nag_real_symm_eigenvalues (f02aac) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_real_symm_eigenvalues (f02aac)


    1  Purpose
    7  Accuracy

1  Purpose

nag_real_symm_eigenvalues (f02aac) calculates all the eigenvalues of a real symmetric matrix.

2  Specification

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

3  Description

nag_real_symm_eigenvalues (f02aac) reduces the real symmetric matrix A  to a real symmetric tridiagonal matrix using Householder's method. The eigenvalues of the tridiagonal matrix are then determined using the QL  algorithm.

4  References

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

5  Arguments

1:     n IntegerInput
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 lower triangle of the n  by n  symmetric matrix A . The elements of the array above the diagonal need not be set.
On exit: the elements of A  below the diagonal are overwritten, and the rest of the array is unchanged.
3:     tda IntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: tdan .
4:     r[n] doubleOutput
On exit: the eigenvalues in ascending order.
5:     fail NagError *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 eigenvalues depends on the sensitivity of the matrix to rounding errors produced in tridiagonalisation. For a detailed error analysis see pages 222 and 235 of Wilkinson and Reinsch (1971).

8  Parallelism and Performance

Not applicable.

9  Further Comments

The time taken by nag_real_symm_eigenvalues (f02aac) is approximately proportional to n 3 .

10  Example

To calculate all the eigenvalues of the real symmetric matrix
0.5 0.0 2.3 -2.6 0.0 0.5 -1.4 -0.7 2.3 -1.4 0.5 0.0 -2.6 -0.7 0.0 0.5 .  

10.1  Program Text

Program Text (f02aace.c)

10.2  Program Data

Program Data (f02aace.d)

10.3  Program Results

Program Results (f02aace.r)

nag_real_symm_eigenvalues (f02aac) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015