C Program for Fixed Point Iteration Method

Fixed point iteration method is commonly known as the iteration method. It is one of the most common methods used to find the real roots of a function. The C program for fixed point iteration method is more particularly useful for locating the real roots of an equation given in the form of an infinite series. So, this method can be used for finding the solution of arithmetic series, geometric series, Taylor’s series and other forms of infinite series.

This method is linearly convergent with somewhat slower rate of convergence, similar to the bisection method. It is based on modification approach to find the fixed point. It is commonly referred to as simple enclosure method or open bracket method.

Like other methods to find the root of a function, the programming effort for Iteration Method in C is easy, short and simple. Just like the Newton-Raphson method, it requires only one initial guess, and the equation is solved by the assumed approximation.

Iterations and modifications are successively continued with the updated approximations of the guess. Iterative method gives good accuracy overall just like the other methods.

Features of Fixed Point Iteration Method:

• Type – open bracket
• No. of initial guesses – 1
• Convergence – linear
• Rate of convergence – fast
• Accuracy – good
• Programming effort – easy
• Approach – modification

Below is a source code in C program for iteration method to find the root of (cosx+2)/3. The desired degree of accuracy in the program can be achieved by continuing the iteration i.e. by increasing the maximum number of iterations.

f(x) = (cos(x) +2)/3

Source Code for Iteration Method in C:

#include<stdio.h>
#include<math.h>
float raj(float);
main()
{
    float a[100],b[100],c=100.0;
    int i=1,j=0;
    b[0]=(cos(0)-3*0+1);
    printf("\nEnter initial guess:\n");
    scanf("%f",&a[0]);
    printf("\n\n The values of iterations are:\n\n ");
    while(c>0.00001)
    {
        a[j+1]=raj(a[j]);
        c=a[j+1]-a[j];
        c=fabs(c);
        printf("%d\t%f\n",j,a[j]);
        j++;

    }
    printf("\n The root of the given function is %f",a[j]);
}
float raj(float x)
{
    float y;
    y=(cos(x)+2)/3;
    return y;
}

Input/Output:

Also see,
Iteration Method Algorithm/Flowchart
Numerical Methods Tutorial Compilation

The fixed point iteration method is found extensively useful in many mathematical formulations and theorems. It is often used to find solutions and approximations to successive guess strategies used in dynamic engineering problems. Some of the popular methods utilizing this method are Newton’s method, Halley’s method, Runge-Kutta methods and the Picard–Lindelöf theorem.

Any questions or queries regarding fixed point iteration method or its C language source code presented here can be discussed and brought up to us from the comments section.