DSTEBZ - compute the eigenvalues of a symmetric tridiagonal matrix T

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( * )

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.

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(N)=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.

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.