SYNOPSIS

SUBROUTINE DLARRD(

RANGE, ORDER, N, VL, VU, IL, IU, GERS, RELTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO )

    

CHARACTER ORDER, RANGE

    

INTEGER IL, INFO, IU, M, N, NSPLIT

    

DOUBLE PRECISION PIVMIN, RELTOL, VL, VU, WL, WU

    

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

    

DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ), W( * ), WERR( * ), WORK( * )

PURPOSE

DLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from DSTEMR.
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'.

GERS (input) DOUBLE PRECISION array, dimension (2*N)

The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)).

RELTOL (input) DOUBLE PRECISION

The minimum relative width of an interval. When an interval is narrower than RELTOL times the larger (in magnitude) endpoint, then it is considered to be sufficiently small, i.e., converged. Note: this should always be at least radix*machine epsilon.

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.

E2 (input) DOUBLE PRECISION array, dimension (N-1)

The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

PIVMIN (input) DOUBLE PRECISION

The minimum pivot allowed in the Sturm sequence for T.

NSPLIT (input) INTEGER

The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.

ISPLIT (input) 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.)

M (output) INTEGER

The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)

W (output) DOUBLE PRECISION array, dimension (N)

On exit, the first M elements of W will contain the eigenvalue approximations. DLARRD computes an interval I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue approximation is given as the interval midpoint W(j)= ( a_j + b_j)/2. The corresponding error is bounded by WERR(j) = abs( a_j - b_j)/2

WERR (output) DOUBLE PRECISION array, dimension (N)

The error bound on the corresponding eigenvalue approximation in W.

WL (output) DOUBLE PRECISION

WU (output) DOUBLE PRECISION The interval (WL, WU] contains all the wanted eigenvalues. If RANGE='V', then WL=VL and WU=VU. If RANGE='A', then WL and WU are the global Gerschgorin bounds on the spectrum. If RANGE='I', then WL and WU are computed by DLAEBZ from the index range specified.

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. (DLARRD may use the remaining N-M elements as workspace.)

INDEXW (output) INTEGER array, dimension (N)

The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= j and IBLOCK(i)=k imply that the i-th eigenvalue W(i) is the j-th eigenvalue in block k.

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

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. Based on contributions by W. Kahan, University of California, Berkeley, USA Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA