Question

Write a C++ program for all the methods (Bisection, Newton-Raphson, Secant, False-Position, and Modified Secant) for locating roots. Make sure that you have clever checks in your program to be warned and stop if you have a divergent solution or stop if the solution is very slowly convergent after a maximum number of iterations.

Answer 1

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1. Bisection Method

#include<bits/stdc++.h>
using namespace std;
#define EPSILON 0.01


double func(double x)
{
        return  2 * x * x * x - 11.7 * x * x + 2 + 17.7 * x - 5;
}

// Prints root of func(x) with error of EPSILON
void bisection(double a, double b)
{
        if (func(a) * func(b) >= 0)
        {
                cout << "You have not assumed right a and b\n";
                return;
        }

        double c = a;
        while ((b - a) >= EPSILON)
        {
                c = (a + b) / 2;

                if (func(c) == 0.0)
                        break;

                else if (func(c)*func(a) < 0)
                        b = c;
                else
                        a = c;
        }
        cout << "The value of root is : " << c;
}

int main()
{
        double a = 0 , b = 4;
        bisection(a, b);
        return 0;
}

2. Newton Raphson Method

#include<bits/stdc++.h>
using namespace std;
#define EPSILON 0.01


double func(double x)
{
        return  2 * x * x * x - 11.7 * x * x  + 17.7 * x - 5;
}

// Prints root of func(x) with error of EPSILON
double derivFunc(double x) {
        6 * x * x - 23.14 * x + 17.7;
}
void newtonRaphson(double x)
{
        double h = func(x) / derivFunc(x);
        while (abs(h) >= EPSILON)
        {
                h = func(x) / derivFunc(x);

                // x(i+1) = x(i) - f(x) / f'(x)
                x = x - h;
        }

        cout << "The value of the root is : " << x;
}

int main()
{
        double x0 = -20; // Initial values assumed
        newtonRaphson(x0);
        return 0;
}

3.Secant Method

#include<bits/stdc++.h>
using namespace std;
const int E =  0.01;


double f(double x)
{
        return  2 * x * x * x - 11.7 * x * x  + 17.7 * x - 5;
}

void secant(float x1, float x2, float E)
{
        float n = 0, xm, x0, c;
        if (f(x1) * f(x2) < 0) {
                do {
                        x0 = (x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1));

                        c = f(x1) * f(x0);
                        x1 = x2;
                        x2 = x0;
                        n++;
                        if (c == 0)
                                break;
                        xm = (x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1));
                } while (fabs(xm - x0) >= E && n <= 100); 

                cout << "Root of the given equation=" << x0 << endl;
                cout << "No. of iterations = " << n << endl;
        } else
                cout << "Can not find a root in the given inteval";
}

// Driver code
int main()
{
        // initializing the values
        float x1 = 0, x2 = 1, E = 0.0001;
        secant(x1, x2, E);
        return 0;
}

4. False-Position Method

#include<bits/stdc++.h>
using namespace std;
const int E =  0.01;
const int MAX_ITER = 100;

double func(double x)
{
        return  2 * x * x * x - 11.7 * x * x  + 17.7 * x - 5;
}
void regulaFalsi(double a, double b)
{
        if (func(a) * func(b) >= 0)
        {
                cout << "You have not assumed right a and b\n";
                return;
        }

        double c = a;  // Initialize result

        for (int i = 0; i < MAX_ITER; i++)
        {
                // Find the point that touches x axis
                c = (a * func(b) - b * func(a)) / (func(b) - func(a));

                // Check if the above found point is root
                if (func(c) == 0)
                        break;

                // Decide the side to repeat the steps
                else if (func(c)*func(a) < 0)
                        b = c;
                else
                        a = c;
        }
        cout << "The value of root is : " << c;
}

// Driver program to test above function
int main()
{
        // Initial values assumed
        double a = 0, b = 4;
        regulaFalsi(a, b);
        return 0;
}