The RegulaFalsi 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 RegulaFalsi Method: Given a continuous function f(x)
|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.
|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 RegulaFalsi Method. Determine an initial estimate of the first positive root within one unit interval. Use Horner's Method for evaluating f(x).