F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08JAF (DSTEV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08JAF (DSTEV) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix $A$.

## 2  Specification

 SUBROUTINE F08JAF ( JOBZ, N, D, E, Z, LDZ, WORK, INFO)
 INTEGER N, LDZ, INFO REAL (KIND=nag_wp) D(*), E(*), Z(LDZ,*), WORK(*) CHARACTER(1) JOBZ
The routine may be called by its LAPACK name dstev.

## 3  Description

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.

## 4  References

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

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix.
Constraint: ${\mathbf{N}}\ge 0$.
3:     D($*$) – 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:     E($*$) – 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:     Z(LDZ,$*$) – 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:     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:     WORK($*$) – 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:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the algorithm failed to converge; $i$ off-diagonal elements of E did not converge to zero.

## 7  Accuracy

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.

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

## 9  Example

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.

### 9.1  Program Text

Program Text (f08jafe.f90)

### 9.2  Program Data

Program Data (f08jafe.d)

### 9.3  Program Results

Program Results (f08jafe.r)