# ?larre2a

Given a tridiagonal matrix, sets small off-diagonal elements to zero and for each unreduced block, finds base representations and eigenvalues.

## Syntax

call slarre2a ( range , n , vl , vu , il , iu , d , e , e2 , rtol1 , rtol2 , spltol , nsplit , isplit , m , dol , dou , needil , neediu , w , werr , wgap , iblock , indexw , gers , sdiam , pivmin , work , iwork , minrgp , info )

call dlarre2a ( range , n , vl , vu , il , iu , d , e , e2 , rtol1 , rtol2 , spltol , nsplit , isplit , m , dol , dou , needil , neediu , w , werr , wgap , iblock , indexw , gers , sdiam , pivmin , work , iwork , minrgp , info )

## Description

To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, ?larre2a sets any "small" off-diagonal elements to zero, and for each unreduced block Ti, it finds

• a suitable shift at one end of the block's spectrum,

• the base representation, Ti - σiI = LiDiLiT, and

• eigenvalues of each LiDiLiT.

### Note

The algorithm obtains a crude picture of all the wanted eigenvalues (as selected by range). However, to reduce work and improve scalability, only the eigenvalues dol to dou are refined. Furthermore, if the matrix splits into blocks, RRRs for blocks that do not contain eigenvalues from dol to dou are skipped. The DQDS algorithm (subroutine ?lasq2) is not used, unlike in the sequential case. Instead, eigenvalues are computed in parallel to some figures using bisection.

Optimization Notice

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804

## Input Parameters

range

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

INTEGER

The order of the matrix. n > 0.

vl, vu

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

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', ?larre2a computes bounds on the desired part of the spectrum.

il, iu

INTEGER

If range='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

1 il iu n.

d

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size n

On entry, the n diagonal elements of the tridiagonal matrix T.

e

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size n

The first (n-1) entries contain the subdiagonal elements of the tridiagonal matrix T; e(n) need not be set.

e2

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size n

The first (n-1) entries contain the squares of the subdiagonal elements of the tridiagonal matrix T; e2(n) need not be set.

rtol1, rtol2

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Parameters for bisection.

An interval [left,right] has converged if right - left < max( rtol1*gap, rtol2*max(|left|,|right|) )

spltol

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

The threshold for splitting.

dol, dou

INTEGER

If the user wants to work on only a selected part of the representation tree, he can specify an index range dol:dou.

Otherwise, the setting dol=1, dou=n should be applied.

Note that dol and dou refer to the order in which the eigenvalues are stored in w.

work

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Workspace array of size 6*n

iwork

INTEGER

Workspace array of size 5*n

minrgp

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

The minimum relative gap threshold to decide whether an eigenvalue or a cluster boundary is reached.

## OUTPUT Parameters

vl, vu

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

If range='V', the lower and upper bounds for the eigenvalues. Eigenvalues less than or equal to vl, or greater than vu, are not returned. vl < vu.

If range='I' or range='A', ?larre2a computes bounds on the desired part of the spectrum.

d

The n diagonal elements of the diagonal matrices Di.

e

e contains the subdiagonal elements of the unit bidiagonal matrices Li. The entries e( isplit(i) ), 1 i nsplit, contain the base points σi on output.

e2

The entries e2( isplit( i ) ), 1 i nsplit have been set to zero.

nsplit

INTEGER

The number of blocks T splits into. 1 nsplit n.

isplit

INTEGER

Array of size 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 block consists of rows/columns isplit(nsplit-1)+1 through isplit(nsplit)=n.

m
INTEGER

The total number of eigenvalues (of all LiDiLiT) found.

needil, neediu
INTEGER

The indices of the leftmost and rightmost eigenvalues of the root node RRR which are needed to accurately compute the relevant part of the representation tree.

w

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size n

The first m elements contain the eigenvalues. The eigenvalues of each of the blocks, LiDiLiT, are sorted in ascending order ( ?larre2a may use the remaining n-m elements as workspace).

Note that immediately after exiting this routine, only the eigenvalues from position dol:dou in w rely on this processor because the eigenvalue computation is done in parallel.

werr

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size n

The error bound on the corresponding eigenvalue in w.

Note that immediately after exiting this routine, only the uncertainties from position dol:dou in werr are reliable on this processor because the eigenvalue computation is done in parallel.

wgap

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size 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

Note that immediately after exiting this routine, only the gaps from position dol:dou in wgap are reliable on this processor because the eigenvalue computation is done in parallel.

iblock

INTEGER Array of size 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, iblock(i)=2 if w(i) belongs to the second block, and so on.

indexw

INTEGER Array of size 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 10th eigenvalue in block 2.

gers

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

Array of size 2*n

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

pivmin

REAL for slarre2a

DOUBLE PRECISION for dlarre2a

The minimum pivot in the sturm sequence for T.

info

INTEGER

= 0: successful exit

> 0: A problem occurred in ?larre2a.

< 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter info for further information.

=-1: Problem in ?larrd2.

=-2: Not enough internal iterations to find base representation.

=-3: Problem in ?larrb2 when computing the refined root representation.

=-4: Problem in ?larrb2 when preforming bisection on the desired part of the spectrum.

= -9 Problem: m < dou-dol+1, that is the code found fewer eigenvalues than it was supposed to.