Talks

The Lanczos algorithm was introduced in 1950 as means of solving eigenvalue problems. Despite its apparent elegance, the algorithm was initially neglected by the scientific community because it was observed to depart from its theoretical properties due to the effects of finite-precision computer arithmetic. The algorithm regained popularity several decades later when it was shown that despite its departure from theory, it nevertheless produces highly accurate eigenvalue estimates.

In my talk, I will briefly introduce the Lanczos algorithm and will present bounds characterizing the quality of eigenvalue estimates generated by the algorithm in exact arithmetic. Then, I will describe the difficulties of producing similar bounds in finite-precision arithmetic, and will present rounding error results, including recent ones, which overcome these difficulties.

We devised an IDR(s)-based SOR method and presented its effectiveness in view of efficiency and robustness by comparison with other iterative methods one year ago. In this talk, we consider the preconditioned IDR(s)-based Residual Reduction (R2) method as an extension of the IDR(s)-based SOR method in view of robust preconditioning. Moreover, we present numerical experiments that clearly show that our proposed IDR(s)-R2 method outperforms other approaches.