Friday, February 11, 2011

Digging for Roots

There is an interesting way of finding polynomial roots, which concerns calculation of eigenvalues for the following matrix (http://mathworld.wolfram.com/PolynomialRoots.html):


After eigenvalues have been found the roots are simply computed as 1/lambda.

This method is rather slow especially for large matrices, but it finds all roots, including complex ones, which is hard to do using conventional scheme "root localization + application of numeric method for finding exact root value".

It's interesting that it's possible to obtain a matrix, which eigenvalues are the same as the polynomial's roots. This matrix is:


It's easy to check that AB = E, where E - is identity martix.


No comments:

Post a Comment