?larre2

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

Syntax

call slarre2 ( range , n , vl , vu , il , iu , d , e , e2 , rtol1 , rtol2 , spltol , nsplit , isplit , m , dol , dou , w , werr , wgap , iblock , indexw , gers , pivmin , work , iwork , info )

call dlarre2 ( range , n , vl , vu , il , iu , d , e , e2 , rtol1 , rtol2 , spltol , nsplit , isplit , m , dol , dou , w , werr , wgap , iblock , indexw , gers , pivmin , work , iwork , info )

Description

To find the desired eigenvalues of a given real symmetric tridiagonal matrix T, ?larre2 sets, via ?larra, "small" off-diagonal elements to zero. For each block Ti, it finds

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

  • the root RRR, Ti - σiI = LiDiLiT, and

  • eigenvalues of each LiDiLiT.

The representations and eigenvalues found are then returned to ?stegr2 to compute the eigenvectors T.

?larre2 is more suitable for parallel computation than the original LAPACK code for computing the root RRR and its eigenvalues. When computing eigenvalues in parallel and the input tridiagonal matrix splits into blocks, ?larre2 can skip over blocks which contain none of the eigenvalues from dol to dou for which the processor is responsible. In extreme cases (such as large matrices consisting of many blocks of small size, e.g. 2x2), the gain can be substantial.

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 slarre2

DOUBLE PRECISION for dlarre2

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.

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 slarre2

DOUBLE PRECISION for dlarre2

Array of size n

The n diagonal elements of the tridiagonal matrix T.

e

REAL for slarre2

DOUBLE PRECISION for dlarre2

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 slarre2

DOUBLE PRECISION for dlarre2

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 slarre2

DOUBLE PRECISION for dlarre2

Parameters for bisection.

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

spltol

REAL for slarre2

DOUBLE PRECISION for dlarre2

The threshold for splitting.

dol, dou

INTEGER

Specifying an index range dol:dou allows the user to work on only a selected part of the representation tree. 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 slarre2

DOUBLE PRECISION for dlarre2

Workspace array of size 6*n

iwork

INTEGER

Workspace array of size 5*n

OUTPUT Parameters

vl, vu

If range='I' or ='A', ?larre2 contains 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, are 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.

w

REAL for slarre2

DOUBLE PRECISION for dlarre2

Array of size n

The first m elements contain the eigenvalues. The eigenvalues of each of the blocks, LiDiLiT, are sorted in ascending order (?larre2 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 might rely on this processor when the eigenvalue computation is done in parallel.

werr

REAL for slarre2

DOUBLE PRECISION for dlarre2

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 might rely on this processor when the eigenvalue computation is done in parallel.

wgap

REAL for slarre2

DOUBLE PRECISION for dlarre2

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 might rely on this processor when 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 slarre2

DOUBLE PRECISION for dlarre2

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 slarre2

DOUBLE PRECISION for dlarre2

The minimum pivot in the sturm sequence for T.

info

INTEGER

= 0: successful exit

> 0: A problem occurred in ?larre2.

< 0: One of the called subroutines signaled an internal problem.

Needs inspection of the corresponding parameter info for further information.

=-1: Problem in ?larrd.

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

=-3: Problem in ?larrb when computing the refined root representation for ?lasq2.

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

=-5: Problem in ?lasq2

=-6: Problem in ?lasq2

See Also

For more complete information about compiler optimizations, see our Optimization Notice.