Date of Award
Doctor of Philosophy
The thesis demonstrates how iterative algorithms for the numerical solution of a nonlinear equation can be deduced from the integral-equation representations of a homotopic imbedding.;As with continuation methods, we begin by imbedding the nonlinear equation in a continuous one-parameter family of problems by means of a homotopy. A root of the original equation is then the end point of the path of the homotopy's zeros which connects the root to an initial estimate. The path is formulated as the solution of a first-order initial value problem.;Instead of numerically solving the differential expression for the end point, as is the case with continuation methods, we first find approximate analytical solutions to the corresponding Volterra integral equations. The approximations are obtained by standard techniques such as Picard's method and the local linearization procedure of Newton. Their end points are then used to define inductively iterative algorithms without memory for the numerical solution of the given nonlinear equation. We also consider approximating the integral operator by means of quadrature rules. Similar iterative algorithms are defined after analytically or numerically approximating the integrand.;Various one-point algorithms, including those of Newton and Olver, have been derived by the analytical approach. With the aid of quadrature rules, multipoint algorithms have been obtained, some of which have supercubic rates of convergence.
Beaudoin, Yves, "The Iterative Solution Of A Nonlinear Equation Using The Integral-equation Formulations Of An Imbedding" (1984). Digitized Theses. 1321.