Table of Contents

Root Finder Secant Method Software Manual

This is a template file for building an entry in the student software manual project. You should use the formatting below to define an entry in your software manual.

Routine Name: root_finder_secant

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 ./secant.out that can be executed.

Description/Purpose: This routine will find the root of a function, \(f\), using two initial guesses \(x0\) and \(x1\) using the secant method of root finding

Input: There are five needed inputs, a function, two initial guesses, a tolerance and the maximum number of iterations.

@tparam T       The type of x0 and tolerance
@tparam F       A function of type T(T)
@param x0       The first initial guess
@param x1       The second initial guess
@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 g = [](double x) { return sin(M_PI * x); };

  auto root = root_finder_secant(f, 4.0, 5.5, 1e-10);
  std::cout << std::setprecision(20) << root << '\n';

  root = root_finder_secant(g, 3.5, 6.5, 1e-10);
  std::cout << std::setprecision(20) << root << '\n';
}

Output from the lines above

1.7320508075688771932

5

explanation of output:

The first line is the root of \(f(x)\) starting with guesses \(4.0\) and \(5.5\).

The second line is the root of \(g(x)\) starting with guesses \(3.5\) and \(6.5\).

Implementation/Code: The following is the code for root_finder_secant

template <typename T, typename F>
T root_finder_secant(F f, T x0, T x1, T tol, const int MAX_ITER = 100)
{
  T x2, fx1;
  for (auto i = 0; i < MAX_ITER; ++i)
  {
    fx1 = f(x1);
    x2 = x1 - fx1 * (x1 - x0) / (fx1 - f(x0));
    if (std::abs(x1 - x0) < tol * std::abs(x1))
    {
      return x2;
    }
    x0 = x1;
    x1 = x2;
  }
  return x1;
}

Last Modified: September 2018