Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

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alt="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 cubic equation (1.2) starting from the same initial guess . The first two columns of Table 1.1 outline the sequences and resulting from the bisection and Newton–Raphson methods, respectively. While the bisection method requires almost 50 iterations to achieve full accuracy, Newton's method does the same job in just 5. In fact, Newton–Raphson's sequence nearly doubles the number of converged digits from one iterate to the next, whereas in the bisection sequence this number grows very slowly (and sometimes even decreases, as seen from to 6). This is better understood when looking at the third and fourth columns of Table 1.1, where we have included the absolute error corresponding to both methods , where is the reference value given in the exact expression (1.3), and whose numerical evaluation with Matlab is .

Table 1.1 Iterates resulting from using bisection

and Newton–Raphson when solving (1.2), with added shading of converged digits.

	Bisection	Newton–Raphson
0	1.5	1.5
1	1.75	1.869 565 215 355 94
2	1.875	1.799 452 405 786 30
3	1.812 5	1.796 327 970 874 37
4	1.781 25	1.796 321 903 282 30
5	1.796 875	1.796 321 903 259 44
6	1.789 062 5	1.796 321 903 259 44
7	1.792 968 75
8	1.794 921 875
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