dstebz(3)

NAME

DSTEBZ - compute the eigenvalues of a symmetric tridiago

nal 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(
* )

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 Tridiag

onal 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 small

est 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 eigen

values. 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 IS

PLIT(2), etc., and the NSPLIT-th consists of rows/columns IS

PLIT(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 re

served 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 ille

gal 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 accu

rate 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 inter

vals. Ideally, a value of 1 should work, but on machines with

sloppy arithmetic, this needs to be larger. The default for pub

licly released versions should be large enough to handle the

worst machine around. Note that this has no effect on accuracy

of the solution.

LAPACK version 3.0 15 June 2000

DSTEBZ(3)