Abstract Pad&approximants for the Solution of a Sytem of Nonlinear Equations

PAD&APPROXIMANTS FOR THE SOLUTION OF A SYTEM OF NONLINEAR EQUATIONS ANNIE Cm-T? and PAUL VAN DER CRUYSSEN University of Antwerp, Department of Mathematics, Universiteitsplein 1, B-2610 Wihijk, Belgium (Receioed January 1981; and in revised form May 1982) Communicated by J. F. Traub Abstract-Let F: R4 *R’ and let x* be a simple root of the system of nonlinear equations F(x) = 0. We will construct several iterative methods, based on the (n, m)-APA (abstract Pad&approximant) or the (n, m)-ARA (abstract rational approximant)[4] for either F (direct one-point interpolation) or its inverse operator G (inverse one-point interpolation). The following methods are special cases: n = 1, m = 0: Newton-iteration (via direct and via inverse interpolation); inverse interpolation with n = 2, m = 0: improvement of the Newton-iteration as indicated by Ehrmann[7]; direct interpolation with n = 1, m = 1: method of tangent hyperbolas[lO], under certain conditions for the ARA. Among other new methods an interesting third-order iterative procedure is constructed via inverse interpolation with n = 1, m = 1:Let F: R4 *R’ and let x* be a simple root of the system of nonlinear equations F(x) = 0. We will construct several iterative methods, based on the (n, m)-APA (abstract Pad&approximant) or the (n, m)-ARA (abstract rational approximant)[4] for either F (direct one-point interpolation) or its inverse operator G (inverse one-point interpolation). The following methods are special cases: n = 1, m = 0: Newton-iteration (via direct and via inverse interpolation); inverse interpolation with n = 2, m = 0: improvement of the Newton-iteration as indicated by Ehrmann[7]; direct interpolation with n = 1, m = 1: method of tangent hyperbolas[lO], under certain conditions for the ARA. Among other new methods an interesting third-order iterative procedure is constructed via inverse interpolation with n = 1, m = 1: xi+r=xi+ 1 Qi ai +3F:-‘F’/oi’ with Fi the 1st Frechet-derivative of F at x,, a, = -F:-‘F; the Newton-correction. F:’ the 2nd Frechetderivative of F at x, and component-wise multiplication and division in Rq. This method is to be preferred to the method of tangent hyperbolas, which is also of third order, since it requires less numerical calculations. In general, the methods derived from the use of the (n, m)-APA or (n, m)-ARA with m 2 I are preferable when F or G have singularities in the neighbourhood of x* or 0 respectively.