SSTEBZ (3lapack)

compute the eigenvalues of a symmetric tridiagonal matrix T

SYNOPSIS

  SUBROUTINE SSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M,
                     NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO )

      CHARACTER      ORDER, RANGE

      INTEGER        IL, INFO, IU, M, N, NSPLIT

      REAL           ABSTOL, VL, VU

      INTEGER        IBLOCK( * ), ISPLIT( * ), IWORK( * )

      REAL           D( * ), E( * ), W( * ), WORK( * )

PURPOSE

  SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T.  The
  user may ask for all eigenvalues, all eigenvalues in the half-open interval
  (VL, VU], or the IL-th through IU-th eigenvalues.

  To avoid overflow, the matrix must be scaled so that its
  largest element is no greater than overflow**(1/2) *
  underflow**(1/4) in absolute value, and for greatest
  accuracy, it should not be much smaller than that.

  See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix",
  Report CS41, Computer Science Dept., Stanford
  University, July 21, 1966.

ARGUMENTS

  RANGE   (input) CHARACTER
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU]
          will be found.  = 'I': ("Index") the IL-th through IU-th
          eigenvalues (of the entire matrix) will be found.

  ORDER   (input) CHARACTER
          = 'B': ("By Block") the eigenvalues will be grouped by split-off
          block (see IBLOCK, ISPLIT) and ordered from smallest to largest
          within the block.  = 'E': ("Entire matrix") the eigenvalues for the
          entire matrix will be ordered from smallest to largest.

  N       (input) INTEGER
          The order of the tridiagonal matrix T.  N >= 0.

  VL      (input) REAL
          VU      (input) REAL If RANGE='V', the lower and upper bounds of
          the interval to be searched for eigenvalues.  Eigenvalues less than
          or equal to VL, or greater than VU, will not be returned.  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 tolerance for the eigenvalues.  An eigenvalue (or
          cluster) is considered to be located if it has been determined to
          lie in an interval whose width is ABSTOL or less.  If ABSTOL is
          less than or equal to zero, then ULP*|T| will be used, where |T|
          means the 1-norm of T.

          Eigenvalues will be computed most accurately when ABSTOL is set to
          twice the underflow threshold 2*SLAMCH('S'), not zero.

  D       (input) REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

  E       (input) REAL array, dimension (N-1)
          The (n-1) off-diagonal elements of the tridiagonal matrix T.

  M       (output) INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.  (See also the
          description of INFO=2,3.)

  NSPLIT  (output) INTEGER
          The number of diagonal blocks in the matrix T.  1 <= NSPLIT <= N.

  W       (output) REAL array, dimension (N)
          On exit, the first M elements of W will contain the eigenvalues.
          (SSTEBZ may use the remaining N-M elements as workspace.)

  IBLOCK  (output) INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the matrix T is
          considered to split into a block diagonal matrix.  On exit, if INFO
          = 0, IBLOCK(i) specifies to which block (from 1 to the number of
          blocks) the eigenvalue W(i) belongs.  (SSTEBZ may use the remaining
          N-M elements as workspace.)

  ISPLIT  (output) INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.  The
          first submatrix consists of rows/columns 1 to ISPLIT(1), the second
          of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the
          NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through
          ISPLIT(NSPLIT)=N.  (Only the first NSPLIT elements will actually be
          used, but since the user cannot know a priori what value NSPLIT
          will have, N words must be reserved for ISPLIT.)

  WORK    (workspace) REAL array, dimension (4*N)

  IWORK   (workspace) INTEGER array, dimension (3*N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  some or all of the eigenvalues failed to converge or
          were not computed:
          =1 or 3: Bisection failed to converge for some eigenvalues; these
          eigenvalues are flagged by a negative block number.  The effect is
          that the eigenvalues may not be as accurate as the absolute and
          relative tolerances.  This is generally caused by unexpectedly
          inaccurate arithmetic.  =2 or 3: RANGE='I' only: Not all of the
          eigenvalues
          IL:IU were found.
          Effect: M < IU+1-IL
          Cause:  non-monotonic arithmetic, causing the Sturm sequence to be
          non-monotonic.  Cure:   recalculate, using RANGE='A', and pick
          out eigenvalues IL:IU.  In some cases, increasing the PARAMETER
          "FUDGE" may make things work.  = 4:    RANGE='I', and the
          Gershgorin interval initially used was too small.  No eigenvalues
          were computed.  Probable cause: your machine has sloppy floating-
          point arithmetic.  Cure: Increase the PARAMETER "FUDGE", recompile,
          and try again.

PARAMETERS

  RELFAC  REAL, default = 2.0e0
          The relative tolerance.  An interval (a,b] lies within "relative
          tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|), where "ulp" is the
          machine precision (distance from 1 to the next larger floating
          point number.)

  FUDGE   REAL, default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally, a
          value of 1 should work, but on machines with sloppy arithmetic,
          this needs to be larger.  The default for publicly released
          versions should be large enough to handle the worst machine around.
          Note that this has no effect on accuracy of the solution.