Determining the roots of a cubic polynomial using N-R method

[shoubie]

Hello! Can anyone explain to me how to use the Newton-Raphson method to get the roots of a cubic polynomial by using functions? It would be greatly appreciated if anyone could point me in the right direction. Thank you. :)

[jsmith]

look up newton raphson on wikipedia.

it even gives you C code to solve the problem

[EDIT: oops -- look on wikipedia, not google]

[shoubie]

Thanks, jsmith! :)

I wrote this:

#include <iostream>
#include <cmath>
using namespace std;
// function prototypes
double linear (double a, double b, double xi1);
double quadratic (double a, double b, double c, double d, double xi1);
double cubic (double a, double b, double c, double d, double xi1);
double NRF1 (double a, double b, double c, double d, double x);
double NRF (double a, double b, double c, double d, double x);

int main ()
{
	char again = 'y';
	while (again == 'y')
	{
	double a (0.0), b(0.0), c (0.0), d (0.0), x (0.0), r1 (0.0), r2 (0.0), sr (0.0), mr (0.0), lr (0.0);
	cout << "Please enter values for a, b, c, and d:" << endl;
	cout << "a: "; cin >> a;
	cout << "b: "; cin >> b;
	cout << "c: "; cin >> c;
	cout << "d: "; cin >> d;

	cout << endl << endl;
	cout << "The roots of " << a << "x^3 + " << b << "x^2 + " << c << "x + " << d << " are ";
	cout << fx1 << ", " << ", " << fx2 << ", " << fx3 << endl;

	x = -2*b / 6*a;
	
	r1 = x - 0.5;
	double f1xr1 = NRF1 (a, b, c, d, r1);
	r2 = x + 0.5;
	double f1xr2 = NRF1 (a, b, c, d, r2);
	
	sr = r1 - 0.5;
	double fx1 = NRF (a, b, c, d, sr);
	mr = (r1 + r2)/2.0;
	double fx2 = NRF (a, b, c, d, mr);
	lr = r2 + 0.5;
	double fx3 = NRF (a, b, c, d, lr);

	cout << endl << endl;
	cout << "Would you like to continue? (y/n): ";
	cin >> again;
	}

	cout << endl << endl;
	cout << "Good bye!" << endl;

	return 0;
}



double NRF1 (double a, double b, double c, double d, double x)
{
	double xi (0.0), xi1 (0.0);
	int count (0);
	
	while ( (abs (xi1 - xi) > 0.001) && (count < 200) )
	{
		xi = xi1;
		xi1 = xi - f1x/f2x;
		count++;
	}
	return xi1;
}

double NRF (double a, double b, double c, double d, double x)
{
	double xi (0.0), xi1 (0.0);
	int count (0);
	
	while ( (abs (xi1 - xi) > 0.001) && (count < 200) )
	{
		xi = xi1;
		xi1 = xi - fx/f1x;
		count++;
	}
	return xi1;
}

double cubic (double a, double b, double c, double d, double xi1)
{
	return a*pow(xi1,3) + b*pow(xi1,2) + c*xi1 + d;
}

double quadratic (double a, double b, double c, double d, double xi1)
{
	return 3.0*a*pow(xi1,2) + 2.0*b*xi1 + c;
}

double linear (double a, double b, double xi1)
{
	return 6.0*xi1 + 2.0*b;
}

Can someone please point out the problem with it?
Thanks!

[zipuch]

line 34: you can`t use variable before initialization( in your case in line 44,46,48 )
line 71,85,: you can`t use undeclared variables

[shoubie]

Fixed it. Thanks, zipuch! :)

[shoubie]

Okay, now I have this:

#include <iostream>
#include <cmath>
using namespace std;
// function prototypes
double linear (double a, double b, double x);
double quadratic (double a, double b, double c, double d, double x);
double cubic (double a, double b, double c, double d, double x);
double NRF1 (double a, double b, double c, double d, double x);
double NRF (double a, double b, double c, double d, double x);

int main ()
{
	char again = 'y';
	while (again == 'y')
	{
	// Variable declarations
	double a (0.0), b(0.0), c (0.0), d (0.0), x (0.0), r1 (0.0), r2 (0.0), sr (0.0), mr (0.0), lr (0.0);
	cout << "Please enter values for a, b, c, and d:" << endl;
	cout << "a: "; cin >> a;
	cout << "b: "; cin >> b;
	cout << "c: "; cin >> c;
	cout << "d: "; cin >> d;

	x = -2*b / 6*a; // root of f''(x)
	
	r1 = x - 0.5;
	double f1xr1 = NRF1 (a, b, c, d, r1); // one root of f'(x)
	r2 = x + 0.5;
	double f1xr2 = NRF1 (a, b, c, d, r2); // other root of f'(x)
	
	sr = r1 - 0.5;
	double fx1 = NRF (a, b, c, d, sr); // first root of f(x)
	mr = (r1 + r2)/2.0;
	double fx2 = NRF (a, b, c, d, mr); // second root of f(x)
	lr = r2 + 0.5;
	double fx3 = NRF (a, b, c, d, lr); // third root of f(x)

	cout << endl << endl;
	cout << "The roots of " << a << "x^3 + " << b << "x^2 + " << c << "x + " << d << " are ";
	cout << fx1 << ", " << fx2 << ", " << fx3 << endl;

	cout << endl << endl;
	cout << "Would you like to continue? (y/n): ";
	cin >> again;
	} // end of loop

	cout << endl << endl;
	cout << "Good bye!" << endl;

	return 0;
}

double cubic (double a, double b, double c, double d, double x)
{
	return a*pow(x,3) + b*pow(x,2) + c*x + d;
}

double quadratic (double a, double b, double c, double d, double x)
{
	return 3.0*a*pow(x,2) + 2.0*b*x + c;
}

double linear (double a, double b, double x)
{
	return 6.0*x + 2.0*b;
}

double NRF1 (double a, double b, double c, double d, double x) // function for the roots of f'(x)
{
	double xi (0.0), xi1 (0.0);
	int count (0);

	do 
	{
		double f1x = quadratic (a, b, c, d, x);
		double f2x = linear (a, b, x);
		xi = xi1;
		xi1 = xi - f1x/f2x;
		count++;
	} while ( (abs (xi1 - xi) > 0.001) && (count < 200) );
	return xi1;
}

double NRF (double a, double b, double c, double d, double x) // function for the root of f(x)
{
	double xi (0.0), xi1 (0.0);
	int count (0);
	
	do 
	{
		double fx = cubic (a, b, c, d, x);
		double f1x = quadratic (a, b, c, d, x);
		xi = xi1;
		xi1 = xi - fx/f1x;
		count++;
	} while ( (abs (xi1 - xi) > 0.001) && (count < 200) );
	return xi1;
}

It compiles now, but it gives me zeroes as the roots, when it shouldn't. Help, please? Thanks!