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zhbevx


 NAME
      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, eigen-
      vectors of a complex Hermitian band matrix A.
      Eigenvalues/vectors can be selected by specifying either a
      range of values or a range of indices for the desired eigen-
      values.

 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.

      AB      (input/output) COMPLEX*16 array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the Hermi-
              tian 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 dur-
              ing 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
              If RANGE='V', the lower bound of the interval to be
              searched for eigenvalues.  Not referenced if RANGE =
              'A' or 'I'.

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

      IL      (input) INTEGER
              If RANGE='I', the index (from smallest to largest)
              of the smallest eigenvalue to be returned.  IL >= 1.
              Not referenced if RANGE = 'A' or 'V'.

      IU      (input) INTEGER
              If RANGE='I', the index (from smallest to largest)
              of the largest eigenvalue to be returned.  min(IL,N)
              <= IU <= N.  Not referenced 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 tridi-
              agonal form.

              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 corresponding to the selected eigenvalues.
              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     (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
              < 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.