ZHBEVX  -  compute  selected  eigenvalues and, optionally,
       eigenvectors of a complex Hermitian band matrix A


SYNOPSIS

       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 )

           CHARACTER      JOBZ, RANGE, UPLO

           INTEGER        IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N

           DOUBLE         PRECISION ABSTOL, VL, VU

           INTEGER        IFAIL( * ), IWORK( * )

           DOUBLE         PRECISION RWORK( * ), W( * )

           COMPLEX*16     AB( LDAB, * ), Q( LDQ, * ), WORK( *  ),
                          Z( LDZ, * )


PURPOSE

       ZHBEVX  computes  selected  eigenvalues  and,  optionally,
       eigenvectors of a complex Hermitian band matrix A.  Eigen­
       values  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.

       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.

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

               N)
               On entry, the upper or lower triangle of the  Her­
               mitian  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    (input) INTEGER
               The leading dimension of the array AB.  LDAB >= KD
               + 1.

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

       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  obtained  by  reducing  AB  to
               tridiagonal form.

               Eigenvalues  will be computed most accurately when

               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) 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 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*16 array, dimension (N)

       RWORK   (workspace) DOUBLE PRECISION 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

               gal value
               >  0:   if INFO = i, then i eigenvectors failed to
               converge.   Their  indices  are  stored  in  array
               IFAIL.


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