Semi-Iterative Methods

Abstract

The semi-iteration comes in three formulations. The first one in Section 8.1 is the most general and associates each semi-iterate with a polynomial. Using the notion of Krylov spaces, we only require that the errors of the semi-iterates $y^{m}$ be elements of the Krylov space $x^{0}+N\mathcal {K}_{m}(AN,r^{0})$. In the second formulation of Section 8.2, the polynomials $p_{m}$ associated with $y^{m}$ are related either by a two-term or by a three-term recursion. Section 8.3 tries to determine the optimal polynomials. Here the result depends on what quantity we want to minimise. Three minimisation problems are discussed. The last formulation is practically solvable and leads to (transformed) Chebyshev polynomials. The corresponding semi-iteration is called the Chebyshev method (cf. §8.3.4). The Chebyshev method improves the order of convergence. Its convergence speed corresponds to the square root of the spectral condition number (cf. §8.3.5). In Section 8.4 the Chebyshev method is applied to the iterations discussed in Part I. In Section 8.5 we describe the ADI method which is not really of the form discussed above, but it might be seen as a generalisation of semi-iterations (replacing scalar parameters by matrix-valued ones).