?larrd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?larrd

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

Syntax

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

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

Include Files

mkl.fi

Description

The routine computes the eigenvalues of a symmetric tridiagonal matrix T to suitable accuracy. This is an auxiliary code to be called from ?stemr. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (vl, vu], or the il-th through iu-th eigenvalues.

To avoid overflow, the matrix must be scaled so that its largest element is no greater than (overflow^1/2*underflow^1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. (For more details see [Kahan66].

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 will be found.

order

CHARACTER.

= 'B': ("By block") the eigenvalues will be grouped by split-off block (see iblock, isplit below) 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 ≥ 1).

vl,vu

REAL for slarrd

DOUBLE PRECISION for dlarrd

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.

If range = 'A' or 'I': not referenced.

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.

If range = 'A' or 'V': not referenced.

gers

REAL for slarrd

DOUBLE PRECISION for dlarrd

Array, DIMENSION (2*n).

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

reltol

REAL for slarrd

DOUBLE PRECISION for dlarrd

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, that is converged. Note: this should always be at least radix*machine epsilon.

d

REAL for slarrd

DOUBLE PRECISION for dlarrd

Array, DIMENSION (n).

Contains n diagonal elements of the tridiagonal matrix T.

e

REAL for slarrd

DOUBLE PRECISION for dlarrd

Array, DIMENSION (n-1).

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

e2

REAL for slarrd

DOUBLE PRECISION for dlarrd

Array, DIMENSION (n-1).

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

pivmin

REAL for slarrd

DOUBLE PRECISION for dlarrd

The minimum pivot in the Sturm sequence for the matrix T.

nsplit

INTEGER.

The number of diagonal blocks the matrix T . 1 ≤ nsplit ≤ n

isplit

INTEGER.

Arrays, DIMENSION (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), and so on, and the nsplit-th consists of rows/columns isplit(nsplit-1)+1 through isplit(nsplit)=n.

(Only the first nsplit elements actually is used, but since the user cannot know a priori value of nsplit, n words must be reserved for isplit.)

work

REAL for slarrd

DOUBLE PRECISION for dlarrd

Workspace array, DIMENSION (4*n).

iwork

INTEGER.

Workspace array, DIMENSION (4*n).

Output Parameters

m

INTEGER.

The actual number of eigenvalues found. 0 ≤ m ≤ n. (See also the description of info=2,3.)

w

REAL for slarrd

DOUBLE PRECISION for dlarrd

Array, DIMENSION (n).

The first m elements of w contain the eigenvalue approximations. ?laprd 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 slarrd

DOUBLE PRECISION for dlarrd

Array, DIMENSION (n).

The error bound on the corresponding eigenvalue approximation in w.

wl, wu

REAL for slarrd

DOUBLE PRECISION for dlarrd

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 ?laebz from the index range specified.

iblock

INTEGER.

Array, DIMENSION (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.

If info = 0, then iblock(i) specifies to which block (from 1 to the number of blocks) the eigenvalue w(i) belongs. (The routine may use the remaining n-m elements as workspace.)

indexw

INTEGER.

Array, DIMENSION (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 has an illegal value

> 0: some or all of the eigenvalues fail to converge or are not computed:

=1 or 3: bisection fail 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 are found.

=4: range='I', and the Gershgorin interval initially used is too small. No eigenvalues are computed.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?larrd