Stationary and almost stationary iterative (k, l)-step methods for linear and nonlinear systems of equations

Summary

We first discuss the solution of a fixed point equationxΦx of a Fréchet differentiable self-mapping Φ by iterative methods of the general form

$$x_m : = \left[ {\Phi \left( {\sum\limits_{i = 0}^{m - 1} {x_i \gamma _{i,m - 1} } } \right) - \sum\limits_{j = 0}^{m - 1} {x_j \beta _{j,m} } } \right]\frac{1}{{\beta _{m,m} }},m = 1,2, \ldots ,$$

defined by two infinite nonsingular upper triangular matricesB=(β _{j, m} ) andC=(γ _{j, m} ) with column sums 1. We show that two such methods, defined byB, C and$\tilde B,\tilde C$, respectively, are in a certain sense equivalent if and only if$C^{ - 1} B = \tilde C^{ - 1} \tilde B$. In particular,$\hat B: = C^{ - 1}$ and the unit matrix (replacingC) define an equivalent so-called semiiterative method. We introduce (k, l)-step methods as those whereB andC have upper bandwidthk andl−1, respectively. They require storing max {k, l} previous iterates only. For stationary methodsB andC have Toeplitz structure except for their first row, which is chosen such that the column sum condition holds. An Euler method, which may require to store all iterates, is equivalent to a (stationary) (k, l)-step method if and only if the underlying conformal mapg is a rational function of the formg(w)=w(μ ₀+...+μ _k ^−k)/ (v ₀+...+v _l−1 w ⁻⁺¹). By choosingg withg(1)=1 such that for some ρ<1 it maps |w|>ρ onto the exterior of some continuumS known to contain the eigen values of the Fréchet derivative of Φ, one obtains a feasible procedure for designing a locally converging stationary (k, l)-step method custom-made for a set of problems.

In the case Φx≔Tx+d, with a linear operatorT, where one wants to solve a linear system of equations, we show that the residual polynomials of a stationary semiiterative method are generalized Faber polynomials with respect to a particular weight function. Using another weight function leads to what we call almost stationary methods. (The classical Chebyshev iteration is an example of such a method.) We define equivalent almost stationary (k, l)-step methods and give a corresponding convergence result.