C++ Newton Raphson method is slower than bisection? - c++

Question

My task is to find the root of a function with both the Newton Raphson and the bisection method within an error margin of 10E-7.
The point of all that is, that we learn that the Newton Raphson method is faster and more effective.
Now for some reason I come to the opposite result. Although I am aware that the initial guess of the root in both methods strongly affects the number of necessary iterations. But I entered a similar guess in both algorithms and my fellow students dont get the result I do.
Bisection method:
#include <iostream>
#include <iomanip>
using namespace std;
//Declaring the given function
double func1(double x) {
return 0.00000000027 * (x - 10000000) - 0.16460351745 * (-1 + ((1000000000) / (x))) * 1 / (sqrt(x));
}
int main() {
std::fixed;
//Initial guess: root ist at 10 to the 7.
double x1 = 10000000;
double x2 = 1000000000;
double eps = 0.0000001;
int i = 0;
double x0[100000];
x0[0] =0;
//Exception handler
if (func1(x1) * func1(x2) > 0) {
cout << "Root is not inside the bracket.";
goto end;
}
goto start;
//Bisection Algorithm
while (abs(x0[i] - x0[i-1]) >= eps) {
start:
i = i + 1;
x0[i] = 0.5 * (x1 + x2);
if (func1(x1) * func1(x0[i]) < 0) {
x2 = x0[i];
}
else {
x1 = x0[i];
}
}
cout << endl << "Bisection Method: " << fixed << setprecision(10) << x0[i] << endl << "Iterations: " << i << endl << endl << endl << endl << endl;
end:
return 0;
}
}
Newton Raphson:
#include <iostream>
#include <iomanip>
using namespace std;
// Declaring the function and its derivative
double func1(double x) {
return 0.00000000027 * (x - 10000000) - 0.16460351745 * (-1 + ((1000000000) / (x))) * 1 / (sqrt(x));
}
double funcderiv1(double x) {
return 0.00000000027+((0.1646035174)/(2*x*x*sqrt(x)))*(30000000-x);
}
int main()
{
std::fixed;
double eps = 1;
double x_start = 10000000;
double c;
int i = 0;
while (eps >= 0.0000001) {
c = x_start - ((func1(x_start)) / (funcderiv1(x_start)));
eps = abs(func1(x_start) / funcderiv1(x_start));
x_start = c;
i = i + 1;
}
cout << fixed << setprecision(5) << "RESULT " << c << endl << " Iterations: " << i << endl;
}
The root is at 17903534.23630
Does anyone know why my bisection method needs 55 iterations while Newton Raphson takes like 82?

Answer 1

For the function
f(x) = A * (x - B) - C * (D / x - 1) / sqrt(x)
A = 0.00000000027
B = 10000000
C = 0.16460351745
D = 1000000000
the correct derivative is:
f'(x) = A - C (x - 3D) / (2 * x * x * sqrt(x))
Compare this with your expression:
g(x) = A - C (x - 3B) / (2 * x * x * sqrt(x))
After fixing the formula (by adding two zeros), your code makes 6 iterations:
RESULT 17903534.23630
Iterations: 6