# NAG Library Routine Document

## 1Purpose

f08jff (dsterf) computes all the eigenvalues of a real symmetric tridiagonal matrix.

## 2Specification

Fortran Interface
 Subroutine f08jff ( n, d, e, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*)
#include <nagmk26.h>
 void f08jff_ (const Integer *n, double d[], double e[], Integer *info)
The routine may be called by its LAPACK name dsterf.

## 3Description

f08jff (dsterf) computes all the eigenvalues of a real symmetric tridiagonal matrix, using a square-root-free variant of the $QR$ algorithm.
The routine uses an explicit shift, and, like f08jef (dsteqr), switches between the $QR$ and $QL$ variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)).

## 4References

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville http://www.netlib.org/lapack/lawnspdf/lawn17.pdf
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the diagonal elements of the tridiagonal matrix $T$.
On exit: the $n$ eigenvalues in ascending order, unless ${\mathbf{info}}>{\mathbf{0}}$ (in which case see Section 6).
3:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the off-diagonal elements of the tridiagonal matrix $T$.
On exit: e is overwritten.
4:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The algorithm has failed to find all the eigenvalues after a total of $30×{\mathbf{n}}$ iterations; $〈\mathit{\text{value}}〉$ elements of e have not converged to zero.

## 7Accuracy

The computed eigenvalues are exact for a nearby matrix $\left(T+E\right)$, where
 $E2 = Oε T2 ,$
and $\epsilon$ is the machine precision.
If ${\lambda }_{i}$ is an exact eigenvalue and ${\stackrel{~}{\lambda }}_{i}$ is the corresponding computed value, then
 $λ~i - λi ≤ c n ε T2 ,$
where $c\left(n\right)$ is a modestly increasing function of $n$.

## 8Parallelism and Performance

f08jff (dsterf) is not threaded in any implementation.

The total number of floating-point operations is typically about $14{n}^{2}$, but depends on how rapidly the algorithm converges. The operations are all performed in scalar mode.
There is no complex analogue of this routine.

## 10Example

This example computes all the eigenvalues of the symmetric tridiagonal matrix $T$, where
 $T = -6.99 -0.44 0.00 0.00 -0.44 7.92 -2.63 0.00 0.00 -2.63 2.34 -1.18 0.00 0.00 -1.18 0.32 .$

### 10.1Program Text

Program Text (f08jffe.f90)

### 10.2Program Data

Program Data (f08jffe.d)

### 10.3Program Results

Program Results (f08jffe.r)