Newton method for computing an inverse - c++

Question

This is following the question I asked in this thread : Link error missing vtable
I defined a class 'function' and two others classes 'polynomial' and 'affine' that inherit from 'function'.
class function {
public:
function(){};
virtual function* clone()const=0;
virtual float operator()(float x)const=0; //gives the image of a number by the function
virtual function* derivative()const=0;
virtual float inverse(float y)const=0;
virtual ~function(){}
};
class polynomial : public function {
protected:
int degree;
private:
float *coefficient;
public:
polynomial(int d);
virtual~polynomial();
virtual function* clone()const;
int get_degree()const;
float operator[](int i)const; //reads coefficient number i
float& operator[](int i); //updates coefficient number i
virtual float operator()(float x)const;
virtual function* derivative()const;
virtual float inverse(float y)const;
};
class affine : public polynomial {
int a;
int b;
//ax+b
public:
affine(int d,float a_, float b_);
function* clone()const;
float operator()(float x)const;
function* derivative()const;
float inverse(float y)const;
~affine(){}
};
Method inverse in polyomial does not seem to work fine. It is based on the Newton method applied to the function x->f(x)-y for fixed y (the element for which we're computing the inverse) and the current polynomial f.
float polynomial::inverse(float y)const
{
int i=0;
float x0=1;
function* deriv=derivative();
float x1=x0+(y-operator()(x0))/(deriv->operator()(x0));
while(i<=100 && abs(x1-x0)>1e-5)
{
x0=x1;
x1=x0+(y-operator()(x0))/(deriv->operator()(x0));
i++;
}
if(abs(x1-x0)<=1e-5)
{
//delete deriv; //I get memory problems when I uncomment this line
return x1;
}
else
{
cout<<"Maximum iteration reached in polynomial method 'inverse'"<<endl;
//delete deriv; //same here
return -1;
}
}
double polynomial::operator()(double x)const
{
double value=0;
for(int i=0;i<=degree;i++) value+=coefficient[i]*pow(x,i);
return value;
}
polynomial* polynomial::derivative()const
{
if(degree==0)
{
return new affine(0,0,0);
}
polynomial* deriv=new polynomial(degree-1);
for(int i=0;i<degree;i++)
deriv[i]=(i+1)*coefficient[i+1];
return deriv;
}
I test this method with p:x->x^3 :
#include "function.h"
int main(int argc, const char * argv[])
{
polynomial p(3);
for(int i=0;i<=2;i++) p[i]=0;
p[3]=1;
cout<<"27^(1/3)="<<p.inverse(27);
return 0;
}
This script outputs 27^(1/3)=Maximum iteration reached in polynomial method 'inverse'
-1 even if I put 10,000 instead of 100. I've read some articles on the internet and it seems that it's a common way to compute the inverse.

Answer 1

the abs function prototype is: int abs(int)
So a test like abs(x1-x0)<=1e-5 won't behave as you expect; you compare a int with a float. In this case the float will be converted to int so it the same as abs(x1-x0)<=0
This is probably why you don't get the expected result - I suggest adding a few more printouts to get to the bottom of things.

Answer 2

Well, the problem was in method 'derivative'. Instead of using the 'operator[]' that I redefined, I used '->coefficient[]' and the main script worked fine for p.inverse(27) (only 14 iterations). I just replaced deriv[i]=(i+1)*coefficient[i+1]; with deriv->coefficient[i]=(i+1)*coefficient[i+1];

Answer 3

Check This Code :
#include<iostream>
#include<cmath>
#include<math.h>
using namespace std;
void c_equation(int choose, double x);
void Processes(double x, double fx1, double fdx1, int choose);
void main()
{
int choose,choose2;
double x;
system("color A");
cout << " " << endl;
cout << "=============================================================" << endl;
cout << "Choose Equation : " << endl;
cout << "_____________________________________" << endl;
cout << "1- x-2sin(x)" << endl;
cout << "2- x^2 + 10 cos(x)" << endl;
cout << "3- e^x - 3x^2" << endl;
cout << " " << endl;
cin >> choose;
cout << "If you have values press 1/ random press 2 :" << endl;
cin >> choose2;
if (choose2 == 1)
{
cout << " " << endl;
cout << "Enter Xo : " << endl;
cin >> x;
c_equation(choose, x);
}
else if (choose2 == 2)
{
x = rand() % 20;
cout << "Xo = " << x << endl;
c_equation(choose, x);
choose2 = NULL;
}
else
{
cout << "Worng Choice !! " << endl;
choose = NULL;
choose2 = NULL;
main();
}
}
void c_equation(int choose, double x)
{
double fx;
double fdx;
double fddx;
double result;
if (choose == 1)
{
fx = x - 2 * sin(x);
fdx = 1 - 2 * cos(x);
fddx = 2 * sin(x);
result = abs((fx * fddx) / pow(fdx, 2));
}
else if (choose == 2)
{
fx = pow(x, 2) + 10 * cos(x);
fdx = 2 * x - 10 * sin(x);
fddx = 2 - 10 * cos(x);
result = abs((fx * fddx) / pow(fdx, 2));
}
else if (choose == 3)
{
fx = exp(x) - 3 * pow(x, 2);
fdx = exp(x) - 6 * x;
fddx = exp(x) - 6;
result = abs((fx * fddx) / pow(fdx, 2));
}
else
{
cout << " " << endl;
}
//------------------------------------------------------------
if (result < 1)
{
cout << "True Equation :) " << endl;
Processes(x, fx, fdx , choose);
}
else
{
system("cls");
cout << "False Equation !!" << endl;
choose = NULL;
x = NULL;
main();
}
}
void Processes(double x, double fx, double fdx , int choose)
{
double xic;
for (int i = 0; i < 3; i++)
{
xic = x - (fx / fdx);
cout << " " << endl;
cout << "Xi = " << x << " " << "F(Xi) = " << fx << " " << " F'(Xi) = " << fdx << " " << " Xi+1 = " << xic << endl;
x = xic;
if (choose == 1)
{
fx = xic - 2 * sin(xic);
fdx = 1 - 2 * cos(xic);
}
else if (choose == 2)
{
fx = pow(xic, 2) + 10 * cos(xic);
fdx = 2 * xic - 10 * sin(xic);
}
else if (choose == 3)
{
fx = exp(xic) - 3 * pow(xic, 2);
fdx = exp(xic) - 6 * xic;
}
}
}