A note on improvement by iteration of approximate solutions of operator equations

Abstract

Let T be a bounded linear operator on a Banach space X = C[0, 1]. Consider the operator equation $$\begin{aligned} u-Tu=f,\;u,f\in X. \end{aligned}$$ Collocation methods are a class of important methods to solve the above equation approximately We consider a modified projection method and an iterated modified projection for the solution of the above operator equation. Let \(\pi _n\) be an interpolatory projection onto the space \(X_n\) of piecewise polynomials of degree \(\le r-1.\) If the collocation points are the Gauss points, then the iterated collocation and iterated modified projection solution is an improvement on the collocation and the modified projection solution. Now let \(\pi _n: X\rightarrow X_n\) be the interpolatory projection with collocation points as the partition points of a uniform partition of [0, 1]. Then iteration fails to improve the order of convergence. But when the approximating space \(X_n\) is the space of continuous functions that are piecewise polynomials of even degree with respect to a uniform partition, then Atkinson (The numerical solutions of integral equations of the second kind, 1997) has shown that iterated collocation solution is an improvement on the collocation solution whereas we show that the iteration improves the order of convergence in the case of modified projection method.