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

NAME

       complexOTHEReigen - complex Other Eigenvalue routines

SYNOPSIS

   Functions
       subroutine cggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)
           CGGGLM
       subroutine chbev (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, RWORK, INFO)
            CHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine chbev_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, INFO)
            CHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine chbevd (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK,
           IWORK, LIWORK, INFO)
            CHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine chbevd_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, RWORK,
           LRWORK, IWORK, LIWORK, INFO)
            CHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine chbevx (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M,
           W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
            CHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine chbevx_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)
            CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine chbgv (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, RWORK, INFO)
           CHBGV
       subroutine chbgvd (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK,
           RWORK, LRWORK, IWORK, LIWORK, INFO)
           CHBGVD
       subroutine chbgvx (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)
           CHBGVX
       subroutine chpev (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, RWORK, INFO)
            CHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine chpevd (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK,
           LIWORK, INFO)
            CHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine chpevx (JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
           RWORK, IWORK, IFAIL, INFO)
            CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine chpgv (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, RWORK, INFO)
           CHPGV
       subroutine chpgvd (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK,
           IWORK, LIWORK, INFO)
           CHPGVD
       subroutine chpgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z,
           LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
           CHPGVX

Detailed Description

       This is the group of complex Other Eigenvalue routines

Function Documentation

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

       Purpose:

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

           X

                     X is COMPLEX array, dimension (M)

           Y

                     Y is COMPLEX array, dimension (P)

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

           WORK

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

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= 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
                     CGEQRF, CGERQF, CUNMQR and CUNMRQ.

                     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 chbev (character JOBZ, character UPLO, integer N, integer KD, complex, dimension(
       ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, * ) Z,
       integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)
        CHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            CHBEV 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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 array, dimension (N)

           RWORK

                     RWORK is REAL 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 chbev_2stage (character JOBZ, character UPLO, integer N, integer KD, complex,
       dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, *
       ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,
       integer INFO)
        CHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors
       for OTHER matrices

       Purpose:

            CHBEV_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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 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 REAL 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 chbevd (character JOBZ, character UPLO, integer N, integer KD, complex, dimension(
       ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, * ) Z,
       integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,
       integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
        CHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            CHBEVD 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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 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 REAL 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 chbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, complex,
       dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) W, complex, dimension( ldz, *
       ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK,
       integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
        CHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for OTHER matrices

       Purpose:

            CHBEVD_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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

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

       Purpose:

            CHBEVX 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 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 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 REAL
                     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 REAL
                     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 REAL
                     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*SLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*SLAMCH('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 REAL array, dimension (N)
                     The first M elements contain the selected eigenvalues in
                     ascending order.

           Z

                     Z is COMPLEX 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 array, dimension (N)

           RWORK

                     RWORK is REAL 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 chbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer
       KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldq, * ) Q,
       integer LDQ, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real,
       dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
       WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer,
       dimension( * ) IFAIL, integer INFO)
        CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for OTHER matrices

       Purpose:

            CHBEVX_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 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 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 REAL
                     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 REAL
                     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 REAL
                     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*SLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*SLAMCH('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 REAL array, dimension (N)
                     The first M elements contain the selected eigenvalues in
                     ascending order.

           Z

                     Z is COMPLEX 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 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 REAL 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 chbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, complex,
       dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB,
       real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( *
       ) WORK, real, dimension( * ) RWORK, integer INFO)
       CHBGV

       Purpose:

            CHBGV 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 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 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 CPBSTF.

           LDBB

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

           W

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

           Z

                     Z is COMPLEX 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 array, dimension (N)

           RWORK

                     RWORK is REAL 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 CPBSTF
                               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 chbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, complex,
       dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * ) BB, integer LDBB,
       real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( *
       ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * )
       IWORK, integer LIWORK, integer INFO)
       CHBGVD

       Purpose:

            CHBGVD 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 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 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 CPBSTF.

           LDBB

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

           W

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

           Z

                     Z is COMPLEX 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 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 REAL 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 CPBSTF
                               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 chbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA,
       integer KB, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( ldbb, * )
       BB, integer LDBB, complex, dimension( ldq, * ) Q, integer LDQ, real VL, real VU, integer
       IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, *
       ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer,
       dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
       CHBGVX

       Purpose:

            CHBGVX 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 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 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 CPBSTF.

           LDBB

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

           Q

                     Q is COMPLEX 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 REAL

                     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 REAL

                     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 REAL
                     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*SLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*SLAMCH('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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 array, dimension (N)

           RWORK

                     RWORK is REAL 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 CPBSTF
                               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 chpev (character JOBZ, character UPLO, integer N, complex, dimension( * ) AP, real,
       dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
       WORK, real, dimension( * ) RWORK, integer INFO)
        CHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            CHPEV 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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

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

           RWORK

                     RWORK is REAL 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 chpevd (character JOBZ, character UPLO, integer N, complex, dimension( * ) AP,
       real, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( *
       ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * )
       IWORK, integer LIWORK, integer INFO)
        CHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            CHPEVD 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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 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 REAL 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 chpevx (character JOBZ, character RANGE, character UPLO, integer N, complex,
       dimension( * ) AP, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real,
       dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * )
       WORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer, dimension( * )
       IFAIL, integer INFO)
        CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            CHPEVX 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 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 REAL
                     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 REAL
                     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 REAL
                     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*SLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*SLAMCH('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 REAL array, dimension (N)
                     If INFO = 0, the selected eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 array, dimension (2*N)

           RWORK

                     RWORK is REAL 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 chpgv (integer ITYPE, character JOBZ, character UPLO, integer N, complex,
       dimension( * ) AP, complex, dimension( * ) BP, real, dimension( * ) W, complex, dimension(
       ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer
       INFO)
       CHPGV

       Purpose:

            CHPGV 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 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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

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

           RWORK

                     RWORK is REAL 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:  CPPTRF or CHPEV returned an error code:
                        <= N:  if INFO = i, CHPEV 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 chpgvd (integer ITYPE, character JOBZ, character UPLO, integer N, complex,
       dimension( * ) AP, complex, dimension( * ) BP, real, dimension( * ) W, complex, dimension(
       ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, real, dimension( * )
       RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
       CHPGVD

       Purpose:

            CHPGVD 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 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 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 REAL array, dimension (N)
                     If INFO = 0, the eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 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 >= 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 REAL 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:  CPPTRF or CHPEVD returned an error code:
                        <= N:  if INFO = i, CHPEVD 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 chpgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N,
       complex, dimension( * ) AP, complex, dimension( * ) BP, real VL, real VU, integer IL,
       integer IU, real ABSTOL, integer M, real, dimension( * ) W, complex, dimension( ldz, * )
       Z, integer LDZ, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer,
       dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
       CHPGVX

       Purpose:

            CHPGVX 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 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 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 REAL

                     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 REAL

                     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 REAL
                     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*SLAMCH('S'), not zero.
                     If this routine returns with INFO>0, indicating that some
                     eigenvectors did not converge, try setting ABSTOL to
                     2*SLAMCH('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 REAL array, dimension (N)
                     On normal exit, the first M elements contain the selected
                     eigenvalues in ascending order.

           Z

                     Z is COMPLEX 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 array, dimension (2*N)

           RWORK

                     RWORK is REAL 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:  CPPTRF or CHPEVX returned an error code:
                        <= N:  if INFO = i, CHPEVX 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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