nag_hermitian_eigensystem (f02axc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_hermitian_eigensystem (f02axc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_hermitian_eigensystem (f02axc) calculates all the eigenvalues and eigenvectors of a complex Hermitian matrix.

2  Specification

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

3  Description

The complex Hermitian matrix A  is first reduced to a real tridiagonal matrix by n-2  unitary transformations and a subsequent diagonal transformation. The eigenvalues and eigenvectors are then derived using the QL  algorithm, an adaptation of the QR  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 ComplexInput
On entry: the elements of the lower triangle of the n  by n  complex Hermitian matrix A . 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]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 eigenvector. 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. See also Section 8.
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_DIAG_IMAG_NON_ZERO
Matrix diagonal element a[value×tda+value]  has nonzero imaginary part.
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 eigenvalues and eigenvectors is dependent on their inherent sensitivity to small changes in the original matrix. For a detailed error analysis see page 235 of Wilkinson and Reinsch (1971).

8  Further Comments

The time taken by nag_hermitian_eigensystem (f02axc) is approximately proportional to n 3 .
The function may be called with the same actual array supplied for a and v, in which case the eigenvectors will overwrite the original matrix A .

9  Example

To calculate the eigenvalues and eigenvectors of the complex Hermitian matrix:
0.50 -1.38 i - 0.00 +0.84 i 1.84 + 1.38 i - 2.08 - 1.56 i 0.00 -1.38 i - 0.50 +0.84 i - 1.12 + 0.84 i -0.56 + 0.42 i 1.84 - 1.38 i - 1.12 - 0.84 i 0.50 +0.84 i - 0.00 +0.84 i 2.08 + 1.56 i -0.56 - 0.42 i 0.00 +0.84 i - 0.50 +0.84 i .

9.1  Program Text

Program Text (f02axce.c)

9.2  Program Data

Program Data (f02axce.d)

9.3  Program Results

Program Results (f02axce.r)


nag_hermitian_eigensystem (f02axc) (PDF version)
f02 Chapter Contents
f02 Chapter Introduction
NAG C Library Manual

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