Numerical Analysis

This is the oldest method for finding the real root of a nonlinear equation f(x) = 0 and closely resembles the bisection method. In this method, also known as regula falsi or the method of chords , we choose two points a and b such that f(a) and f(b) are of opposite signs. Hence, a root must lie in between these points. Now, the equation of the chord joining the two points [a, f(a)] and [b, f(b)] is given by {y – f(a)}/{x – a} = {f(b) – f(a)}/{b – a} The method consists in replacing the part of the curve between the points [a, f(a)] and [b, f(b)] by means of the chord joining these points, and taking the point of intersection of the chord with the x-axis as an approximation to the root. The point of intersection in the present case is obtained by putting y = 0 in first equation. Thus, we obtain x 1 = a – {(b – a)f(a)}/{f(b) – f(a)} = {af(b) – bf(a)}/{f(b) – f(a)} Which is the first approximation to the root of f(x) = 0. If now f(x 1 ) and f(a) are of opposite sign...