complex16OTHEReigen(3)

NAME
complex16OTHEReigen - complex16 Other Eigenvalue routines

SYNOPSIS

  Functions
subroutine zggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
ZGGGLM
subroutine zhbev (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, rwork, info)
ZHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbev_2stage (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, info)
ZHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbevd (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbevd_2stage (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)
ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhbgv (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)
ZHBGV
subroutine zhbgvd (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHBGVD
subroutine zhbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
ZHBGVX
subroutine zhpev (jobz, uplo, n, ap, w, z, ldz, work, rwork, info)
ZHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhpevd (jobz, uplo, n, ap, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhpevx (jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine zhpgv (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, rwork, info)
ZHPGV
subroutine zhpgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHPGVD
subroutine zhpgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)
ZHPGVX

Detailed Description

This is the group of complex16 Other Eigenvalue routines

Function Documentation

  subroutine zggglm (integer n, integer m, integer p, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) d, complex*16, dimension( * ) x, complex*16, dimension( * ) y, complex*16, dimension( * ) work, integer lwork, integer info)

ZGGGLM

Purpose:

     ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
             minimize || y ||_2   subject to   d = A*x + B*y
                 x
     where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
     given N-vector. It is assumed that M <= N <= M+P, and
                rank(A) = M    and    rank( A B ) = N.
     Under these assumptions, the constrained equation is always
     consistent, and there is a unique solution x and a minimal 2-norm
     solution y, which is obtained using a generalized QR factorization
     of the matrices (A, B) given by
        A = Q*(R),   B = Q*T*Z.
              (0)
     In particular, if matrix B is square nonsingular, then the problem
     GLM is equivalent to the following weighted linear least squares
     problem
                  minimize || inv(B)*(d-A*x) ||_2
                      x
     where inv(B) denotes the inverse of B.

Parameters
    N

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

    M

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

    P

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

    A

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

    LDA

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

    B

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

    LDB

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

    D

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

    X

              X is COMPLEX*16 array, dimension (M)

    Y

              Y is COMPLEX*16 array, dimension (P)
              On exit, X and Y are the solutions of the GLM problem.

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,N+M+P).
              For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
              where NB is an upper bound for the optimal blocksizes for
              ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              = 1:  the upper triangular factor R associated with A in the
                    generalized QR factorization of the pair (A, B) is
                    singular, so that rank(A) < M; the least squares
                    solution could not be computed.
              = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                    factor T associated with B in the generalized QR
                    factorization of the pair (A, B) is singular, so that
                    rank( A B ) < N; the least squares solution could not
                    be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 183 of file zggglm.f.

  subroutine zhbev (character jobz, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

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

Parameters
    JOBZ

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

    UPLO

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

    N

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

    KD

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

    AB

              AB is COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermitian band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

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

    W

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

    Z

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

    LDZ

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

    WORK

              WORK is COMPLEX*16 array, dimension (N)

    RWORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 150 of file zhbev.f.

  subroutine zhbev_2stage (character jobz, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZHBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

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

Parameters
    JOBZ

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

    UPLO

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

    N

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

    KD

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

    AB

              AB is COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermitian band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The length of the array WORK. LWORK >= 1, when N <= 1;
              otherwise
              If JOBZ = 'N' and N > 1, LWORK must be queried.
                                       LWORK = MAX(1, dimension) where
                                       dimension = (2KD+1)*N + KD*NTHREADS
                                       where KD is the size of the band.
                                       NTHREADS is the number of threads used when
                                       openMP compilation is enabled, otherwise =1.
              If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

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

Definition at line 209 of file zhbev_2stage.f.

  subroutine zhbevd (character jobz, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of
     a complex Hermitian band matrix A.  If eigenvectors are desired, it
     uses a divide and conquer algorithm.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    JOBZ

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

    UPLO

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

    N

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

    KD

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

    AB

              AB is COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermitian band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK must be at least 1.
              If JOBZ = 'N' and N > 1, LWORK must be at least N.
              If JOBZ = 'V' and N > 1, LWORK must be at least 2*N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK must be at least 1.
              If JOBZ = 'N' and N > 1, LRWORK must be at least N.
              If JOBZ = 'V' and N > 1, LRWORK must be at least
                            1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
              If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 213 of file zhbevd.f.

  subroutine zhbevd_2stage (character jobz, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     ZHBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
     a complex Hermitian band matrix A using the 2stage technique for
     the reduction to tridiagonal.  If eigenvectors are desired, it
     uses a divide and conquer algorithm.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    JOBZ

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

    UPLO

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

    N

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

    KD

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

    AB

              AB is COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermitian band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The length of the array WORK. LWORK >= 1, when N <= 1;
              otherwise
              If JOBZ = 'N' and N > 1, LWORK must be queried.
                                       LWORK = MAX(1, dimension) where
                                       dimension = (2KD+1)*N + KD*NTHREADS
                                       where KD is the size of the band.
                                       NTHREADS is the number of threads used when
                                       openMP compilation is enabled, otherwise =1.
              If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK must be at least 1.
              If JOBZ = 'N' and N > 1, LRWORK must be at least N.
              If JOBZ = 'V' and N > 1, LRWORK must be at least
                            1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ = 'N' or N <= 1, LIWORK must be at least 1.
              If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N .
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

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

Definition at line 257 of file zhbevd_2stage.f.

  subroutine zhbevx (character jobz, character range, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

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

Parameters
    JOBZ

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

    RANGE

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

    UPLO

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

    N

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

    KD

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

    AB

              AB is COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermitian band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.

    LDAB

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

    Q

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

    LDQ

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

    VL

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

    VU

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

    IL

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

    IU

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

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              The absolute error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AB to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*DLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*DLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

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

    W

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

    Z

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

    LDZ

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

    WORK

              WORK is COMPLEX*16 array, dimension (N)

    RWORK

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

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 264 of file zhbevx.f.

  subroutine zhbevx_2stage (character jobz, character range, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

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

Parameters
    JOBZ

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

    RANGE

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

    UPLO

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

    N

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

    KD

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

    AB

              AB is COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermitian band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.

    LDAB

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

    Q

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

    LDQ

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

    VL

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

    VU

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

    IL

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

    IU

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

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              The absolute error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AB to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*DLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*DLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

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

    W

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

    Z

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

    LDZ

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

    WORK

              WORK is COMPLEX*16 array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              The length of the array WORK. LWORK >= 1, when N <= 1;
              otherwise
              If JOBZ = 'N' and N > 1, LWORK must be queried.
                                       LWORK = MAX(1, dimension) where
                                       dimension = (2KD+1)*N + KD*NTHREADS
                                       where KD is the size of the band.
                                       NTHREADS is the number of threads used when
                                       openMP compilation is enabled, otherwise =1.
              If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

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

Definition at line 323 of file zhbevx_2stage.f.

  subroutine zhbgv (character jobz, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHBGV

Purpose:

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

Parameters
    JOBZ

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

    UPLO

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

    N

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

    KA

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

    KB

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

    AB

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

    LDAB

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

    BB

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

    LDBB

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

    W

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

    Z

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

    LDZ

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

    WORK

              WORK is COMPLEX*16 array, dimension (N)

    RWORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 181 of file zhbgv.f.

  subroutine zhbgvd (character jobz, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHBGVD

Purpose:

     ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors
     of a complex generalized Hermitian-definite banded eigenproblem, of
     the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
     and banded, and B is also positive definite.  If eigenvectors are
     desired, it uses a divide and conquer algorithm.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    JOBZ

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

    UPLO

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

    N

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

    KA

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

    KB

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

    AB

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

    LDAB

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

    BB

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

    LDBB

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= N.
              If JOBZ = 'V' and N > 1, LWORK >= 2*N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK >= 1.
              If JOBZ = 'N' and N > 1, LRWORK >= N.
              If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 249 of file zhbgvd.f.

  subroutine zhbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHBGVX

Purpose:

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

Parameters
    JOBZ

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

    RANGE

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

    UPLO

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

    N

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

    KA

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

    KB

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

    AB

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

    LDAB

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

    BB

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

    LDBB

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

    Q

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

    LDQ

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

    VL

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

    VU

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

    IL

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

    IU

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

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              The absolute error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AP to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*DLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*DLAMCH('S').

    M

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

    W

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

    Z

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

    LDZ

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

    WORK

              WORK is COMPLEX*16 array, dimension (N)

    RWORK

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

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 297 of file zhbgvx.f.

  subroutine zhpev (character jobz, character uplo, integer n, complex*16, dimension( * ) ap, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

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

Parameters
    JOBZ

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

    UPLO

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

    N

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

    AP

              AP is COMPLEX*16 array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, AP is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the diagonal
              and first superdiagonal of the tridiagonal matrix T overwrite
              the corresponding elements of A, and if UPLO = 'L', the
              diagonal and first subdiagonal of T overwrite the
              corresponding elements of A.

    W

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

    Z

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

    LDZ

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

    WORK

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

    RWORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 136 of file zhpev.f.

  subroutine zhpevd (character jobz, character uplo, integer n, complex*16, dimension( * ) ap, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     ZHPEVD computes all the eigenvalues and, optionally, eigenvectors of
     a complex Hermitian matrix A in packed storage.  If eigenvectors are
     desired, it uses a divide and conquer algorithm.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    JOBZ

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

    UPLO

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

    N

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

    AP

              AP is COMPLEX*16 array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, AP is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the diagonal
              and first superdiagonal of the tridiagonal matrix T overwrite
              the corresponding elements of A, and if UPLO = 'L', the
              diagonal and first subdiagonal of T overwrite the
              corresponding elements of A.

    W

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

    Z

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

    LDZ

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of array WORK.
              If N <= 1,               LWORK must be at least 1.
              If JOBZ = 'N' and N > 1, LWORK must be at least N.
              If JOBZ = 'V' and N > 1, LWORK must be at least 2*N.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK must be at least 1.
              If JOBZ = 'N' and N > 1, LRWORK must be at least N.
              If JOBZ = 'V' and N > 1, LRWORK must be at least
                        1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
              If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 198 of file zhpevd.f.

  subroutine zhpevx (character jobz, character range, character uplo, integer n, complex*16, dimension( * ) ap, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

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

Parameters
    JOBZ

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

    RANGE

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

    UPLO

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

    N

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

    AP

              AP is COMPLEX*16 array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, AP is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the diagonal
              and first superdiagonal of the tridiagonal matrix T overwrite
              the corresponding elements of A, and if UPLO = 'L', the
              diagonal and first subdiagonal of T overwrite the
              corresponding elements of A.

    VL

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

    VU

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

    IL

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

    IU

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

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              The absolute error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AP to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*DLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*DLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    RWORK

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

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 237 of file zhpevx.f.

  subroutine zhpgv (integer itype, character jobz, character uplo, integer n, complex*16, dimension( * ) ap, complex*16, dimension( * ) bp, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHPGV

Purpose:

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

Parameters
    ITYPE

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

    JOBZ

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

    UPLO

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

    N

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

    AP

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

    BP

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    RWORK

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 163 of file zhpgv.f.

  subroutine zhpgvd (integer itype, character jobz, character uplo, integer n, complex*16, dimension( * ) ap, complex*16, dimension( * ) bp, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHPGVD

Purpose:

     ZHPGVD computes all the eigenvalues and, optionally, the eigenvectors
     of a complex generalized Hermitian-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
     B are assumed to be Hermitian, stored in packed format, and B is also
     positive definite.
     If eigenvectors are desired, it uses a divide and conquer algorithm.
     The divide and conquer algorithm makes very mild assumptions about
     floating point arithmetic. It will work on machines with a guard
     digit in add/subtract, or on those binary machines without guard
     digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
     Cray-2. It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    ITYPE

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

    JOBZ

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

    UPLO

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

    N

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

    AP

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

    BP

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

    W

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

    Z

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

    LDZ

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= N.
              If JOBZ = 'V' and N > 1, LWORK >= 2*N.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

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

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK >= 1.
              If JOBZ = 'N' and N > 1, LRWORK >= N.
              If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 229 of file zhpgvd.f.

  subroutine zhpgvx (integer itype, character jobz, character range, character uplo, integer n, complex*16, dimension( * ) ap, complex*16, dimension( * ) bp, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHPGVX

Purpose:

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

Parameters
    ITYPE

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

    JOBZ

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

    RANGE

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

    UPLO

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

    N

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

    AP

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

    BP

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

    VL

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

    VU

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

    IL

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

    IU

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

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              The absolute error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AP to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*DLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*DLAMCH('S').

    M

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

    W

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

    Z

              Z is COMPLEX*16 array, dimension (LDZ, N)
              If JOBZ = 'N', then Z is not referenced.
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**H*B*Z = I;
              if ITYPE = 3, Z**H*inv(B)*Z = I.
              If an eigenvector fails to converge, then that column of Z
              contains the latest approximation to the eigenvector, and the
              index of the eigenvector is returned in IFAIL.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

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

    WORK

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

    RWORK

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

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 274 of file zhpgvx.f.

Author

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