DSTEVX  -  compute  selected  eigenvalues and, optionally,
       eigenvectors of a real symmetric tridiagonal matrix A


SYNOPSIS

       SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU,  IL,  IU,
                          ABSTOL,  M,  W,  Z,  LDZ,  WORK, IWORK,
                          IFAIL, INFO )

           CHARACTER      JOBZ, RANGE

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

           DOUBLE         PRECISION ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           DOUBLE         PRECISION D( * ), E( * ), W( * ), WORK(
                          * ), Z( LDZ, * )


PURPOSE

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


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.

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

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

       E       (input/output) DOUBLE PRECISION array, dimension
               (N)
               On entry, the (n-1) subdiagonal  elements  of  the
               tridiagonal  matrix  A  in elements 1 to N-1 of E;
               E(N)  need  not  be  set.   On  exit,  E  may   be

               over/underflow in computing the eigenvalues.

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

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

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

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

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

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

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

       Z       (output) DOUBLE PRECISION array, dimension (LDZ,
               max(1,M) )
               If  JOBZ  =  'V',  then  if  INFO = 0, the first M
               columns of Z contain the orthonormal  eigenvectors
               of  the  matrix  A  corresponding  to the selected
               eigenvalues, with the i-th column of Z holding the

               tor fails to converge (INFO > 0), then that column
               of  Z  contains  the  latest  approximation to the
               eigenvector, and the index of the  eigenvector  is
               returned  in  IFAIL.  If JOBZ = 'N', then Z is not
               referenced.  Note: the user must  ensure  that  at
               least  max(1,M)  columns are supplied in the array
               Z; if RANGE = 'V', the exact value  of  M  is  not
               known  in advance and an upper bound must be used.

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

       WORK    (workspace) DOUBLE PRECISION array, dimension
               (5*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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