# ?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 )

## 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 `(overflow1/2*underflow1/4)` 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 `Ij = (aj, bj]` that includes eigenvalue `j`. The eigenvalue approximation is given as the interval midpoint `w(j)= (aj+bj)/2`. The corresponding error is bounded by `werr(j) = abs(aj-bj)/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: Auxiliary Routines