DSYEVX (3lapack)

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

SYNOPSIS

  SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M,
                     W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )

      CHARACTER      JOBZ, RANGE, UPLO

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

      DOUBLE         PRECISION ABSTOL, VL, VU

      INTEGER        IFAIL( * ), IWORK( * )

      DOUBLE         PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE

  DSYEVX computes selected eigenvalues and, optionally, eigenvectors of a
  real symmetric 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.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading N-
          by-N upper triangular part of A contains the upper triangular part
          of the matrix A.  If UPLO = 'L', the leading N-by-N lower
          triangular part of A contains the lower triangular part of the
          matrix A.  On exit, the lower triangle (if UPLO='L') or the upper
          triangle (if UPLO='U') of A, including the diagonal, is destroyed.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= 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 A 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)
          On normal exit, the first M elements contain the selected
          eigenvalues 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
          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/output) DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of the array WORK.  LWORK >= max(1,8*N).  For optimal
          efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for DSYTRD
          returned by ILAENV.

  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.