Question

**Write In JAVA Please!!**

2. Write a program to apply the Modified Newton's Method Tn-1 to the equation ()3r2 +4 0 starting with o 3. Use m and 2 and make separate numerical runs. In each case, set the maximum number of iterations to 25, but stop the computation when the backward error, i.e. f(n), is less than 10-12, Print all intermediate points xn and backward errors f(xn). Verify the convergence rates of your numerical solutions in both cases. (For m 1, you should observe linear convergence. For m-2, you should get quadratic convergence.) Use double precision in your computation. 15 points)

Answer 1

public class Newton_method {
static double f(double x)
{
return x*x*x-3*x*x+4;
}
static double f_d(double x)
{ return 3*x*x-6*x;
}
public static void main(String args[])
{
int max_ite=25;
double x0=3;
int m=1;
double xn=x0;
System.out.println("When x0=3 amd m=1");
for( int i=1;i<=max_ite;i++)
{double error=Newton_method.f(xn);
xn=xn-(m*Newton_method.f(xn)/Newton_method.f_d(xn));
System.out.println("x"+i+"="+xn+" backward error="+error);
if(Newton_method.f(xn)<Math.pow(10,-12))
break;
}

m=2;
xn=x0;
System.out.println("When x0=3 amd m=2");
for( int i=1;i<=max_ite;i++)
{
double error=Newton_method.f(xn);
xn=xn-(m*Newton_method.f(xn)/Newton_method.f_d(xn));
System.out.println("x"+i+"="+xn+" backward error="+error);
if(Newton_method.f(xn)<Math.pow(10,-12))
break;
}
}
}