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Regula FalsiEdit

In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection metho
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Regula falsi method to find the roots of a function.

d and the secant method.

A method of calculating an unknown quantity by first making an estimate and then using this and the properties of the unknown to obtain it. Also known as rule of false position.

The MethodEdit

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 off(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.


Regula

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)).

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