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sspevx


 NAME
      SSPEVX - compute selected eigenvalues and, optionally,
      eigenvectors of a real symmetric matrix A in packed storage

 SYNOPSIS
      SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
                         ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
                         INFO )

          CHARACTER      JOBZ, RANGE, UPLO

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

          REAL           ABSTOL, VL, VU

          INTEGER        IFAIL( * ), IWORK( * )

          REAL           AP( * ), W( * ), WORK( * ), Z( LDZ, * )

 PURPOSE
      SSPEVX computes selected eigenvalues and, optionally, eigen-
      vectors of a real symmetric matrix A in packed storage.
      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.

      AP      (input/output) REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the sym-
              metric matrix A, packed columnwise in a linear
              array.  The j-th column of A is stored in the array
              AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) =
              A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-
              1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

              On exit, AP is overwritten by values generated dur-
              ing the reduction to tridiagonal form.  If UPLO =
              'U', the diagonal and first superdiagonal of the
              tridiagonal matrix T overwrite the corresponding
              elements of A, and if UPLO = 'L', the diagonal and
              first subdiagonal of T overwrite the corresponding
              elements of A.

      VL      (input) REAL
              If RANGE='V', the lower bound of the interval to be
              searched for eigenvalues.  Not referenced if RANGE =
              'A' or 'I'.

      VU      (input) REAL
              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) 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 AP 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) REAL array, dimension (N)

              If INFO = 0, the selected eigenvalues in ascending
              order.

      Z       (output) REAL 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) REAL array, dimension (8*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.