Preface

This section gives an introduction to a particular class of iterative root finding methods, called open methods. As opposed to closed or bracketing methods (discussed in the previous section), open methods do not necessarily bracket a root; therefore, they may diverge as computation progresses, but when they do converge, open methods are usually so much faster than bracketing methods. The root finding methods are applied for solving nonlinear equations of the form f(x) = 0. These are the problems of applied mathematics that occur most often for which explicit solutions are not available. We present some iterative algorithms for the determination of roots of these equations to any specified degree of accuracy. There are two large classes of iterative numerical methods for root finding, namely those that either involve a usage of derivatives or not.

We begin with Newton's method and its discretization---the secant method, which provide root-finding algorithms that use a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. The section also contains some iterative algorithms to find a root of a nonlinear equation f(x) = 0 of order two and larger.

Return to Mathematica page
Return to the main page (APMA0330)
Return to the Part 1 (Plotting)
Return to the Part 2 (First Order ODEs)
Return to the Part 3 (Numerical Methods)
Return to the Part 4 (Second and Higher Order ODEs)
Return to the Part 5 (Series and Recurrences)
Return to the Part 6 (Laplace Transform)
Return to the Part 7 (Boundary Value Problems)