LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
sstebz.f File Reference

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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:
 [in] 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.``` [in] 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.``` [in] N ``` N is INTEGER The order of the tridiagonal matrix T. N >= 0.``` [in] VL ` VL is REAL` [in] 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'.``` [in] IL ` IL is INTEGER` [in] 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'.``` [in] 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.``` [in] D ``` D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T.``` [in] E ``` E is REAL array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T.``` [out] M ``` M is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)``` [out] NSPLIT ``` NSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.``` [out] 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.)``` [out] 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.)``` [out] 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.)``` [out] WORK ` WORK is REAL array, dimension (4*N)` [out] IWORK ` IWORK is INTEGER array, dimension (3*N)` [out] 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.```
Date:
November 2011

Definition at line 262 of file sstebz.f.

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