Falseposition

Calculates the zeros of a function using the Regula-Falsi method.

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Interface
Falseposition
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Interface

C++

Falseposition

doublefalseposition(	double	(`f`)(double)[function pointer]*
	double	`x0` `= -1E+7`
	double	`x1` `= 1E+7`
	double	`eps` `= 1E-10`
	double	`maxit` `= 1000`	)

The false position method, on contrast with Newton's method, in which the calculation of the derivative $\inline f'(x_n)$ was required, replaces it with its approximation

$f'(x_n) = \frac {f(x_n) - f(x_{n - 1})} {x_n - x_{n - 1}}$

(2)

Given $\inline x_0$ and $\inline x_1$ , the main algorithm is based on the following recurrence relation

$x_{n + 1} = \frac {x_n f(x_{n - 1}) - x_{n - 1} f(x_n)} {f(x_{n - 1}) - f(x_n)} \qquad n \geq 1$

(3)

This method has rate of convergence $\inline \frac {1 + \sqrt{5}} {2} \approx 1.618$ , therefore inferior to the method of Newton. However it is more accurate than Newton because only one function evaluation is required, $\inline f(x_n)$ , $\inline f(x_{n - 1})$ being calculated in the previous iteration, while Newton requires two, $\inline f(x_n)$ and $\inline f'(x_n)$ .

To give you a better idea on the way this method works, the following graph shows different iterations in the approximation process. Here is the associated list of pairs chosen at consecutive steps

$(a_0, b_0) \quad (a_1, b_0) \quad (a_2, b_0)$

(4)

MISSING IMAGE!

1/falseposition-378.png cannot be found in /users/1/falseposition-378.png. Please contact the submission author.

This algorithm finds the roots of the user-defined function f starting with an initial interval [x0, x1] and iterating the sequence above until either the accuracy eps is achieved or the maximum number of iterations maxit is exceeded.

Example:

#include <codecogs/maths/rootfinding/falseposition.h>
 
#include <iostream>
#include <iomanip>
#include <cmath>
 
// user-defined function
double f(double x) {
    return cos(x);
}
 
int main() 
{
  double x = Maths::RootFinding::falseposition(f, 1, 3);
 
  std::cout << "The calculated zero is X = " << std::setprecision(12) << x << std::endl;
  std::cout << "The associated ordinate value is Y = " << f(x) << std::endl;
  return 0;
}

Output:

The calculated zero is X = 1.70614146372
The associated ordinate value is Y = -0.134932299622

References:

Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
F.R. Ruckdeschel, "BASIC Scientific Subroutines", Vol. II, BYTE/McGRAWW-HILL, 1981

Parameters

f	the user-defined function
x0	Default value = -1E+7
x1	Default value = 1E+7
eps	Default value = 1E-10
maxit	Default value = 1000

Authors

Lucian Bentea (August 2005)

Source Code

Source code is available when you buy a Commercial licence.

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