.

           P

                     P is INTEGER
                     The number of columns of the matrix B.  P >= N-M.

           A

                     A is COMPLEX*16 array, dimension (LDA,M)
                     On entry, the N-by-M matrix A.
                     On exit, the upper triangular part of the array A contains
                     the M-by-M upper triangular matrix R.

           LDA

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

           B

                     B is COMPLEX*16 array, dimension (LDB,P)
                     On entry, the N-by-P matrix B.
                     On exit, if N <= P, the upper triangle of the subarray
                     B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
                     if N > P, the elements on and above the (N-P)th subdiagonal
                     contain the N-by-P upper trapezoidal matrix T.

           LDB

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

           D

                     D is COMPLEX*16 array, dimension (N)
                     On entry, D is the left hand side of the GLM equation.
                     On exit, D is destroyed.

           X

                     X is COMPLEX*16 array, dimension (M)

           Y

                     Y is COMPLEX*16 array, dimension (P)

                     On exit, X and Y are the solutions of the GLM problem.

           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 >= max(1,N+M+P).
                     For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
                     where NB is an upper bound for the optimal blocksizes for
                     ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.

                     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.
                     = 1:  the upper triangular factor R associated with A in the
                           generalized QR factorization of the pair (A, B) is
                           singular, so that rank(A) < M; the least squares
                           solution could not be computed.
                     = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                           factor T associated with B in the generalized QR
                           factorization of the pair (A, B) is singular, so that
                           rank( A B ) < N; the least squares solution could not
                           be computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhbev (character JOBZ, character UPLO, integer N, integer KD, complex*16,
       dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16,
       dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision,
       dimension( * ) RWORK, integer INFO)
        ZHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            ZHBEV computes all the eigenvalues and, optionally, eigenvectors of
            a complex Hermitian band matrix A.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           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 matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, 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).

                     On exit, AB is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the first
                     superdiagonal and the diagonal of the tridiagonal matrix T
                     are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
                     the diagonal and first subdiagonal of T are returned in the
                     first two rows of AB.

           LDAB

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

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
                     eigenvectors of the matrix A, with the i-th column of Z
                     holding the eigenvector associated with W(i).
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = i, the algorithm failed to converge; i
                           off-diagonal elements of an intermediate tridiagonal
                           form did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhbev_2stage (character JOBZ, character UPLO, integer N, integer KD, complex*16,
       dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16,
       dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double
       precision, dimension( * ) RWORK, integer INFO)
        ZHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors
       for OTHER matrices

       Purpose:

            ZHBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
            a complex Hermitian band matrix A using the 2stage technique for
            the reduction to tridiagonal.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.
                             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.

           KD

                     KD is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, 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).

                     On exit, AB is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the first
                     superdiagonal and the diagonal of the tridiagonal matrix T
                     are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
                     the diagonal and first subdiagonal of T are returned in the
                     first two rows of AB.

           LDAB

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

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
                     eigenvectors of the matrix A, with the i-th column of Z
                     holding the eigenvector associated with W(i).
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

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

           LWORK

                     LWORK is INTEGER
                     The length of the array WORK. LWORK >= 1, when N <= 1;
                     otherwise
                     If JOBZ = 'N' and N > 1, LWORK must be queried.
                                              LWORK = MAX(1, dimension) where
                                              dimension = (2KD+1)*N + KD*NTHREADS
                                              where KD is the size of the band.
                                              NTHREADS is the number of threads used when
                                              openMP compilation is enabled, otherwise =1.
                     If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = i, the algorithm failed to converge; i
                           off-diagonal elements of an intermediate tridiagonal
                           form did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             All details about the 2stage techniques are available in:

             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 zhbevd (character JOBZ, character UPLO, integer N, integer KD, complex*16,
       dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16,
       dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double
       precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer
       LIWORK, integer INFO)
        ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of
            a complex Hermitian band matrix A.  If eigenvectors are desired, it
            uses a divide and conquer algorithm.

            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           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 matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, 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).

                     On exit, AB is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the first
                     superdiagonal and the diagonal of the tridiagonal matrix T
                     are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
                     the diagonal and first subdiagonal of T are returned in the
                     first two rows of AB.

           LDAB

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

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
                     eigenvectors of the matrix A, with the i-th column of Z
                     holding the eigenvector associated with W(i).
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           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.
                     If N <= 1,               LWORK must be at least 1.
                     If JOBZ = 'N' and N > 1, LWORK must be at least N.
                     If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array,
                                                    dimension (LRWORK)
                     On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

           LRWORK

                     LRWORK is INTEGER
                     The dimension of array RWORK.
                     If N <= 1,               LRWORK must be at least 1.
                     If JOBZ = 'N' and N > 1, LRWORK must be at least N.
                     If JOBZ = 'V' and N > 1, LRWORK must be at least
                                   1 + 5*N + 2*N**2.

                     If LRWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of array IWORK.
                     If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
                     If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK 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, the algorithm failed to converge; i
                           off-diagonal elements of an intermediate tridiagonal
                           form did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, complex*16,
       dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, complex*16,
       dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double
       precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer
       LIWORK, integer INFO)
        ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for OTHER matrices

       Purpose:

            ZHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
            a complex Hermitian band matrix A using the 2stage technique for
            the reduction to tridiagonal.  If eigenvectors are desired, it
            uses a divide and conquer algorithm.

            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.
                             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.

           KD

                     KD is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, 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).

                     On exit, AB is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the first
                     superdiagonal and the diagonal of the tridiagonal matrix T
                     are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
                     the diagonal and first subdiagonal of T are returned in the
                     first two rows of AB.

           LDAB

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

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
                     eigenvectors of the matrix A, with the i-th column of Z
                     holding the eigenvector associated with W(i).
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           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 the array WORK. LWORK >= 1, when N <= 1;
                     otherwise
                     If JOBZ = 'N' and N > 1, LWORK must be queried.
                                              LWORK = MAX(1, dimension) where
                                              dimension = (2KD+1)*N + KD*NTHREADS
                                              where KD is the size of the band.
                                              NTHREADS is the number of threads used when
                                              openMP compilation is enabled, otherwise =1.
                     If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array,
                                                    dimension (LRWORK)
                     On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.

           LRWORK

                     LRWORK is INTEGER
                     The dimension of array RWORK.
                     If N <= 1,               LRWORK must be at least 1.
                     If JOBZ = 'N' and N > 1, LRWORK must be at least N.
                     If JOBZ = 'V' and N > 1, LRWORK must be at least
                                   1 + 5*N + 2*N**2.

                     If LRWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of array IWORK.
                     If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
                     If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK 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, the algorithm failed to converge; i
                           off-diagonal elements of an intermediate tridiagonal
                           form did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             All details about the 2stage techniques are available in:

             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 zhbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD,
       complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldq, * ) Q,
       integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double
       precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension(
       ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * )
       RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
        ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
            of a complex Hermitian band matrix A.  Eigenvalues and eigenvectors
            can be selected by specifying either a range of values or a range of
            indices for the desired eigenvalues.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           RANGE

                     RANGE is CHARACTER*1
                     = 'A': all eigenvalues will be found;
                     = 'V': all eigenvalues in the half-open interval (VL,VU]
                            will be found;
                     = 'I': the IL-th through IU-th eigenvalues will be found.

           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 matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, 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).

                     On exit, AB is overwritten by values generated during the
                     reduction to tridiagonal form.

           LDAB

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

           Q

                     Q is COMPLEX*16 array, dimension (LDQ, N)
                     If JOBZ = 'V', the N-by-N unitary matrix used in the
                                     reduction to tridiagonal form.
                     If JOBZ = 'N', the array Q is not referenced.

           LDQ

                     LDQ is INTEGER
                     The leading dimension of the array Q.  If JOBZ = 'V', then
                     LDQ >= max(1,N).

           VL

                     VL is DOUBLE PRECISION
                     If RANGE='V', the lower bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           VU

                     VU is DOUBLE PRECISION
                     If RANGE='V', the upper bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL

                     IL is INTEGER
                     If RANGE='I', the index of the
                     smallest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           IU

                     IU is INTEGER
                     If RANGE='I', the index of the
                     largest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           ABSTOL

                     ABSTOL is DOUBLE PRECISION
                     The absolute error tolerance for the eigenvalues.
                     An approximate eigenvalue is accepted as converged
                     when it is determined to lie in an interval [a,b]
                     of width less than or equal to

                             ABSTOL + EPS *   max( |a|,|b| ) ,

                     where EPS is the machine precision.  If ABSTOL is less than
                     or equal to zero, then  EPS*|T|  will be used in its place,
                     where |T| is the 1-norm of the tridiagonal matrix obtained
                     by reducing AB to tridiagonal form.

                     Eigenvalues will be computed most accurately when ABSTOL is
                     set to twice the underflow threshold 2*DLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*DLAMCH('S').

                     See 'Computing Small Singular Values of Bidiagonal Matrices
                     with Guaranteed High Relative Accuracy,' by Demmel and
                     Kahan, LAPACK Working Note #3.

           M

                     M is INTEGER
                     The total number of eigenvalues found.  0 <= M <= N.
                     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     The first M elements contain the selected eigenvalues in
                     ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
                     If JOBZ = 'V', then if INFO = 0, the first M columns of Z
                     contain the orthonormal eigenvectors of the matrix A
                     corresponding to the selected eigenvalues, with the i-th
                     column of Z holding the eigenvector associated with W(i).
                     If an eigenvector fails to converge, then that column of Z
                     contains the latest approximation to the eigenvector, and the
                     index of the eigenvector is returned in IFAIL.
                     If JOBZ = 'N', then Z is not referenced.
                     Note: the user must ensure that at least max(1,M) columns are
                     supplied in the array Z; if RANGE = 'V', the exact value of M
                     is not known in advance and an upper bound must be used.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (7*N)

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

                     IFAIL is INTEGER array, dimension (N)
                     If JOBZ = 'V', then if INFO = 0, the first M elements of
                     IFAIL are zero.  If INFO > 0, then IFAIL contains the
                     indices of the eigenvectors that failed to converge.
                     If JOBZ = 'N', then IFAIL is not referenced.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, then i eigenvectors failed to converge.
                           Their indices are stored in array IFAIL.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer
       KD, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldq, * ) Q,
       integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double
       precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16, dimension(
       ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision,
       dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
       integer INFO)
        ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for OTHER matrices

       Purpose:

            ZHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
            of a complex Hermitian band matrix A using the 2stage technique for
            the reduction to tridiagonal.  Eigenvalues and eigenvectors
            can be selected by specifying either a range of values or a range of
            indices for the desired eigenvalues.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.
                             Not available in this release.

           RANGE

                     RANGE is CHARACTER*1
                     = 'A': all eigenvalues will be found;
                     = 'V': all eigenvalues in the half-open interval (VL,VU]
                            will be found;
                     = 'I': the IL-th through IU-th eigenvalues will be found.

           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 matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, 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).

                     On exit, AB is overwritten by values generated during the
                     reduction to tridiagonal form.

           LDAB

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

           Q

                     Q is COMPLEX*16 array, dimension (LDQ, N)
                     If JOBZ = 'V', the N-by-N unitary matrix used in the
                                     reduction to tridiagonal form.
                     If JOBZ = 'N', the array Q is not referenced.

           LDQ

                     LDQ is INTEGER
                     The leading dimension of the array Q.  If JOBZ = 'V', then
                     LDQ >= max(1,N).

           VL

                     VL is DOUBLE PRECISION
                     If RANGE='V', the lower bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           VU

                     VU is DOUBLE PRECISION
                     If RANGE='V', the upper bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL

                     IL is INTEGER
                     If RANGE='I', the index of the
                     smallest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           IU

                     IU is INTEGER
                     If RANGE='I', the index of the
                     largest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           ABSTOL

                     ABSTOL is DOUBLE PRECISION
                     The absolute error tolerance for the eigenvalues.
                     An approximate eigenvalue is accepted as converged
                     when it is determined to lie in an interval [a,b]
                     of width less than or equal to

                             ABSTOL + EPS *   max( |a|,|b| ) ,

                     where EPS is the machine precision.  If ABSTOL is less than
                     or equal to zero, then  EPS*|T|  will be used in its place,
                     where |T| is the 1-norm of the tridiagonal matrix obtained
                     by reducing AB to tridiagonal form.

                     Eigenvalues will be computed most accurately when ABSTOL is
                     set to twice the underflow threshold 2*DLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*DLAMCH('S').

                     See 'Computing Small Singular Values of Bidiagonal Matrices
                     with Guaranteed High Relative Accuracy,' by Demmel and
                     Kahan, LAPACK Working Note #3.

           M

                     M is INTEGER
                     The total number of eigenvalues found.  0 <= M <= N.
                     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     The first M elements contain the selected eigenvalues in
                     ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
                     If JOBZ = 'V', then if INFO = 0, the first M columns of Z
                     contain the orthonormal eigenvectors of the matrix A
                     corresponding to the selected eigenvalues, with the i-th
                     column of Z holding the eigenvector associated with W(i).
                     If an eigenvector fails to converge, then that column of Z
                     contains the latest approximation to the eigenvector, and the
                     index of the eigenvector is returned in IFAIL.
                     If JOBZ = 'N', then Z is not referenced.
                     Note: the user must ensure that at least max(1,M) columns are
                     supplied in the array Z; if RANGE = 'V', the exact value of M
                     is not known in advance and an upper bound must be used.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (LWORK)

           LWORK

                     LWORK is INTEGER
                     The length of the array WORK. LWORK >= 1, when N <= 1;
                     otherwise
                     If JOBZ = 'N' and N > 1, LWORK must be queried.
                                              LWORK = MAX(1, dimension) where
                                              dimension = (2KD+1)*N + KD*NTHREADS
                                              where KD is the size of the band.
                                              NTHREADS is the number of threads used when
                                              openMP compilation is enabled, otherwise =1.
                     If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (7*N)

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

                     IFAIL is INTEGER array, dimension (N)
                     If JOBZ = 'V', then if INFO = 0, the first M elements of
                     IFAIL are zero.  If INFO > 0, then IFAIL contains the
                     indices of the eigenvectors that failed to converge.
                     If JOBZ = 'N', then IFAIL is not referenced.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, then i eigenvectors failed to converge.
                           Their indices are stored in array IFAIL.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             All details about the 2stage techniques are available in:

             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 zhbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB,
       complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB,
       integer LDBB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z,
       integer LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK,
       integer INFO)
       ZHBGV

       Purpose:

            ZHBGV computes all the eigenvalues, and optionally, the eigenvectors
            of a complex generalized Hermitian-definite banded eigenproblem, of
            the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
            and banded, and B is also positive definite.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           KA

                     KA is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'. KA >= 0.

           KB

                     KB is INTEGER
                     The number of superdiagonals of the matrix B if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'. KB >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the Hermitian band
                     matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

                     On exit, the contents of AB are destroyed.

           LDAB

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

           BB

                     BB is COMPLEX*16 array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the Hermitian band
                     matrix B, stored in the first kb+1 rows of the array.  The
                     j-th column of B is stored in the j-th column of the array BB
                     as follows:
                     if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
                     if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

                     On exit, the factor S from the split Cholesky factorization
                     B = S**H*S, as returned by ZPBSTF.

           LDBB

                     LDBB is INTEGER
                     The leading dimension of the array BB.  LDBB >= KB+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors, with the i-th column of Z holding the
                     eigenvector associated with W(i). The eigenvectors are
                     normalized so that Z**H*B*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= N.

           WORK

                     WORK is COMPLEX*16 array, dimension (N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (3*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, and i is:
                        <= N:  the algorithm failed to converge:
                               i off-diagonal elements of an intermediate
                               tridiagonal form did not converge to zero;
                        > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
                               returned INFO = i: B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB,
       complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( ldbb, * ) BB,
       integer LDBB, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z,
       integer LDZ, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension(
       * ) RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
       ZHBGVD

       Purpose:

            ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors
            of a complex generalized Hermitian-definite banded eigenproblem, of
            the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
            and banded, and B is also positive definite.  If eigenvectors are
            desired, it uses a divide and conquer algorithm.

            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           KA

                     KA is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'. KA >= 0.

           KB

                     KB is INTEGER
                     The number of superdiagonals of the matrix B if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'. KB >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the Hermitian band
                     matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

                     On exit, the contents of AB are destroyed.

           LDAB

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

           BB

                     BB is COMPLEX*16 array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the Hermitian band
                     matrix B, stored in the first kb+1 rows of the array.  The
                     j-th column of B is stored in the j-th column of the array BB
                     as follows:
                     if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
                     if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

                     On exit, the factor S from the split Cholesky factorization
                     B = S**H*S, as returned by ZPBSTF.

           LDBB

                     LDBB is INTEGER
                     The leading dimension of the array BB.  LDBB >= KB+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors, with the i-th column of Z holding the
                     eigenvector associated with W(i). The eigenvectors are
                     normalized so that Z**H*B*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= N.

           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.
                     If N <= 1,               LWORK >= 1.
                     If JOBZ = 'N' and N > 1, LWORK >= N.
                     If JOBZ = 'V' and N > 1, LWORK >= 2*N**2.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
                     On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.

           LRWORK

                     LRWORK is INTEGER
                     The dimension of array RWORK.
                     If N <= 1,               LRWORK >= 1.
                     If JOBZ = 'N' and N > 1, LRWORK >= N.
                     If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.

                     If LRWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of array IWORK.
                     If JOBZ = 'N' or N <= 1, LIWORK >= 1.
                     If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK 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, and i is:
                        <= N:  the algorithm failed to converge:
                               i off-diagonal elements of an intermediate
                               tridiagonal form did not converge to zero;
                        > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
                               returned INFO = i: B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

   subroutine zhbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA,
       integer KB, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension(
       ldbb, * ) BB, integer LDBB, complex*16, dimension( ldq, * ) Q, integer LDQ, double
       precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL,
       integer M, double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer
       LDZ, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer,
       dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
       ZHBGVX

       Purpose:

            ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors
            of a complex generalized Hermitian-definite banded eigenproblem, of
            the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
            and banded, and B is also positive definite.  Eigenvalues and
            eigenvectors can be selected by specifying either all eigenvalues,
            a range of values or a range of indices for the desired eigenvalues.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           RANGE

                     RANGE is CHARACTER*1
                     = 'A': all eigenvalues will be found;
                     = 'V': all eigenvalues in the half-open interval (VL,VU]
                            will be found;
                     = 'I': the IL-th through IU-th eigenvalues will be found.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           KA

                     KA is INTEGER
                     The number of superdiagonals of the matrix A if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'. KA >= 0.

           KB

                     KB is INTEGER
                     The number of superdiagonals of the matrix B if UPLO = 'U',
                     or the number of subdiagonals if UPLO = 'L'. KB >= 0.

           AB

                     AB is COMPLEX*16 array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the Hermitian band
                     matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

                     On exit, the contents of AB are destroyed.

           LDAB

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

           BB

                     BB is COMPLEX*16 array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the Hermitian band
                     matrix B, stored in the first kb+1 rows of the array.  The
                     j-th column of B is stored in the j-th column of the array BB
                     as follows:
                     if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
                     if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

                     On exit, the factor S from the split Cholesky factorization
                     B = S**H*S, as returned by ZPBSTF.

           LDBB

                     LDBB is INTEGER
                     The leading dimension of the array BB.  LDBB >= KB+1.

           Q

                     Q is COMPLEX*16 array, dimension (LDQ, N)
                     If JOBZ = 'V', the n-by-n matrix used in the reduction of
                     A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
                     and consequently C to tridiagonal form.
                     If JOBZ = 'N', the array Q is not referenced.

           LDQ

                     LDQ is INTEGER
                     The leading dimension of the array Q.  If JOBZ = 'N',
                     LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

           VL

                     VL is DOUBLE PRECISION

                     If RANGE='V', the lower bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           VU

                     VU is DOUBLE PRECISION

                     If RANGE='V', the upper bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL

                     IL is INTEGER

                     If RANGE='I', the index of the
                     smallest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           IU

                     IU is INTEGER

                     If RANGE='I', the index of the
                     largest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           ABSTOL

                     ABSTOL is DOUBLE PRECISION
                     The absolute error tolerance for the eigenvalues.
                     An approximate eigenvalue is accepted as converged
                     when it is determined to lie in an interval [a,b]
                     of width less than or equal to

                             ABSTOL + EPS *   max( |a|,|b| ) ,

                     where EPS is the machine precision.  If ABSTOL is less than
                     or equal to zero, then  EPS*|T|  will be used in its place,
                     where |T| is the 1-norm of the tridiagonal matrix obtained
                     by reducing AP to tridiagonal form.

                     Eigenvalues will be computed most accurately when ABSTOL is
                     set to twice the underflow threshold 2*DLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*DLAMCH('S').

           M

                     M is INTEGER
                     The total number of eigenvalues found.  0 <= M <= N.
                     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors, with the i-th column of Z holding the
                     eigenvector associated with W(i). The eigenvectors are
                     normalized so that Z**H*B*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= N.

           WORK

                     WORK is COMPLEX*16 array, dimension (N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (7*N)

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

                     IFAIL is INTEGER array, dimension (N)
                     If JOBZ = 'V', then if INFO = 0, the first M elements of
                     IFAIL are zero.  If INFO > 0, then IFAIL contains the
                     indices of the eigenvectors that failed to converge.
                     If JOBZ = 'N', then IFAIL is not referenced.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, and i is:
                        <= N:  then i eigenvectors failed to converge.  Their
                               indices are stored in array IFAIL.
                        > N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
                               returned INFO = i: B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

   subroutine zhpev (character JOBZ, character UPLO, integer N, complex*16, dimension( * ) AP,
       double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ,
       complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)
        ZHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            ZHPEV computes all the eigenvalues and, optionally, eigenvectors of a
            complex Hermitian matrix in packed storage.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           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.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, AP is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the diagonal
                     and first superdiagonal of the tridiagonal matrix T overwrite
                     the corresponding elements of A, and if UPLO = 'L', the
                     diagonal and first subdiagonal of T overwrite the
                     corresponding elements of A.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
                     eigenvectors of the matrix A, with the i-th column of Z
                     holding the eigenvector associated with W(i).
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (max(1, 2*N-1))

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = i, the algorithm failed to converge; i
                           off-diagonal elements of an intermediate tridiagonal
                           form did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhpevd (character JOBZ, character UPLO, integer N, complex*16, dimension( * ) AP,
       double precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ,
       complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK,
       integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
        ZHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            ZHPEVD computes all the eigenvalues and, optionally, eigenvectors of
            a complex Hermitian matrix A in packed storage.  If eigenvectors are
            desired, it uses a divide and conquer algorithm.

            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           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.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, AP is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the diagonal
                     and first superdiagonal of the tridiagonal matrix T overwrite
                     the corresponding elements of A, and if UPLO = 'L', the
                     diagonal and first subdiagonal of T overwrite the
                     corresponding elements of A.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
                     eigenvectors of the matrix A, with the i-th column of Z
                     holding the eigenvector associated with W(i).
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

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

           LWORK

                     LWORK is INTEGER
                     The dimension of array WORK.
                     If N <= 1,               LWORK must be at least 1.
                     If JOBZ = 'N' and N > 1, LWORK must be at least N.
                     If JOBZ = 'V' and N > 1, LWORK must be at least 2*N.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the required sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
                     On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

           LRWORK

                     LRWORK is INTEGER
                     The dimension of array RWORK.
                     If N <= 1,               LRWORK must be at least 1.
                     If JOBZ = 'N' and N > 1, LRWORK must be at least N.
                     If JOBZ = 'V' and N > 1, LRWORK must be at least
                               1 + 5*N + 2*N**2.

                     If LRWORK = -1, then a workspace query is assumed; the
                     routine only calculates the required sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of array IWORK.
                     If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
                     If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the required sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK 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, the algorithm failed to converge; i
                           off-diagonal elements of an intermediate tridiagonal
                           form did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhpevx (character JOBZ, character RANGE, character UPLO, integer N, complex*16,
       dimension( * ) AP, double precision VL, double precision VU, integer IL, integer IU,
       double precision ABSTOL, integer M, double precision, dimension( * ) W, complex*16,
       dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double precision,
       dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL,
       integer INFO)
        ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            ZHPEVX computes selected eigenvalues and, optionally, eigenvectors
            of a complex Hermitian matrix A in packed storage.
            Eigenvalues/vectors can be selected by specifying either a range of
            values or a range of indices for the desired eigenvalues.

       Parameters
           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           RANGE

                     RANGE is CHARACTER*1
                     = 'A': all eigenvalues will be found;
                     = 'V': all eigenvalues in the half-open interval (VL,VU]
                            will be found;
                     = 'I': the IL-th through IU-th eigenvalues will be found.

           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.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, AP is overwritten by values generated during the
                     reduction to tridiagonal form.  If UPLO = 'U', the diagonal
                     and first superdiagonal of the tridiagonal matrix T overwrite
                     the corresponding elements of A, and if UPLO = 'L', the
                     diagonal and first subdiagonal of T overwrite the
                     corresponding elements of A.

           VL

                     VL is DOUBLE PRECISION
                     If RANGE='V', the lower bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           VU

                     VU is DOUBLE PRECISION
                     If RANGE='V', the upper bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL

                     IL is INTEGER
                     If RANGE='I', the index of the
                     smallest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           IU

                     IU is INTEGER
                     If RANGE='I', the index of the
                     largest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           ABSTOL

                     ABSTOL is DOUBLE PRECISION
                     The absolute error tolerance for the eigenvalues.
                     An approximate eigenvalue is accepted as converged
                     when it is determined to lie in an interval [a,b]
                     of width less than or equal to

                             ABSTOL + EPS *   max( |a|,|b| ) ,

                     where EPS is the machine precision.  If ABSTOL is less than
                     or equal to zero, then  EPS*|T|  will be used in its place,
                     where |T| is the 1-norm of the tridiagonal matrix obtained
                     by reducing AP to tridiagonal form.

                     Eigenvalues will be computed most accurately when ABSTOL is
                     set to twice the underflow threshold 2*DLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*DLAMCH('S').

                     See 'Computing Small Singular Values of Bidiagonal Matrices
                     with Guaranteed High Relative Accuracy,' by Demmel and
                     Kahan, LAPACK Working Note #3.

           M

                     M is INTEGER
                     The total number of eigenvalues found.  0 <= M <= N.
                     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the selected eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
                     If JOBZ = 'V', then if INFO = 0, the first M columns of Z
                     contain the orthonormal eigenvectors of the matrix A
                     corresponding to the selected eigenvalues, with the i-th
                     column of Z holding the eigenvector associated with W(i).
                     If an eigenvector fails to converge, then that column of Z
                     contains the latest approximation to the eigenvector, and
                     the index of the eigenvector is returned in IFAIL.
                     If JOBZ = 'N', then Z is not referenced.
                     Note: the user must ensure that at least max(1,M) columns are
                     supplied in the array Z; if RANGE = 'V', the exact value of M
                     is not known in advance and an upper bound must be used.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (2*N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (7*N)

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

                     IFAIL is INTEGER array, dimension (N)
                     If JOBZ = 'V', then if INFO = 0, the first M elements of
                     IFAIL are zero.  If INFO > 0, then IFAIL contains the
                     indices of the eigenvectors that failed to converge.
                     If JOBZ = 'N', then IFAIL is not referenced.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, then i eigenvectors failed to converge.
                           Their indices are stored in array IFAIL.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhpgv (integer ITYPE, character JOBZ, character UPLO, integer N, complex*16,
       dimension( * ) AP, complex*16, dimension( * ) BP, double precision, dimension( * ) W,
       complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, double
       precision, dimension( * ) RWORK, integer INFO)
       ZHPGV

       Purpose:

            ZHPGV computes all the eigenvalues and, optionally, the eigenvectors
            of a complex generalized Hermitian-definite eigenproblem, of the form
            A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
            Here A and B are assumed to be Hermitian, stored in packed format,
            and B is also positive definite.

       Parameters
           ITYPE

                     ITYPE is INTEGER
                     Specifies the problem type to be solved:
                     = 1:  A*x = (lambda)*B*x
                     = 2:  A*B*x = (lambda)*x
                     = 3:  B*A*x = (lambda)*x

           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, the contents of AP are destroyed.

           BP

                     BP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     B, packed columnwise in a linear array.  The j-th column of B
                     is stored in the array BP as follows:
                     if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
                     if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

                     On exit, the triangular factor U or L from the Cholesky
                     factorization B = U**H*U or B = L*L**H, in the same storage
                     format as B.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors.  The eigenvectors are normalized as follows:
                     if ITYPE = 1 or 2, Z**H*B*Z = I;
                     if ITYPE = 3, Z**H*inv(B)*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (max(1, 2*N-1))

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (max(1, 3*N-2))

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  ZPPTRF or ZHPEV returned an error code:
                        <= N:  if INFO = i, ZHPEV failed to converge;
                               i off-diagonal elements of an intermediate
                               tridiagonal form did not convergeto zero;
                        > N:   if INFO = N + i, for 1 <= i <= n, then the leading
                               minor of order i of B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zhpgvd (integer ITYPE, character JOBZ, character UPLO, integer N, complex*16,
       dimension( * ) AP, complex*16, dimension( * ) BP, double precision, dimension( * ) W,
       complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) WORK, integer
       LWORK, double precision, dimension( * ) RWORK, integer LRWORK, integer, dimension( * )
       IWORK, integer LIWORK, integer INFO)
       ZHPGVD

       Purpose:

            ZHPGVD computes all the eigenvalues and, optionally, the eigenvectors
            of a complex generalized Hermitian-definite eigenproblem, of the form
            A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
            B are assumed to be Hermitian, stored in packed format, and B is also
            positive definite.
            If eigenvectors are desired, it uses a divide and conquer algorithm.

            The divide and conquer algorithm makes very mild assumptions about
            floating point arithmetic. It will work on machines with a guard
            digit in add/subtract, or on those binary machines without guard
            digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
            Cray-2. It could conceivably fail on hexadecimal or decimal machines
            without guard digits, but we know of none.

       Parameters
           ITYPE

                     ITYPE is INTEGER
                     Specifies the problem type to be solved:
                     = 1:  A*x = (lambda)*B*x
                     = 2:  A*B*x = (lambda)*x
                     = 3:  B*A*x = (lambda)*x

           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, the contents of AP are destroyed.

           BP

                     BP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     B, packed columnwise in a linear array.  The j-th column of B
                     is stored in the array BP as follows:
                     if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
                     if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

                     On exit, the triangular factor U or L from the Cholesky
                     factorization B = U**H*U or B = L*L**H, in the same storage
                     format as B.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
                     eigenvectors.  The eigenvectors are normalized as follows:
                     if ITYPE = 1 or 2, Z**H*B*Z = I;
                     if ITYPE = 3, Z**H*inv(B)*Z = I.
                     If JOBZ = 'N', then Z is not referenced.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

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

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If N <= 1,               LWORK >= 1.
                     If JOBZ = 'N' and N > 1, LWORK >= N.
                     If JOBZ = 'V' and N > 1, LWORK >= 2*N.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the required sizes of the WORK, RWORK and
                     IWORK arrays, returns these values as the first entries of
                     the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
                     On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

           LRWORK

                     LRWORK is INTEGER
                     The dimension of array RWORK.
                     If N <= 1,               LRWORK >= 1.
                     If JOBZ = 'N' and N > 1, LRWORK >= N.
                     If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.

                     If LRWORK = -1, then a workspace query is assumed; the
                     routine only calculates the required sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of array IWORK.
                     If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
                     If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the required sizes of the WORK, RWORK
                     and IWORK arrays, returns these values as the first entries
                     of the WORK, RWORK and IWORK arrays, and no error message
                     related to LWORK or LRWORK or LIWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  ZPPTRF or ZHPEVD returned an error code:
                        <= N:  if INFO = i, ZHPEVD failed to converge;
                               i off-diagonal elements of an intermediate
                               tridiagonal form did not convergeto zero;
                        > N:   if INFO = N + i, for 1 <= i <= n, then the leading
                               minor of order i of B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

   subroutine zhpgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N,
       complex*16, dimension( * ) AP, complex*16, dimension( * ) BP, double precision VL, double
       precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double
       precision, dimension( * ) W, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16,
       dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * )
       IWORK, integer, dimension( * ) IFAIL, integer INFO)
       ZHPGVX

       Purpose:

            ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
            of a complex generalized Hermitian-definite eigenproblem, of the form
            A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
            B are assumed to be Hermitian, stored in packed format, and B is also
            positive definite.  Eigenvalues and eigenvectors can be selected by
            specifying either a range of values or a range of indices for the
            desired eigenvalues.

       Parameters
           ITYPE

                     ITYPE is INTEGER
                     Specifies the problem type to be solved:
                     = 1:  A*x = (lambda)*B*x
                     = 2:  A*B*x = (lambda)*x
                     = 3:  B*A*x = (lambda)*x

           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute eigenvalues only;
                     = 'V':  Compute eigenvalues and eigenvectors.

           RANGE

                     RANGE is CHARACTER*1
                     = 'A': all eigenvalues will be found;
                     = 'V': all eigenvalues in the half-open interval (VL,VU]
                            will be found;
                     = 'I': the IL-th through IU-th eigenvalues will be found.

           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangles of A and B are stored;
                     = 'L':  Lower triangles of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

                     On exit, the contents of AP are destroyed.

           BP

                     BP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     B, packed columnwise in a linear array.  The j-th column of B
                     is stored in the array BP as follows:
                     if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
                     if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

                     On exit, the triangular factor U or L from the Cholesky
                     factorization B = U**H*U or B = L*L**H, in the same storage
                     format as B.

           VL

                     VL is DOUBLE PRECISION

                     If RANGE='V', the lower bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           VU

                     VU is DOUBLE PRECISION

                     If RANGE='V', the upper bound of the interval to
                     be searched for eigenvalues. VL < VU.
                     Not referenced if RANGE = 'A' or 'I'.

           IL

                     IL is INTEGER

                     If RANGE='I', the index of the
                     smallest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           IU

                     IU is INTEGER

                     If RANGE='I', the index of the
                     largest eigenvalue to be returned.
                     1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
                     Not referenced if RANGE = 'A' or 'V'.

           ABSTOL

                     ABSTOL is DOUBLE PRECISION
                     The absolute error tolerance for the eigenvalues.
                     An approximate eigenvalue is accepted as converged
                     when it is determined to lie in an interval [a,b]
                     of width less than or equal to

                             ABSTOL + EPS *   max( |a|,|b| ) ,

                     where EPS is the machine precision.  If ABSTOL is less than
                     or equal to zero, then  EPS*|T|  will be used in its place,
                     where |T| is the 1-norm of the tridiagonal matrix obtained
                     by reducing AP to tridiagonal form.

                     Eigenvalues will be computed most accurately when ABSTOL is
                     set to twice the underflow threshold 2*DLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*DLAMCH('S').

           M

                     M is INTEGER
                     The total number of eigenvalues found.  0 <= M <= N.
                     If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

           W

                     W is DOUBLE PRECISION array, dimension (N)
                     On normal exit, the first M elements contain the selected
                     eigenvalues in ascending order.

           Z

                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     If JOBZ = 'N', then Z is not referenced.
                     If JOBZ = 'V', then if INFO = 0, the first M columns of Z
                     contain the orthonormal eigenvectors of the matrix A
                     corresponding to the selected eigenvalues, with the i-th
                     column of Z holding the eigenvector associated with W(i).
                     The eigenvectors are normalized as follows:
                     if ITYPE = 1 or 2, Z**H*B*Z = I;
                     if ITYPE = 3, Z**H*inv(B)*Z = I.

                     If an eigenvector fails to converge, then that column of Z
                     contains the latest approximation to the eigenvector, and the
                     index of the eigenvector is returned in IFAIL.
                     Note: the user must ensure that at least max(1,M) columns are
                     supplied in the array Z; if RANGE = 'V', the exact value of M
                     is not known in advance and an upper bound must be used.

           LDZ

                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

           WORK

                     WORK is COMPLEX*16 array, dimension (2*N)

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (7*N)

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

                     IFAIL is INTEGER array, dimension (N)
                     If JOBZ = 'V', then if INFO = 0, the first M elements of
                     IFAIL are zero.  If INFO > 0, then IFAIL contains the
                     indices of the eigenvectors that failed to converge.
                     If JOBZ = 'N', then IFAIL is not referenced.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  ZPPTRF or ZHPEVX returned an error code:
                        <= N:  if INFO = i, ZHPEVX failed to converge;
                               i eigenvectors failed to converge.  Their indices
                               are stored in array IFAIL.
                        > N:   if INFO = N + i, for 1 <= i <= n, then the leading
                               minor of order i of B is not positive definite.
                               The factorization of B could not be completed and
                               no eigenvalues or eigenvectors were computed.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Author

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