nag_real_eigensystem (f02agc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_real_eigensystem (f02agc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_eigensystem (f02agc) calculates all the eigenvalues and eigenvectors of a real unsymmetric matrix.

2  Specification

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

3  Description

The matrix A  is first balanced and then reduced to upper Hessenberg form using real stabilised elementary similarity transformations. The eigenvalues and eigenvectors of the Hessenberg matrix are calculated using the QR  algorithm. The eigenvectors of the Hessenberg matrix are back-transformed to give the eigenvectors of the original matrix A .

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:     v[n×tdv]ComplexOutput
Note: the i,jth element of the matrix V is stored in v[i-1×tdv+j-1].
On exit: the eigenvectors, stored by columns. The i th column corresponds to the i th eigenvalue. The eigenvectors are normalized so that the sum of the squares of the moduli of the elements is equal to 1 and the element of largest modulus is real. This ensures that real eigenvalues have real eigenvectors.
6:     tdvIntegerInput
On entry: the stride separating matrix column elements in the array v.
Constraint: tdvn .
7:     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.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, tda=value  while n=value . These arguments must satisfy tdan .
On entry, tdv=value  while n=value . These arguments must satisfy tdvn .
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, n=value.
Constraint: n1.
NE_TOO_MANY_ITERATIONS
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 390 Wilkinson and Reinsch (1971).

8  Further Comments

The time taken by nag_real_eigensystem (f02agc) is approximately proportional to n 3 .

9  Example

To calculate all the eigenvalues and eigenvectors 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 .

9.1  Program Text

Program Text (f02agce.c)

9.2  Program Data

Program Data (f02agce.d)

9.3  Program Results

Program Results (f02agce.r)


nag_real_eigensystem (f02agc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012