SRC/sstebz.f(3)

NAME
SRC/sstebz.f

SYNOPSIS

  Functions/Subroutines
subroutine sstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
SSTEBZ

Function/Subroutine Documentation

  subroutine sstebz (character range, character order, integer n, real vl, real vu, integer il, integer iu, real abstol, real, dimension( * ) d, real, dimension( * ) e, integer m, integer nsplit, real, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSTEBZ

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.

Parameters
    RANGE

              RANGE is CHARACTER*1
              = '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

              ORDER is CHARACTER*1
              = '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

              N is INTEGER
              The order of the tridiagonal matrix T.  N >= 0.

    VL

              VL is REAL
              If RANGE='V', the lower bound 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'.

    VU

              VU is REAL
              If RANGE='V', the upper bound 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

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue 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'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue 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

              ABSTOL is 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

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

    E

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

    M

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

    NSPLIT

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

    W

              W is 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

              IBLOCK is 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

              ISPLIT is 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

              WORK is REAL array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (3*N)

    INFO

              INFO is 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.

Internal 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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 270 of file sstebz.f.

Author

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