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#include <math.h>
#include <vector>
#include <string>
#include <iostream.h>
using namespace std;
double Get_f_of_y_values(int i, vector<double>& t_values, vector<double>& y_values)
{
double Get_f_of_y_values;
Get_f_of_y_values = ((- 8.0 * t_values[i]) / (1 + (t_values[i] * t_values[i]))) * y_values[i];
return Get_f_of_y_values;
}
double Get_f_of_x_values(int i, int j, vector<double>& t_values, vector<double>& x_values)
{
double Get_f_of_x_values;
Get_f_of_x_values = ((-8.0 * t_values[i + 1]) / (1 + (t_values[i + 1] * t_values[i + 1]))) * x_values[j];
return Get_f_of_x_values;
}
double Get_g_of_x_values(int i, int j, vector<double>& f_of_y_values, vector<double>& f_of_x_values, vector<double>& t_values,
vector<double>& x_values, vector<double>& y_values, double h)
{
double Get_g_of_x_values;
Get_g_of_x_values = (0.5 * h * (f_of_y_values[i] + f_of_x_values[j])) + y_values[i] - x_values[j];
return Get_g_of_x_values;
}
double Get_g_primed_of_x_values(int i, vector<double>& t_values, double h)
{
double Get_g_primed_of_x_values;
Get_g_primed_of_x_values = (0.5 * h * (( - 8.0 * t_values[i + 1])/( 1 + (t_values[ i + 1 ] * t_values[i + 1])))) - 1;
return Get_g_primed_of_x_values;
}
double Newton_Raphson(int j, vector<double>& g_values, vector<double>& g_primed_values, vector<double>& x_values)
{
double Newton_Raphson;
Newton_Raphson = x_values[j] - (g_values[j]/g_primed_values[j]);
return Newton_Raphson;
}
double Newton_Raphson_Error(int j, vector<double>& x_values)
{
double Newton_Raphson_Error;
Newton_Raphson_Error = fabs(x_values[j + 1] - x_values[j]);
return Newton_Raphson_Error;
}
int main(){
//define vectors
vector<double> f_of_x_values;
vector<double> f_of_y_values;
vector<double> g_values;
vector<double> g_primed_values;
vector<double> t_values;
vector<double> x_values;
vector<double> y_values;
//define stepsize
double newton_raphson_error;
double N = 10.0;
int n = 20;
int i, j;
double h = N / n;
//define vector size
f_of_x_values.reserve(n+1);
f_of_y_values.reserve(n+1);
g_values.resize(j+1);
g_primed_values.resize(j+1);
t_values.reserve(n+1);
x_values.resize(j+1);
y_values.reserve(n+1);
//initialise vectors
f_of_y_values[0]=0;
f_of_x_values[0]=0;
g_values[0]=0;
g_primed_values[0]=0;
t_values[0]=0;
x_values[0]=1;
y_values[0]=1;
//begin computation
for ( i = 0; i < n; i++ ){
t_values[i] = i * h;
f_of_y_values[i] = Get_f_of_y_values(i, t_values, y_values);
for ( j = 0; Newton_Raphson_Error(j, x_values) < 0.01; j++ ){
f_of_x_values[j] = Get_f_of_x_values(i, j, t_values, x_values);
g_values[j] = Get_g_of_x_values(i, j, f_of_y_values, f_of_x_values, t_values, x_values, y_values, h);
g_primed_values[j] = Get_g_primed_of_x_values(j, t_values, h);
x_values[j+1] = Newton_Raphson(j, g_values, g_primed_values, x_values);
newton_raphson_error = Newton_Raphson_Error(j, x_values);
cout << x_values[j+1] << endl;
}
}
return 0;
}
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