nag_real_symm_eigensystem (f02abc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_real_symm_eigensystem (f02abc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_symm_eigensystem (f02abc) calculates all the eigenvalues and eigenvectors of a real symmetric matrix.

2  Specification

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

3  Description

nag_real_symm_eigensystem (f02abc) reduces the real symmetric matrix A  to a real symmetric tridiagonal matrix by Householder's method. The eigenvalues and eigenvectors are calculated 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:     nIntegerInput
On entry: n , the order of the matrix A .
Constraint: n1 .
2:     a[n×tda]const doubleInput
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. See also Section 8
3:     tdaIntegerInput
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:     v[n×tdv]doubleOutput
Note: the i,jth element of the matrix V is stored in v[i-1×tdv+j-1].
On exit: the normalized eigenvectors, stored by columns; the i th column corresponds to the i th eigenvalue. The eigenvectors are normalized so that the sum of squares of the elements is equal to 1.
6:     tdvIntegerInput
On entry: the stride separating matrix column elements in the array v.
Constraint: tdvn .
7:     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 eigenvectors are always accurately orthogonal but the accuracy of the individual eigenvectors is dependent on their inherent sensitivity to changes in the original matrix. For a detailed error analysis see pages  222 and 235 of Wilkinson and Reinsch (1971).

8  Further Comments

The time taken by nag_real_symm_eigensystem (f02abc) is approximately proportional to n 3 .
The function may be called with the same actual array supplied for arguments a and v, in which case the eigenvectors will overwrite the original matrix.

9  Example

To calculate all the eigenvalues and eigenvectors 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 .

9.1  Program Text

Program Text (f02abce.c)

9.2  Program Data

Program Data (f02abce.d)

9.3  Program Results

Program Results (f02abce.r)


nag_real_symm_eigensystem (f02abc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG C Library Manual

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