Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

Fundamentals of Numerical Mathematics for Physicists and Engineers

href="#fb3_img_img_302de88e-aea1-5c0f-be85-ded3acb3a1c1.png" alt="images"/>, usually called the residuals. Since this quantity should approach zero when images

converges to a root, the reader may wonder why images

should not be used as a measure for stopping the iterations. The reason is that the residual is not always a reliable measure (although it is often used in practice to decide convergence). We will address this question later in this chapter, when we introduce the concept of condition number of a root.

1.3 Newton's Method

Newton's method, also known as Newton–Raphson method,⁷ is probably the most important root‐finding algorithm of numerical mathematics. Its formulation naturally leads to a wide variety of other root‐finding algorithms and is easily adaptable to solving systems of nonlinear equations.⁸ To formulate this method we will assume that images is differentiable and that the root is close to some initial guess images . To get a better estimation of the root we assume that, close to images , the function images can be approximated by its tangent line images at the point images plotted in gray below. In other words, we locally approximate images by its first order Taylor expansion images at images :

Graph depicts a curve intersecting at a point at the x-axis and the root is between x1 and x0.

(1.6) equation

where images is the value of the derivative of images at images . The motivation for approximating images by images is simple: if images is a reasonably good approximation of images , then the solution of images should be a reasonable good estimation of the solution of images . Let images be the solution of images satisfying

(1.7) equation

The resulting abscissa images where images intercepts the images ‐axis is the new (hopefully more accurate) estimation of the root. The process can be repeated approximating images by its Taylor expansion images at images (straight gray line below the curve) in order to obtain an even better estimation images satisfying

(1.8) equation

and so on, leading to the iterative formula:

Newton's Iteration:

(1.9) equation

A simple implementation of Newton's method can be found in the next code, whose structure essentially differs from the one seen in the bisection in two main aspects. First, the code needs an initial guess images as starting point, instead of an interval. Second, in addition to the string name corresponding to the M‐file function fun, the code also needs the one associated with images in the argument dfun. Providing the M‐file for the exact derivative entails both analytical differentiation and its subsequent programming, both being error‐prone tasks. While this is the correct procedure, it is sometimes advisable to let the computer do the job by approximating the value of images by the ratio

(1.10) equation

where images is a small increment ( images in our code). This approximation of the derivative, along with the suitability of the chosen images , will be properly justified later in the chapter devoted to numerical differentiation.⁹ The reader may check that there are no noticeable differences when using either the exact or the approximate derivative (1.10).

% Code 2: Newton's method for solving f(x) = 0 % Input: 1. a: initial guess % 2. tol: tolerance so that abs(x_k+1 - x_k) < tol % 3. itmax: maximum number of iterations allowed % 4. fun: function's name % 5. dfun: derivative's name % (If dfun = ‘0’, then f'(x) is approximated) % Output: 1. xk: resulting sequence % 2. res: resulting residuals % 3. it: number of required iterations function [xk,res,it] = newton(a,tol,itmax,fun,dfun) xk = [a]; fk = feval(fun,xk); res = abs(fk); it = 0; tolk = res(1); dx = 1e-8 ; while it < itmax & tolk > tol if dfun == ‘0’ dfk = (feval(fun,xk(end)+dx)-fk)/dx; else dfk = feval(dfun,xk(end)); end xk = [xk, xk(end) - fk/dfk]; fk = feval(fun,xk(end)); res = [res abs(fk)]; tolk = abs(xk(end)-xk(end-1)); it = it + 1; end

Let us compare the performance of Newton's method with the bisection by applying both algorithms to solve the

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