CHBEVX (3lapack)

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

SYNOPSIS

  SUBROUTINE CHBEVX( 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

      REAL           ABSTOL, VL, VU

      INTEGER        IFAIL( * ), IWORK( * )

      REAL           RWORK( * ), W( * )

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

PURPOSE

  CHBEVX 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 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 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) 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
          referenced 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 AB 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.

  W       (output) REAL array, dimension (N)
          The first M elements contain 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 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 array, dimension (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 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.