e.doroskevic e.doroskevic - 4 months ago 15
Java Question

How-to apply Newton-Raphson method to find roots of a quintic function

Description



I have developed an algorithm implementing Newton-Raphson method to find a root of a quintic function. The result which I must mirror is
303.6
. However, my implementation fails to measure up.

Data



Parameters


g = 9.81;
Ds = 0.198;
uj = 805.9;
W = 0.0557;



Equation


0.024*((gDs/uj^2)^(1/3))(Y^(5/3)) + 0.2*(Y^(2/3)) - ((2.85/W)^(2/3)) = 0


Where derivative for
Y
:


(0.04*d^(1/3)⋅g(1/3)⋅y^(2/3)) / u(2/3) + 2/15*y^(1/3)


Solving root for
Y


Code



import java.lang.*;
public class InvokeNewton {
public static void main(String argv[]) {
double del = 1e-5,
double xx = 0 ;
double dx =0,
double x= Math.PI/2;
int k = 0;
while (Math.abs(xx-x) > del && k<10 && f(x)!=0) {
dx = f(x)/d(x);
xx=x;
x =x - dx;
k++;

System.out.println("Iteration number: " + k);
System.out.println("Root obtained: " + x);
System.out.println("Estimated error: " + Math.abs(xx-x));
}
}

// Method to provide function f(x)

public static double f(double x) {
return 0.024*(Math.pow(g * Ds / Math.pow(uj, 2.0),(1.0/3.0)) * (Math.pow(Y,5.0/3.0))+ 0.2*(Math.pow(Y,2.0/3.0)) - (Math.pow((2.85/W)(2.0/3.0))));
}

// Method to provide the derivative f'(x).

public static double d(double x) {
return (0.04*Math.pow(Ds,1.0/3.0)*Math.pow(Y,2.0/3.0)) / Math.pow*uj,2.0/3.0) + 2 / 15*Math.pow(Y,1.0/3.0);
}

}


Output



Iteration number: 1
Root obtained: 3.65373153496716
Estimated error: 2.0829352081722634
Iteration number: 2
Root obtained: 5.2246000232674215
Estimated error: 1.5708684883002615
Iteration number: 3
Root obtained: 6.618389759316356
Estimated error: 1.3937897360489346
Iteration number: 4
Root obtained: 7.906164279270034
Estimated error: 1.287774519953678
Iteration number: 5
Root obtained: 9.119558352547333
Estimated error: 1.213394073277299
Iteration number: 6
Root obtained: 10.27633029334909
Estimated error: 1.1567719408017574
Iteration number: 7
Root obtained: 11.387769167896339
Estimated error: 1.1114388745472485
Iteration number: 8
Root obtained: 12.461641418739712
Estimated error: 1.0738722508433725
Iteration number: 9
Root obtained: 13.503592201954325
Estimated error: 1.041950783214613
Iteration number: 10
Root obtained: 14.517895007865569
Estimated error: 1.0143028059112442


The equation and the derivative has been checked and double-checked, however I am still not getting the desired output

Reference



Newton Method

Answer

The code has been modified accordingly. The code provides comments to simplify modification to your specific case.

    public class RootFinder {

    // Method to be called to calculate the root 

    public void InvokeNewton(){
        double del = 1e-10;                                  // Delta or precision; 
        double xx = 0 ;                                      // Storage for previous root

        double dx = 0;                                       // Storage to hold a derivative of a pre-defined function;
        double x= Math.sqrt(beta/alpha);                     // Initial guess;

        while (Math.abs(xx-x) >= del &&  f(x)!= 0.0) {       // Math.abs(xx-x) - Estimated error;
            dx = f(x)/d(x);                                  // Derivative
            xx=x;                                            // New xx value set to previous root for compersion;
            x = x - dx;                                      // Root obtained;
        }
    }// Method to provide function f(x)
                          // !!! Provide your function bellow !!!
    public double f(double x) {
        return (alpha * (Math.pow(x,5.0/3.0)) + 0.2*(Math.pow(x,2.0/3.0)) - beta);
    }

    // Method to provide the derivative f'(x).
                          // !!!Provide the prime derivative of your function bellow!!!
    public double d(double x) {
        return (0.04*Math.pow(jf.getSourceDiameter(),1.0/3.0)*Math.pow(x,2.0/3.0)) / Math.pow(jf.getJetVelocity(),2.0/3.0) + 2.0 /  15.0*Math.pow(x,1.0/3.0);
    }
}