Finding the roots of a nonlinear equation using C++ - c++

Question

What method would I use to find the roots of f(x) = 5x(e^-mod(x))cos(x) + 1 ? I have being trying the durand-kerner method but I can't get it to work. Are there any easier ways of doing it?
Here is the my code using the durand-kerner method
#include <iostream>
#include <complex>
#include <math.h>
using namespace std;
typedef complex<double> dcmplx;
dcmplx f(dcmplx x)
{
// the function we are interested in
double a4 = 5;
double a0 = 1;
return (a4 * x * exp(-x) * cos(x) )+ a0;
}
int main()
{
dcmplx p(.9,2);
dcmplx q(.1, .5);
dcmplx r(.7,1);
dcmplx s(.3, .5);
dcmplx p0, q0, r0, s0;
int max_iterations = 100;
bool done = false;
int i=0;
while (i<max_iterations && done == false)
{
p0 = p;
q0 = q;
r0 = r;
s0 = s;
p = p0 - f(p0)/((p0-q)*(p0-r)*(p0-s));
q = q0 - f(q0)/((q0-p)*(q0-r)*(q0-s));
r = r0 - f(r0)/((r0-p)*(r0-q)*(r0-s0));
s = s0 - f(s0)/((s0-p)*(s0-q)*(s0-r));
// if convergence within small epsilon, declare done
if (abs(p-p0)<1e-5 && abs(q-q0)<1e-5 && abs(r-r0)<1e-5 && abs(s-s0)<1e-5)
done = true;
i++;
}
cout<<"roots are :\n";
cout << p << "\n";
cout << q << "\n";
cout << r << "\n";
cout << s << "\n";
cout << "number steps taken: "<< i << endl;
return 0;
}

Answer 1

This approach makes use of the bisection method, and the fact that you can do a little math to find an upper bound for the highest zero, respectively.
Reproduced at http://ideone.com/fFLjsh
#include <iostream>
#include <iomanip>
#include <cmath>
#include <vector>
#include <utility>
#define MINX (-20.0f)
//MAXX Happens to be an upper bound for all zeroes of the function...
#define MAXX (1.0f)
#define NUM_INTERVALS (1000000)
#define NUM_BISECTION_ITERATIONS (30)
using namespace std;
double f(double x){
return 5 * x * exp(-x) * cos(x) + 1;
}
double bisection_method(double x0, double x1){
for (unsigned int i = 0; i < NUM_BISECTION_ITERATIONS; i++){
double midpoint = 0.5*x0 + 0.5*x1;
f(x0) * f(midpoint) < 0 ? x1 = midpoint : x0 = midpoint;
}
return 0.5*x0 + 0.5*x1;
}
int main(int argc, char** argv){
vector<pair<double, double>> relevant_intervals;
for (unsigned int i = 0; i < NUM_INTERVALS - 1; i++){
double x0 = MINX + (MAXX - MINX) / NUM_INTERVALS * (i);
double x1 = MINX + (MAXX - MINX) / NUM_INTERVALS * (i + 1);
if (f(x0) * f(x1) < 0)
relevant_intervals.push_back(make_pair(x0, x1));
}
cout << fixed << setprecision(20);
for (const auto & x : relevant_intervals){
cout << "One solution is approximately " << bisection_method(x.first, x.second) << endl;
}
}

Answer 2

I am not familiar with Durand-Kerner method, but according to Wiki http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method, it is suitable only for solving polynomial equations. Note the line in the wiki page: "The explanation is for equations of degree four. It is easily generalized to other degrees."
Your equation is not polynomial. The numerical solution will probably not converge.
Regardless of that your function f returns wrong formula: return a4 * x * exp(-abs(x)) * cos(x) + a0; (you forgot about complex modulo, i.e. abs)
and your iterations seem also wrong. They should read:
p = p0 - f(p0)/((p0-q0)*(p0-r0)*(p0-s0));
q = q0 - f(q0)/((q0-p)*(q0-r0)*(q0-s0));
r = r0 - f(r0)/((r0-p)*(r0-q)*(r0-s0));
s = s0 - f(s0)/((s0-p)*(s0-q)*(s0-r));
but even if you make these changes, the solution will not converge - it will be oscilating. The reason is probably as written above - the method is not suitable for this type of equation.