 ### REGULA–FALSI METHOD

##### Please note that the material on this website is not intended to be exhaustive. This is intended as a summary and supplementary material to the required textbook.

The Regula–Falsi Method is a numerical method for estimating the roots of a polynomial f(x).   A value x replaces the midpoint in the Bisection Method and serves as the new approximation of a root of f(x).   The objective is to make convergence faster.   Assume that f(x) is continuous.

Algorithm for the Regula–Falsi Method: Given a continuous function f(x)

1. Find points a and b such that a < b and f(a) * f(b) < 0.
2. Take the interval [a, b] and determine the next value of x1.
3. If f(x1) = 0 then x1 is an exact root, else if f(x1) * f(b) < 0 then let a = x1, else if f(a) * f(x1) < 0 then let b = x1.
4. Repeat steps 2 & 3 until f(xi) = 0 or |f(xi)| £ DOA, where DOA stands for degree of accuracy. Observe that

EC / BC = E / AB
[ x – a ] / [ b – a ] = [ f(x) – f(a) ] / [ f(b) – f(a) ]
x – a = [ b – a ] [ 0 – f(a) ] / [ f(b) – f(a) ]
x = a + [ b – a ] [ – f(a) ] / [ f(b) – f(a) ]
x = a – [ b – a ] f(a) / [ f(b) – f(a) ]

Note that the line segment drawn from f(a) to f(b) is called the interpolation line.

Graphically, if the root is in [ a, xi ], then the next interpolation line is drawn between ( a, f(a) ) and ( xi, f(xi) ); otherwise, if the root is in [ xi, b ], then the next interpolation line is drawn between ( xi, f(xi) ) and (b, f(b)). EXAMPLE:   Consider f(x) = x3 + 3x – 5, where [ a = 1, b = 2 ] and DOA = 0.001.

i a x b f(a) f(x) f(b)
1 1 1.1 2 – 1 – 0.369 9
2 1.1 1.13544668587896 2 – 0.369 – 0.129797592130931 9
3 1.13544668587896 1.14773797024856 2 – 0.129797592130931 – 0.0448680509813286 9
4 1.14773797024856 1.15196570867269 2 – 0.0448680509813286 – 0.0154155863909917 9
5 1.15196570867269 1.15341577448 2 – 0.0154155863909917 – 0.0052852985292482 9
6 1.15341577448 1.15391264384212 2 – 0.0052852985292482 – 0.00181077883487646 9
7 1.15391264384212 1.15408284038531 2 – 0.00181077883487646 – 0.000620231485743084 9

PROBLEM:   Develop an algorithm, expressed as a NSD, that will find an estimate of the first positive root of a given polynomial f(x) within a certain degree of accuracy DOA using the Regula–Falsi Method.   Determine an initial estimate of the first positive root within one unit interval.   Use Horner's Method for evaluating f(x).