?larrd2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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?larrd2

Computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Syntax

call slarrd2( range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, dol, dou, info )

call dlarrd2( range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, dol, dou, info )

Description

?larrd2 computes the eigenvalues of a symmetric tridiagonal matrix T to limited initial accuracy. This is an auxiliary code to be called from larre2a.

?larrd2 has been created using the LAPACK code larrd which itself stems from stebz. The motivation for creating ?larrd2 is efficiency: When computing eigenvalues in parallel and the input tridiagonal matrix splits into blocks, ?larrd2 can skip over blocks which contain none of the eigenvalues from DOL to DOU for which the processor responsible. In extreme cases (such as large matrices consisting of many blocks of small size like 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

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

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

INTEGER

The order of the tridiagonal matrix T. n >= 0.

vl, vu

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

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, iu

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

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

Array of size 2*n

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

reltol

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

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

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

Array of size n

The n diagonal elements of the tridiagonal matrix T.

e

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

Array of size n-1

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

e2

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

Array of size n-1

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

pivmin

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

The minimum pivot allowed in the sturm sequence for T.

nsplit

INTEGER

The number of diagonal blocks in the matrix T.

1 ≤nsplit≤n.

isplit

INTEGERArray of size 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 submatrix 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.)

work

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

(workspace) Array of size 4*n

iwork

(workspace) INTEGER Array of size 3*n

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.

OUTPUT Parameters

m

INTEGER

The actual number of eigenvalues found. 0 ≤m≤n.

(See also the description of info=2,3.)

w

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

Array of size n

On exit, the first m elements of w will contain the eigenvalue approximations. ?larrd2 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

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

Array of size n

The error bound on the corresponding eigenvalue approximation in w.

wl, wu

REAL for slarrd2

DOUBLE PRECISION for dlarrd2

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 SLAEBZ from the

index range specified.

iblock

INTEGERArray of size 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. (?larrd2 may use the remaining n-m elements as workspace.)

indexw

INTEGERArray of size 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.

info

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.
=2 or 3: range='I' only: Not all of the eigenvalues il:iu were found.
= 4: range='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed.

Parent topic: ScaLAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?larrd2

Syntax

Description

Input Parameters

OUTPUT Parameters

See Also