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.
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.
