SLARRE(3) find the desired eigenvalues of a given real symmetric tridiagonal matrix T, SLARRE sets any small off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum,

## SYNOPSIS

SUBROUTINE SLARRE(
RANGE, N, VL, VU, IL, IU, D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN, WORK, IWORK, INFO )

IMPLICIT NONE

CHARACTER RANGE

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

REAL PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU

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

REAL D( * ), E( * ), E2( * ), GERS( * ), W( * ),WERR( * ), WGAP( * ), WORK( * )

## PURPOSE

To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, SLARRE sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds (a) a suitable shift at one end of the block's spectrum, (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and (c) eigenvalues of each L_i D_i L_i^T.
The representations and eigenvalues found are then used by SSTEMR to compute the eigenvectors of T.
The accuracy varies depending on whether bisection is used to find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to conpute all and then discard any unwanted one.
As an added benefit, SLARRE also outputs the n
Gerschgorin intervals for the matrices L_i D_i L_i^T.

## 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.
N (input) INTEGER
The order of the matrix. N > 0.
VL (input/output) REAL
VU (input/output) REAL If RANGE='V', the lower and upper bounds for the eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. If RANGE='I' or ='A', SLARRE computes bounds on the desired part of the spectrum.
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.
D (input/output) REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal matrix T. On exit, the N diagonal elements of the diagonal matrices D_i.
E (input/output) REAL array, dimension (N)
On entry, the first (N-1) entries contain the subdiagonal elements of the tridiagonal matrix T; E(N) need not be set. On exit, E contains the subdiagonal elements of the unit bidiagonal matrices L_i. The entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT, contain the base points sigma_i on output.
E2 (input/output) REAL array, dimension (N)
On entry, the first (N-1) entries contain the SQUARES of the subdiagonal elements of the tridiagonal matrix T; E2(N) need not be set. On exit, the entries E2( ISPLIT( I ) ), 1 <= I <= NSPLIT, have been set to zero
RTOL1 (input) REAL
RTOL2 (input) REAL Parameters for bisection. RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) SPLTOL (input) REAL The threshold for splitting.
NSPLIT (output) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
ISPLIT (output) INTEGER array, dimension (N)
The splitting points, at which T breaks up into blocks. The first block 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.
M (output) INTEGER
The total number of eigenvalues (of all L_i D_i L_i^T) found.
W (output) REAL array, dimension (N)
The first M elements contain the eigenvalues. The eigenvalues of each of the blocks, L_i D_i L_i^T, are sorted in ascending order ( SLARRE may use the remaining N-M elements as workspace).
WERR (output) REAL array, dimension (N)
The error bound on the corresponding eigenvalue in W.
WGAP (output) REAL array, dimension (N)
The separation from the right neighbor eigenvalue in W. The gap is only with respect to the eigenvalues of the same block as each block has its own representation tree. Exception: at the right end of a block we store the left gap
IBLOCK (output) INTEGER array, dimension (N)
The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc.
INDEXW (output) INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
GERS (output) REAL array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)).
PIVMIN (output) DOUBLE PRECISION
The minimum pivot in the Sturm sequence for T.
WORK (workspace) REAL array, dimension (6*N)
Workspace.
IWORK (workspace) INTEGER array, dimension (5*N)
Workspace.
INFO (output) INTEGER
= 0: successful exit
> 0: A problem occured in SLARRE.
< 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information.
=-1: Problem in SLARRD.
= 2: No base representation could be found in MAXTRY iterations. Increasing MAXTRY and recompilation might be a remedy. =-3: Problem in SLARRB when computing the refined root representation for SLASQ2. =-4: Problem in SLARRB when preforming bisection on the desired part of the spectrum. =-5: Problem in SLASQ2.
=-6: Problem in SLASQ2. Further Details element growth and consequently define all their eigenvalues to high relative accuracy. =============== Based on contributions by 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