CHPEVX  -  compute  selected  eigenvalues and, optionally,
       eigenvectors of a complex Hermitian  matrix  A  in  packed
       storage


SYNOPSIS

       SUBROUTINE CHPEVX( JOBZ,  RANGE,  UPLO, N, AP, VL, VU, IL,
                          IU, ABSTOL, M, W, Z, LDZ, WORK,  RWORK,
                          IWORK, IFAIL, INFO )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, IU, LDZ, M, N

           REAL           ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           REAL           RWORK( * ), W( * )

           COMPLEX        AP( * ), WORK( * ), Z( LDZ, * )


PURPOSE

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


ARGUMENTS

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

       RANGE   (input) 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    (input) CHARACTER*1
               = 'U':  Upper triangle of A is stored;
               = 'L':  Lower triangle of A is stored.

       N       (input) INTEGER
               The order of the matrix A.  N >= 0.

       AP      (input/output) COMPLEX array, dimension
               (N*(N+1)/2)
               On entry, the upper or lower triangle of the  Her­
               mitian  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 +

               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      (input) REAL
               VU       (input)  REAL If RANGE='V', the lower and
               upper bounds of the interval to  be  searched  for
               eigenvalues.  VL  < VU.  Not referenced if RANGE =
               'A' or 'I'.

       IL      (input) INTEGER
               IU      (input) INTEGER If RANGE='I', the  indices
               (in  ascending  order) of the smallest and largest
               eigenvalues to be returned.  1 <= IL <= IU  <=  N,
               if  N > 0; IL = 1 and IU = 0 if N = 0.  Not refer­
               enced if RANGE = 'A' or 'V'.

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

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

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

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

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

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

               If INFO = 0, the selected eigenvalues in ascending
               order.

       Z       (output) COMPLEX array, dimension (LDZ, max(1,M))
               If  JOBZ  =  'V',  then  if  INFO = 0, the first M
               columns of Z contain the orthonormal  eigenvectors
               of  the  matrix  A  corresponding  to the selected
               eigenvalues, with the i-th column of Z holding the
               eigenvector associated with W(i).  If an eigenvec­
               tor fails to converge, then that column of Z  con­
               tains 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     (input) INTEGER
               The  leading  dimension of the array Z.  LDZ >= 1,
               and if JOBZ = 'V', LDZ >= max(1,N).

       WORK    (workspace) COMPLEX array, dimension (2*N)

       RWORK   (workspace) REAL array, dimension (7*N)

       IWORK   (workspace) INTEGER array, dimension (5*N)

       IFAIL   (output) INTEGER array, dimension (N)
               If JOBZ = 'V', then if INFO = 0, the first M  ele­
               ments  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    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               >  0:   if INFO = i, then i eigenvectors failed to
               converge.   Their  indices  are  stored  in  array
               IFAIL.


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