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False position method

False position method - is a root-finding algorithm that uses a succession of roots of secant lines combined with bisection method to approximate a root of a function f.
Timur2014-06-25 09:50:21

Brief false position method description can be found below the calculator

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False position method

False position method is a root-finding algorithm that combines features from the bisection method and the Secant method.

As in secant method, we use the root of secant line (the value of x such that y=0) to compute next root approximation for function f.

The derivation of recurrence relation is the same as in secant method:

Suppose we have starting values x0 and x1, with function values f(x0) and f(x1).
The secant line has equation

\frac{y - f(x_1)}{f(x_1)-f(x_0)}=\frac{x - x_1}{x_1-x_0}

The root of secant line (where у=0) hence

x = x_1 - \frac{x_1 - x_0}{f(x_1)-f(x_0)}f(x_1)

The above formula is also used in the secant method, but the secant method always retains the last two computed points, while the false position method retains two points which certainly bracket a root.
Graphical interpretation can be seen below.

Metodo_delle_secanti.gif

Source

As can be seen from the recurrence relation, the false position method requires two initial values, x0 and x1, which should bracket the root. But unlike the bisection method, the width of the bracket does not tend to zero with iterations.

The tolerance condition can be either:

f(x_k)< \epsilon — function value is less than ε.

\left|x_k-x_{k-1}\right| < \epsilon — difference between two subsequent хk is less than ε. Note that this is the difference between two calculated subsequent xk, not the end-points of the interval.

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