Table of Contents

Root Finder Newton’s Software Manual

Routine Name: root_finder_newton

Author: Philip Nelson

Language: C++. The code can be compiled using the GNU C++ compiler (gcc). A make file is included to compile an example program

For example,

make

will produce an executable ./newton.out that can be executed.

Description/Purpose: This routine will find the root of a function \(f\) starting at \(x\) using newton’s method of root finding.

Input: there are five needed inputs, a function, it’s derivative, the starting point, the tolerance, and a maximum number of iterations.

@tparam T       The type of x0 and tolerance
@tparam F       A function of type T(T)
@tparam Fprime  A function of type T(T)
@param x0       The starting point
@param tol      The Tolerance
@param MAX_ITER The maximum iterations

Output: This routine returns the root of the given function \(f\).

Usage/Example:

The following is an example using two functions, \(f(x) = x^2 - 3\) and \(g(x) = sin(\pi \cdot x)\).

int main()
{
  auto f = [](double x) { return x * x - 3; };
  auto fprime = [](double x) { return 2 * x; };

  auto g = [](double x) { return sin(M_PI * x); };
  auto gprime = [](double x) { return M_PI * cos(M_PI * x); };

  auto root = root_finder_newton(f, fprime, 3.0, 1e-100, 100);
  std::cout << std::setprecision(15) << root << std::endl;

  root = root_finder_newton(g, gprime, 4.75, 1e-100, 100);
  std::cout << std::setprecision(15) << root << std::endl;
}

Output from the lines above

1.73205080756888

5

explanation of output: The first line is the root of \(f(x)\) starting at \(3\).

The second line is the root if \(g(x)\) starting at \(4.75\)

Implementation/Code: The following is the code for root_finder_newton

template <typename T, typename F, typename Fprime>
T root_finder_newton(F f, Fprime fprime, T x0, T tol, const int MAX_ITER = 100)
{
  T x1;

  for (auto i = 0; i < MAX_ITER; ++i)
  {
    x1 = x0 - f(x0) / fprime(x0);
    if (std::abs(x1 - x0) < tol * std::abs(x1))
    {
      break;
    }
    x0 = x1;
  }

  return x1;
}

Last Modified: September 2018