ZHBEVX(3f)            LAPACK driver routine (version 3.1)           ZHBEVX(3f)



NAME
       ZHBEVX - 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.  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.

       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 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    (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
               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 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       (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 eigenvalues 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
               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 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    (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.



 LAPACK driver routine (version 3November 2006                      ZHBEVX(3f)
