Parameter Free Methods

Abstract

In this chapter we present a survey of so-called parameter free polynomial iteration method. These are schemes

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa % aaleaacaWGUbaabeaakiaaykW7cqGH9aqpcaWGWbWaaSbaaSqaaiaa % d6gaaeqaaOGaaiikaiaadgeacaGGPaGaamOCamaaBaaaleaacaaIWa % aabeaakiaacYcacaaMc8UaamiCamaaBaaaleaacaWGUbaabeaakiaa % cIcacaaIWaGaaiykaiaaykW7cqGH9aqpcaaMc8UaaGymaaaa!4C0C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${r_n}\, = {p_n}(A){r_0},\,{p_n}(0)\, = \,1$$

where the residual polynomial p _n is constructed during the iteration process based on certain orthogonality relations. The methods require as input only the system matrix A, the right hand side f, and a starting guess x ₀. The name “parameter free” is motivated by the fact that no further information (or parameter) is necessary to start the process. Incidentally, Householder [83] calls these schemes methods of projection.