Newton Raphson question

[aurimas13]

Hello, I am trying to write a program that gives me a root of the function using Newton Rhapson method. Currently I am trying to write it in the int main() function to make it work. Afterwards I want to transfer the iteration to root function as thats what my assignment asks. A problem is this : write equation f(x) = 0 solution program. Show program's success by finding the root of function f(x) = 1 - x^2. The program should be made of three functions int main(), void rood(double& x, double eps) and double fk(double x). So, below is what I wrote. It doesn't seem to be a hard problem, but when I run the program I get the root to be equal to nan , which I know is a mistake. Can someone point me out to the right direction where I might be doing it wrongly trying to first write an iteration in function int main()? Thank you for all the help. All suggestions are welcome. Also, how I can find a derivative of a function without writing it as a different function? Is it possible?

#include <iostream>
#include <stdio.h>
#include <cmath>

using namespace std;

void root(double& x, double eps);

double fk(double x);

double fk_p(double x);

int main()
{
  // Data: accuracy eps and some x

  double x, x_1, eps;
  printf("x_1 = ");
  scanf("%lf", &x_1);
  printf("eps = ");
  scanf("%lf", &eps);
  
  do
  {
      x = x_1;
      x_1 = x - (fk(x) / fk_p(x));
      printf("%lf \n", x_1);
  } while (abs(x_1 - x) >= eps );
  printf("Exact root is: %lf\n", x_1);

  //root(x, eps);

  // Result: x
}

void root(double& x, double eps)
{

//Iteration  

 double f = fk(x);  // Function

}

double fk(double x)
{
  return 1 - x * x;
}

double fk_p(double x)
{
  return 2 * x;
}

[aurimas13]

I changed my function and it works now, but I am curious to know how to write derivatives in C++ without calling a function? Is it possible in my case?

Code:

#include <iostream>
#include <stdio.h>
#include <cmath>

using namespace std;

void root(double& x, double eps);

double fk(double x);

double fk_p(double x);

int main()
{
  // Data: accuracy eps and some x

  double x, eps;
  printf("x = ");
  scanf("%lf", &x);
  printf("eps = ");
  scanf("%lf", &eps);

  root(x, eps);
  printf("Exact root is: %lf\n", x);
}

void root(double& x, double eps)
{
    // Iteration
    double x_1;

    do
    {
        x_1 = x;
        x = x_1 - fk(x_1) / fk_p(x_1);
        printf("%lf \t %lf \n", x, eps);
    } while (abs(x - x_1) >= eps );
}

double fk(double x)
{
  return (1.0 - x * x);
}

double fk_p(double x)
{
  return (- 2.0 * x);
}

[lastchance]

You can get an estimate of the derivative from the actual function itself by finite differences (relating directly to the definition of a derivative):

df/dx ~ ( f(x+dx) - f(x-dx) ) / ( 2 * dx )

where dx is "small".

In code, replace your current derivative function by, e.g.

double fk_p(double x)
{
  const double SMALL = 1.0e-10;
  return ( fk( x + SMALL ) - fk( x - SMALL ) ) / ( 2.0 * SMALL );
}

[aurimas13]

Thanks, lastchance. However, I would like to keep fk(x) function and remove fk_p() which basically gives the derivative of fk(x). I want the derivative to be calculated in root() function without fk_p(x) function available or used. Any ideas?

[jonnin]

if you know the function you can just do it on paper and code the dx.

for example if you have ax*x + bx + c
you can code in
2*a*x + b yourself. This is probably the best way when the function is known.

[lastchance]

I would like to keep fk(x) function and remove fk_p() which basically gives the derivative of fk(x). I want the derivative to be calculated in root() function without fk_p(x) function available or used. Any ideas?

So, in root(), where you update x, just put

  const double SMALL = 1.0e-10;
  double derivative = ( fk( x + SMALL ) - fk( x - SMALL ) ) / ( 2.0 * SMALL );
  x = x_1 - fk(x_1) / derivative;

Then you don't need function fk_p() at all and you only have to change fk() if the function changes.