nag_real_eigenvalues (f02afc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_real_eigenvalues (f02afc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_eigenvalues (f02afc) calculates all the eigenvalues of a real unsymmetric matrix.

2  Specification

#include <nag.h>
#include <nagf02.h>
void  nag_real_eigenvalues (Integer n, double a[], Integer tda, Complex r[], Integer iter[], NagError *fail)

3  Description

The matrix A  is first balanced and then reduced to upper Hessenberg form using stabilised elementary similarity transformations. The eigenvalues are then found using the QR  algorithm for real Hessenberg matrices.

4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

5  Arguments

1:     nIntegerInput
On entry: n , the order of the matrix A .
Constraint: n1 .
2:     a[n×tda]doubleInput/Output
Note: the i,jth element of the matrix A is stored in a[i-1×tda+j-1].
On entry: the n  by n  matrix A .
On exit: a is overwritten.
3:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: tdan .
4:     r[n]ComplexOutput
On exit: the eigenvalues.
5:     iter[n]IntegerOutput
On exit: iter[i-1]  contains the number of iterations used to find the i th eigenvalue. If iter[i-1]  is negative, the i th eigenvalue is the second of a pair found simultaneously.
Note: the eigenvalues are found in reverse order, starting with the n th.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, tda=value  while n=value . These arguments must satisfy tdan .
Dynamic memory allocation failed.
On entry, n=value.
Constraint: n1.
More than value iterations are required to isolate all the eigenvalues.

7  Accuracy

The accuracy of the results depends on the original matrix and the multiplicity of the roots. For a detailed error analysis see pages 352 and 367 Wilkinson and Reinsch (1971).

8  Parallelism and Performance

Not applicable.

9  Further Comments

The time taken by nag_real_eigenvalues (f02afc) is approximately proportional to n 3 .

10  Example

To calculate all the eigenvalues of the real matrix
1.5 0.1 4.5 -1.5 -22.5 3.5 12.5 -2.5 -2.5 0.3 4.5 -2.5 -2.5 0.1 4.5 2.5 .

10.1  Program Text

Program Text (f02afce.c)

10.2  Program Data

Program Data (f02afce.d)

10.3  Program Results

Program Results (f02afce.r)

nag_real_eigenvalues (f02afc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014