dstebz(3S)

NAME
     DSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix T

SYNOPSIS
     SUBROUTINE DSTEBZ( 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

         DOUBLE         PRECISION ABSTOL, VL, VU

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

         DOUBLE         PRECISION D( * ), E( * ), W( * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DSTEBZ 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) DOUBLE PRECISION
             VU      (input) DOUBLE PRECISION 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) DOUBLE PRECISION
             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*DLAMCH('S'), not zero.

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

     E       (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
             On exit, the first M elements of W will contain the eigenvalues.
             (DSTEBZ 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.  (DSTEBZ 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) DOUBLE PRECISION 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  DOUBLE PRECISION, 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   DOUBLE PRECISION, 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.
SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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