sstebz.f man page

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

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

sstebz(3) is an alias of sstebz.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK