?larre2

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 11/07/2023

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?larre2

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

Syntax

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

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

Include Files

mkl_scalapack.h

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 T_i, it finds

a suitable shift at one end of the block's spectrum,
the root RRR, T_i - σ_iI = L_iD_iL_i^T, and
eigenvalues of each L_iD_iL_i^T.

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.

Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex. Notice revision #20201201

Product and Performance Information

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.

Notice revision #20201201

Input Parameters

range

= 'A': ("All") all eigenvalues will be found.

= 'V': ("Value") all eigenvalues in the half-open interval (vl, vu] will be found.

= 'I': ("Index") eigenvalues of the entire matrix with the indices in a given range will be found.

n

The order of the matrix. n > 0.

vl, vu

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

If range='I', the indices (in ascending order) of the smallest eigenvalue, to be returned in w[il-1], and largest eigenvalue, to be returned in w[iu-1].

1 ≤il≤iu≤n.

d

Array of size n

The n diagonal elements of the tridiagonal matrix T.

e

Array of size n

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

e2

Array of size n

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

rtol1, rtol2

Parameters for bisection.

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

spltol

The threshold for splitting.

dol, dou

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

Workspace array of size 6*n

iwork

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 D_i.

e

e contains the subdiagonal elements of the unit bidiagonal matrices L_i. The entries e[isplit[i]], 0 ≤i<nsplit, contain the base points σ_i+1 on output.

e2

The entries e2[isplit[i]], 0≤i<nsplit, are set to zero.

nsplit

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

isplit

Array of size n

The splitting points, at which T breaks up into blocks.

The first block consists of rows/columns 1 to isplit[0], the second of rows/columns isplit[0]+1 through isplit[1], etc., and the nsplit-th block consists of rows/columns isplit[nsplit-2]+1 through isplit[nsplit-1]=n.

m

The total number of eigenvalues (of all L_iD_iL_i^T) found.

w

Array of size n

The first m elements contain the eigenvalues. The eigenvalues of each of the blocks, L_iD_iL_i^T, are sorted in ascending order (?larre2 may use the remaining n-m elements as workspace).

Note that immediately after exiting this function, only the eigenvalues in wwith indices in range dol-1:dou-1 might rely on this processor when the eigenvalue computation is done in parallel.

werr

Array of size n

The error bound on the corresponding eigenvalue in w.

Note that immediately after exiting this function, only the uncertainties in werrwith indices in range dol-1:dou-1 might rely on this processor when the eigenvalue computation is done in parallel.

wgap

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 function, only the gaps in wgapwith indices in range dol-1:dou-1 might rely on this processor when the eigenvalue computation is done in parallel.

iblock

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

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+1)-th eigenvalue w[i] is the 10th eigenvalue in block 2.

gers

Array of size 2*n

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

pivmin

The minimum pivot in the sturm sequence for T.

info

= 0: successful exit

> 0: A problem occurred in ?larre2.

< 0: One of the called functions 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

Parent topic: ScaLAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?larre2

Syntax

Include Files

Description

Input Parameters

OUTPUT Parameters

See Also