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

NAME

       doubleOTHEReigen - double

SYNOPSIS

   Functions
       subroutine dbdsvdx (UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, NS, S, Z, LDZ, WORK,
           IWORK, INFO)
           DBDSVDX
       subroutine dggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)
           DGGGLM
       subroutine dsbev (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, INFO)
            DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dsbev_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, INFO)
            DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine dsbevd (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK,
           INFO)
            DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dsbevd_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK,
           LIWORK, INFO)
            DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine dsbevx (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M,
           W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
            DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dsbevx_2stage (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU,
           ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
            DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
           eigenvectors for OTHER matrices
       subroutine dsbgv (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO)
           DSBGV
       subroutine dsbgvd (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK,
           IWORK, LIWORK, INFO)
           DSBGVD
       subroutine dsbgvx (JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL,
           IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
           DSBGVX
       subroutine dspev (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO)
            DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dspevd (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
            DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dspevx (JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
           IWORK, IFAIL, INFO)
            DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dspgv (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO)
           DSPGV
       subroutine dspgvd (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK,
           INFO)
           DSPGVD
       subroutine dspgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z,
           LDZ, WORK, IWORK, IFAIL, INFO)
           DSPGVX
       subroutine dstev (JOBZ, N, D, E, Z, LDZ, WORK, INFO)
            DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dstevd (JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
            DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dstevr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ,
           WORK, LWORK, IWORK, LIWORK, INFO)
            DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices
       subroutine dstevx (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
           IWORK, IFAIL, INFO)
            DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for OTHER matrices

Detailed Description

       This is the group of double Other Eigenvalue routines

Function Documentation

   subroutine dbdsvdx (character UPLO, character JOBZ, character RANGE, integer N, double
       precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL,
       double precision VU, integer IL, integer IU, integer NS, double precision, dimension( * )
       S, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
       WORK, integer, dimension( * ) IWORK, integer INFO)
       DBDSVDX

       Purpose:

             DBDSVDX computes the singular value decomposition (SVD) of a real
             N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT,
             where S is a diagonal matrix with non-negative diagonal elements
             (the singular values of B), and U and VT are orthogonal matrices
             of left and right singular vectors, respectively.

             Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ]
             and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the
             singular value decompositon of B through the eigenvalues and
             eigenvectors of the N*2-by-N*2 tridiagonal matrix

                   |  0  d_1                |
                   | d_1  0  e_1            |
             TGK = |     e_1  0  d_2        |
                   |         d_2  .   .     |
                   |              .   .   . |

             If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then
             (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) /
             sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and
             P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].

             Given a TGK matrix, one can either a) compute -s,-v and change signs
             so that the singular values (and corresponding vectors) are already in
             descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder
             the values (and corresponding vectors). DBDSVDX implements a) by
             calling DSTEVX (bisection plus inverse iteration, to be replaced
             with a version of the Multiple Relative Robust Representation
             algorithm. (See P. Willems and B. Lang, A framework for the MR^3
             algorithm: theory and implementation, SIAM J. Sci. Comput.,
             35:740-766, 2013.)

       Parameters
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  B is upper bidiagonal;
                     = 'L':  B is lower bidiagonal.

           JOBZ

                     JOBZ is CHARACTER*1
                     = 'N':  Compute singular values only;
                     = 'V':  Compute singular values and singular vectors.

           RANGE

                     RANGE is CHARACTER*1
                     = 'A': all singular values will be found.
                     = 'V': all singular values in the half-open interval [VL,VU)
                            will be found.
                     = 'I': the IL-th through IU-th singular values will be found.

           N

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

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     The n diagonal elements of the bidiagonal matrix B.

           E

                     E is DOUBLE PRECISION array, dimension (max(1,N-1))
                     The (n-1) superdiagonal elements of the bidiagonal matrix
                     B in elements 1 to N-1.

           VL

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

           VU

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

           IL

                     IL is INTEGER
                     If RANGE='I', the index of the
                     smallest singular value to be returned.
                     1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
                     Not referenced if RANGE = 'A' or 'V'.

           IU

                     IU is INTEGER
                     If RANGE='I', the index of the
                     largest singular value to be returned.
                     1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
                     Not referenced if RANGE = 'A' or 'V'.

           NS

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

           S

                     S is DOUBLE PRECISION array, dimension (N)
                     The first NS elements contain the selected singular values in
                     ascending order.

           Z

                     Z is DOUBLE PRECISION array, dimension (2*N,K)
                     If JOBZ = 'V', then if INFO = 0 the first NS columns of Z
                     contain the singular vectors of the matrix B corresponding to
                     the selected singular values, with U in rows 1 to N and V
                     in rows N+1 to N*2, i.e.
                     Z = [ U ]
                         [ V ]
                     If JOBZ = 'N', then Z is not referenced.
                     Note: The user must ensure that at least K = NS+1 columns are
                     supplied in the array Z; if RANGE = 'V', the exact value of
                     NS 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(2,N*2).

           WORK

                     WORK is DOUBLE PRECISION array, dimension (14*N)

           IWORK

                     IWORK is INTEGER array, dimension (12*N)
                     If JOBZ = 'V', then if INFO = 0, the first NS elements of
                     IWORK are zero. If INFO > 0, then IWORK contains the indices
                     of the eigenvectors that failed to converge in DSTEVX.

           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
                              in DSTEVX. The indices of the eigenvectors
                              (as returned by DSTEVX) are stored in the
                              array IWORK.
                           if INFO = N*2 + 1, an internal error occurred.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dggglm (integer N, integer M, integer P, double precision, dimension( lda, * ) A,
       integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision,
       dimension( * ) D, double precision, dimension( * ) X, double precision, dimension( * ) Y,
       double precision, dimension( * ) WORK, integer LWORK, integer INFO)
       DGGGLM

       Purpose:

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

           X

                     X is DOUBLE PRECISION array, dimension (M)

           Y

                     Y is DOUBLE PRECISION array, dimension (P)

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

           WORK

                     WORK is DOUBLE PRECISION 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
                     DGEQRF, SGERQF, DORMQR and SORMRQ.

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

       Purpose:

            DSBEV computes all the eigenvalues and, optionally, eigenvectors of
            a real symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

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

           LDAB

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

           W

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

           Z

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

           INFO

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dsbev_2stage (character JOBZ, character UPLO, integer N, integer KD, double
       precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W,
       double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
       WORK, integer LWORK, integer INFO)
        DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors
       for OTHER matrices

       Purpose:

            DSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
            a real symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

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

           LDAB

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 + N
                                              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 size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, 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 dsbevd (character JOBZ, character UPLO, integer N, integer KD, double precision,
       dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double
       precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK,
       integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
        DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
            a real symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

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

           LDAB

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 DOUBLE PRECISION array,
                                                    dimension (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 > 2, LWORK must be at least 2*N.
                     If JOBZ  = 'V' and N > 2, LWORK must be at least
                                    ( 1 + 5*N + 2*N**2 ).

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK and IWORK
                     arrays, returns these values as the first entries of the WORK
                     and IWORK arrays, and no error message related to LWORK 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 the array IWORK.
                     If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
                     If JOBZ  = 'V' and N > 2, 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 and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK 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 dsbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, double
       precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W,
       double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
       WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
        DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for OTHER matrices

       Purpose:

            DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
            a real symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric band
                     matrix A, stored in the first KD+1 rows of the array.  The
                     j-th column of A is stored in the j-th column of the array AB
                     as follows:
                     if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
                     if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

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

           LDAB

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 + N
                                              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 and IWORK
                     arrays, returns these values as the first entries of the WORK
                     and IWORK arrays, and no error message related to LWORK 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 the array IWORK.
                     If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
                     If JOBZ  = 'V' and N > 2, 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 and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK 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 dsbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD,
       double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldq,
       * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU,
       double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision,
       dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer,
       dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
        DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSBEVX computes selected eigenvalues and, optionally, eigenvectors
            of a real symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric 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.

           Q

                     Q is DOUBLE PRECISION array, dimension (LDQ, N)
                     If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                                    reduction to tridiagonal form.
                     If JOBZ = 'N', the array Q is not referenced.

           LDQ

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

           VL

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

           VU

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

           IL

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

           IU

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

           ABSTOL

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

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

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

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

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

           M

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

           W

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

           Z

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

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

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

           INFO

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dsbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer
       KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension(
       ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU,
       double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision,
       dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
       integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
        DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for OTHER matrices

       Purpose:

            DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
            of a real symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric 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.

           Q

                     Q is DOUBLE PRECISION array, dimension (LDQ, N)
                     If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                                    reduction to tridiagonal form.
                     If JOBZ = 'N', the array Q is not referenced.

           LDQ

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

           VL

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

           VU

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

           IL

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

           IU

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

           ABSTOL

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

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

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

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

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

           M

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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, 7*N, dimension) where
                                              dimension = (2KD+1)*N + KD*NTHREADS + 2*N
                                              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 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.

           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 dsbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, double
       precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * )
       BB, integer LDBB, double precision, dimension( * ) W, double precision, dimension( ldz, *
       ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)
       DSBGV

       Purpose:

            DSBGV computes all the eigenvalues, and optionally, the eigenvectors
            of a real generalized symmetric-definite banded eigenproblem, of
            the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
            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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the symmetric 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**T*S, as returned by DPBSTF.

           LDBB

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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**T*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 DOUBLE PRECISION array, dimension (3*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  if INFO = i, and i is:
                        <= N:  the algorithm failed to converge:
                               i off-diagonal elements of an intermediate
                               tridiagonal form did not converge to zero;
                        > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
                               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 dsbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, double
       precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * )
       BB, integer LDBB, double precision, dimension( * ) W, double precision, dimension( ldz, *
       ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer,
       dimension( * ) IWORK, integer LIWORK, integer INFO)
       DSBGVD

       Purpose:

            DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
            of a real generalized symmetric-definite banded eigenproblem, of the
            form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric 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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the symmetric 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(ka+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**T*S, as returned by DPBSTF.

           LDBB

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 Z**T*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 DOUBLE PRECISION 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 >= 2*N.
                     If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.

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

           IWORK

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

           LIWORK

                     LIWORK is INTEGER
                     The dimension of the 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 and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK 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 DPBSTF
                               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 dsbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA,
       integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision,
       dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( ldq, * ) Q, integer
       LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision
       ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, *
       ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,
       integer, dimension( * ) IFAIL, integer INFO)
       DSBGVX

       Purpose:

            DSBGVX computes selected eigenvalues, and optionally, eigenvectors
            of a real generalized symmetric-definite banded eigenproblem, of
            the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
            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 DOUBLE PRECISION array, dimension (LDAB, N)
                     On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (LDBB, N)
                     On entry, the upper or lower triangle of the symmetric 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(ka+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**T*S, as returned by DPBSTF.

           LDBB

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

           Q

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

           LDQ

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

           VL

                     VL is DOUBLE PRECISION

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

           VU

                     VU is DOUBLE PRECISION

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

           IL

                     IL is INTEGER

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

           IU

                     IU is INTEGER

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

           ABSTOL

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

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

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

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

           M

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 Z**T*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 DOUBLE PRECISION array, dimension (7*N)

           IWORK

                     IWORK is INTEGER array, dimension (5*N)

           IFAIL

                     IFAIL is INTEGER array, dimension (M)
                     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 eigenvalues 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
                     <= N: if INFO = i, then i eigenvectors failed to converge.
                             Their indices are stored in IFAIL.
                     > N:  DPBSTF returned an error code; i.e.,
                           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 dspev (character JOBZ, character UPLO, integer N, double precision, dimension( * )
       AP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer
       LDZ, double precision, dimension( * ) WORK, integer INFO)
        DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSPEV computes all the eigenvalues and, optionally, eigenvectors of a
            real symmetric matrix A 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

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

           W

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

           Z

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

           INFO

                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  if INFO = i, 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 dspevd (character JOBZ, character UPLO, integer N, double precision, dimension( * )
       AP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer
       LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
       integer LIWORK, integer INFO)
        DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSPEVD computes all the eigenvalues and, optionally, eigenvectors
            of a real symmetric 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

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

           W

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

           Z

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

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If N <= 1,               LWORK must be at least 1.
                     If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
                     If JOBZ = 'V' and N > 1, LWORK must be at least
                                                            1 + 6*N + N**2.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the required sizes of the WORK and IWORK
                     arrays, returns these values as the first entries of the WORK
                     and IWORK arrays, and no error message related to LWORK 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 the 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 and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK 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 dspevx (character JOBZ, character RANGE, character UPLO, integer N, double
       precision, dimension( * ) AP, double precision VL, double precision VU, integer IL,
       integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double
       precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK,
       integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
        DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSPEVX computes selected eigenvalues and, optionally, eigenvectors
            of a real symmetric 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

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

           VL

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

           VU

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

           IL

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

           IU

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

           ABSTOL

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

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

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

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

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

           M

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (8*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 dspgv (integer ITYPE, character JOBZ, character UPLO, integer N, double precision,
       dimension( * ) AP, double precision, dimension( * ) BP, double precision, dimension( * )
       W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
       WORK, integer INFO)
       DSPGV

       Purpose:

            DSPGV computes all the eigenvalues and, optionally, the eigenvectors
            of a real generalized symmetric-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 symmetric, 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric 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**T*U or B = L*L**T, in the same storage
                     format as B.

           W

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

           Z

                     Z is DOUBLE PRECISION 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**T*B*Z = I;
                     if ITYPE = 3, Z**T*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 DOUBLE PRECISION array, dimension (3*N)

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value
                     > 0:  DPPTRF or DSPEV returned an error code:
                        <= N:  if INFO = i, DSPEV 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 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 dspgvd (integer ITYPE, character JOBZ, character UPLO, integer N, double precision,
       dimension( * ) AP, double precision, dimension( * ) BP, double precision, dimension( * )
       W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
       WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
       DSPGVD

       Purpose:

            DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
            of a real generalized symmetric-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 symmetric, 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric 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**T*U or B = L*L**T, in the same storage
                     format as B.

           W

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

           Z

                     Z is DOUBLE PRECISION 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**T*B*Z = I;
                     if ITYPE = 3, Z**T*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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the required LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If N <= 1,               LWORK >= 1.
                     If JOBZ = 'N' and N > 1, LWORK >= 2*N.
                     If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the required sizes of the WORK and IWORK
                     arrays, returns these values as the first entries of the WORK
                     and IWORK arrays, and no error message related to LWORK 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 the 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 and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK 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:  DPPTRF or DSPEVD returned an error code:
                        <= N:  if INFO = i, DSPEVD 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 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 dspgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N,
       double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision
       VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M,
       double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ,
       double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension(
       * ) IFAIL, integer INFO)
       DSPGVX

       Purpose:

            DSPGVX computes selected eigenvalues, and optionally, eigenvectors
            of a real generalized symmetric-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 symmetric, stored in packed storage, 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 triangle of A and B are stored;
                     = 'L':  Lower triangle of A and B are stored.

           N

                     N is INTEGER
                     The order of the matrix pencil (A,B).  N >= 0.

           AP

                     AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric 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**T*U or B = L*L**T, in the same storage
                     format as B.

           VL

                     VL is DOUBLE PRECISION

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

           VU

                     VU is DOUBLE PRECISION

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

           IL

                     IL is INTEGER

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

           IU

                     IU is INTEGER

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

           ABSTOL

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

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

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

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

           M

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

           W

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

           Z

                     Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
                     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**T*B*Z = I;
                     if ITYPE = 3, Z**T*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 DOUBLE PRECISION array, dimension (8*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:  DPPTRF or DSPEVX returned an error code:
                        <= N:  if INFO = i, DSPEVX 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

   subroutine dstev (character JOBZ, integer N, double precision, dimension( * ) D, double
       precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double
       precision, dimension( * ) WORK, integer INFO)
        DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSTEV computes all eigenvalues and, optionally, eigenvectors of a
            real symmetric tridiagonal matrix A.

       Parameters
           JOBZ

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

           N

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

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     On entry, the n diagonal elements of the tridiagonal matrix
                     A.
                     On exit, if INFO = 0, the eigenvalues in ascending order.

           E

                     E is DOUBLE PRECISION array, dimension (N-1)
                     On entry, the (n-1) subdiagonal elements of the tridiagonal
                     matrix A, stored in elements 1 to N-1 of E.
                     On exit, the contents of E are destroyed.

           Z

                     Z is DOUBLE PRECISION 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 D(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 DOUBLE PRECISION array, dimension (max(1,2*N-2))
                     If JOBZ = 'N', WORK 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, the algorithm failed to converge; i
                           off-diagonal elements of E did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dstevd (character JOBZ, integer N, double precision, dimension( * ) D, double
       precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double
       precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer
       LIWORK, integer INFO)
        DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSTEVD computes all eigenvalues and, optionally, eigenvectors of a
            real symmetric tridiagonal matrix. 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.

           N

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

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     On entry, the n diagonal elements of the tridiagonal matrix
                     A.
                     On exit, if INFO = 0, the eigenvalues in ascending order.

           E

                     E is DOUBLE PRECISION array, dimension (N-1)
                     On entry, the (n-1) subdiagonal elements of the tridiagonal
                     matrix A, stored in elements 1 to N-1 of E.
                     On exit, the contents of E are destroyed.

           Z

                     Z is DOUBLE PRECISION 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 D(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 DOUBLE PRECISION array,
                                                    dimension (LWORK)
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If JOBZ  = 'N' or N <= 1 then LWORK must be at least 1.
                     If JOBZ  = 'V' and N > 1 then LWORK must be at least
                                    ( 1 + 4*N + N**2 ).

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal sizes of the WORK and IWORK
                     arrays, returns these values as the first entries of the WORK
                     and IWORK arrays, and no error message related to LWORK 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 the array IWORK.
                     If JOBZ  = 'N' or N <= 1 then LIWORK must be at least 1.
                     If JOBZ  = 'V' and N > 1 then 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 and
                     IWORK arrays, returns these values as the first entries of
                     the WORK and IWORK arrays, and no error message related to
                     LWORK 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 E did not converge to zero.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dstevr (character JOBZ, character RANGE, integer N, double precision, dimension( *
       ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer
       IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W,
       double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ,
       double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK,
       integer LIWORK, integer INFO)
        DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSTEVR computes selected eigenvalues and, optionally, eigenvectors
            of a real symmetric tridiagonal matrix T.  Eigenvalues and
            eigenvectors can be selected by specifying either a range of values
            or a range of indices for the desired eigenvalues.

            Whenever possible, DSTEVR calls DSTEMR to compute the
            eigenspectrum using Relatively Robust Representations.  DSTEMR
            computes eigenvalues by the dqds algorithm, while orthogonal
            eigenvectors are computed from various 'good' L D L^T representations
            (also known as Relatively Robust Representations). Gram-Schmidt
            orthogonalization is avoided as far as possible. More specifically,
            the various steps of the algorithm are as follows. For the i-th
            unreduced block of T,
               (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
                    is a relatively robust representation,
               (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
                   relative accuracy by the dqds algorithm,
               (c) If there is a cluster of close eigenvalues, 'choose' sigma_i
                   close to the cluster, and go to step (a),
               (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
                   compute the corresponding eigenvector by forming a
                   rank-revealing twisted factorization.
            The desired accuracy of the output can be specified by the input
            parameter ABSTOL.

            For more details, see 'A new O(n^2) algorithm for the symmetric
            tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,
            Computer Science Division Technical Report No. UCB//CSD-97-971,
            UC Berkeley, May 1997.

            Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
            on machines which conform to the ieee-754 floating point standard.
            DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
            when partial spectrum requests are made.

            Normal execution of DSTEMR may create NaNs and infinities and
            hence may abort due to a floating point exception in environments
            which do not handle NaNs and infinities in the ieee standard default
            manner.

       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.
                     For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
                     DSTEIN are called

           N

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

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     On entry, the n diagonal elements of the tridiagonal matrix
                     A.
                     On exit, D may be multiplied by a constant factor chosen
                     to avoid over/underflow in computing the eigenvalues.

           E

                     E is DOUBLE PRECISION array, dimension (max(1,N-1))
                     On entry, the (n-1) subdiagonal elements of the tridiagonal
                     matrix A in elements 1 to N-1 of E.
                     On exit, E may be multiplied by a constant factor chosen
                     to avoid over/underflow in computing the eigenvalues.

           VL

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

           VU

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

           IL

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

           IU

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

           ABSTOL

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

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

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

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

                     If high relative accuracy is important, set ABSTOL to
                     DLAMCH( 'Safe minimum' ).  Doing so will guarantee that
                     eigenvalues are computed to high relative accuracy when
                     possible in future releases.  The current code does not
                     make any guarantees about high relative accuracy, but
                     future releases will. See J. Barlow and J. Demmel,
                     'Computing Accurate Eigensystems of Scaled Diagonally
                     Dominant Matrices', LAPACK Working Note #7, for a discussion
                     of which matrices define their eigenvalues to high relative
                     accuracy.

           M

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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).
                     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).

           ISUPPZ

                     ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
                     The support of the eigenvectors in Z, i.e., the indices
                     indicating the nonzero elements in Z. The i-th eigenvector
                     is nonzero only in elements ISUPPZ( 2*i-1 ) through
                     ISUPPZ( 2*i ).
                     Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

           WORK

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

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,20*N).

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

           LIWORK

                     LIWORK is INTEGER
                     The dimension of the array IWORK.  LIWORK >= max(1,10*N).

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

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Contributors:
           Inderjit Dhillon, IBM Almaden, USA
            Osni Marques, LBNL/NERSC, USA
            Ken Stanley, Computer Science Division, University of California at Berkeley, USA

   subroutine dstevx (character JOBZ, character RANGE, integer N, double precision, dimension( *
       ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer
       IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W,
       double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * )
       WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
        DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       OTHER matrices

       Purpose:

            DSTEVX computes selected eigenvalues and, optionally, eigenvectors
            of a real symmetric tridiagonal 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.

           N

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

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     On entry, the n diagonal elements of the tridiagonal matrix
                     A.
                     On exit, D may be multiplied by a constant factor chosen
                     to avoid over/underflow in computing the eigenvalues.

           E

                     E is DOUBLE PRECISION array, dimension (max(1,N-1))
                     On entry, the (n-1) subdiagonal elements of the tridiagonal
                     matrix A in elements 1 to N-1 of E.
                     On exit, E may be multiplied by a constant factor chosen
                     to avoid over/underflow in computing the eigenvalues.

           VL

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

           VU

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

           IL

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

           IU

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

           ABSTOL

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

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

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

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

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

           M

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

           W

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

           Z

                     Z is DOUBLE PRECISION 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 (INFO > 0), 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 DOUBLE PRECISION array, dimension (5*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.

Author

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