A new Lanczos type algorithm for system of linear equations
For researchers and practitioners solving linear systems, this algorithm offers improved stability over existing Lanczos-type methods, though it is an incremental improvement.
The paper introduces a new Lanczos-type algorithm for solving systems of linear equations that uses a recurrence relation involving higher-degree polynomials, resulting in superior stability compared to existing methods. Numerical tests on standard problems show it outperforms algorithms like A5/B10, A8/B10, and Arnoldi's method.
Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal polynomials of low degree. In this paper, we consider a recurrence relation that has not been studied before and which involves a relatively higher degree polynomial. Interestingly, it leads to an algorithm that shows superior stability when compared to existing Lanczos-type algorithms. This new algorithm is derived and described. It is then compared to the best known algorithms of this type, namely A5/B10, A8/B10, as well as Arnoldi's algorithm, on a set of standard test problems. Numerical results are included.