Rayleigh quotient iteration

[Proektsoftbg]

Rayleigh quotient iteration is an extremely fast iterative algorithm for finding the smallest eigenvalue and the corresponging eigenvector of a matrix. It converges cubically. So, it can be very useful to quickly estimate the first natural vibration frequence and mode shape. As you can see from the example below, that even the first iteration provides a very good approximation.

Here is the Calcpad VM code to compute the Rayleigh quotient iterations of a matrix:
Rayleigh quotient iteration.zip

[idealkindom]

It worth a detailed review and study.

[Proektsoftbg]

Yes, it is a very good and simple classical method.

What I am personally interested now is the parallel block Lanczos method for extracting some eigenvalues of large sparce matrices, like this one:
http://matwbn.icm.edu.pl/ksiazki/amc/amc16/amc1629.pdf

The current QL algorithm always find all eigenvalues/vectors and works for matrices with several thousand elements in reasonable time. The Lanczos method can find the first n eigenvalalues/vectors for much larger symmetric sparce matrices.
The problem is that it is not very stable, looses precision and thence - orthogonality. This needs some additinal reorthogonalization and other things that make the algorithm more complicated. However, I am considering to include it in Calcpad as an alternative method.