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NAME

       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 (characterRANGE, characterORDER, integerN, realVL, realVU, integerIL,
       integerIU, realABSTOL, real, dimension( * )D, real, dimension( * )E, integerM,
       integerNSPLIT, real, dimension( * )W, integer, dimension( * )IBLOCK, integer, dimension( *
       )ISPLIT, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
       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

           VU

                     VU is 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

                     IL is INTEGER

           IU

                     IU is 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

                     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.

       Date:
           November 2011

       Definition at line 262 of file sstebz.f.

Author

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