# NAG Library Routine Document

## 1Purpose

f08jaf (dstev) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f08jaf ( jobz, n, d, e, z, ldz, work, info)
 Integer, Intent (In) :: n, ldz Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), z(ldz,*), work(*) Character (1), Intent (In) :: jobz
#include <nagmk26.h>
 void f08jaf_ (const char *jobz, const Integer *n, double d[], double e[], double z[], const Integer *ldz, double work[], Integer *info, const Charlen length_jobz)
The routine may be called by its LAPACK name dstev.

## 3Description

f08jaf (dstev) computes all the eigenvalues and, optionally, all the eigenvectors of $A$ using a combination of the $QR$ and $QL$ algorithms, with an implicit shift.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{jobz}$ – Character(1)Input
On entry: indicates whether eigenvectors are computed.
${\mathbf{jobz}}=\text{'N'}$
Only eigenvalues are computed.
${\mathbf{jobz}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{jobz}}=\text{'N'}$ or $\text{'V'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{d}\left(*\right)$ – 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 $n$ diagonal elements of the tridiagonal matrix $A$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the eigenvalues in ascending order.
4:     $\mathbf{e}\left(*\right)$ – 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 $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
On exit: the contents of e are destroyed.
5:     $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobz}}=\text{'V'}$, then if ${\mathbf{info}}={\mathbf{0}}$, z contains the orthonormal eigenvectors of the matrix $A$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{d}}\left(i\right)$.
If ${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
6:     $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08jaf (dstev) is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldz}}\ge 1$.
7:     $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}-2\right)$.
On exit: if ${\mathbf{jobz}}=\text{'N'}$, work is not referenced.
8:     $\mathbf{info}$ – 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 failed to converge; $〈\mathit{\text{value}}〉$ off-diagonal elements of e did not converge to zero.

## 7Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08jaf (dstev) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jaf (dstev) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is proportional to ${n}^{2}$ if ${\mathbf{jobz}}=\text{'N'}$ and is proportional to ${n}^{3}$ if ${\mathbf{jobz}}=\text{'V'}$.

## 10Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
 $A = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.

### 10.1Program Text

Program Text (f08jafe.f90)

### 10.2Program Data

Program Data (f08jafe.d)

### 10.3Program Results

Program Results (f08jafe.r)