Newton-Raphson iteration

Define the function  f(x) = cos(x) - x

The equation  f(x) = 0   cannot be solved by mathematical analysis. In order to solve an equation like this we have to use a numerical method, typically on a computer. An often used method is called Newton-Raphson iteration.

The principle is this:

• guess an x-value called x₁ which you expect to be near the solution.
• calculate   y₁ = f(x₁)
• calculate the derivative   y₁' = f'(x₁) = -sin(x₁) - 1.
• approximate the function f(x) with the straight line which is the tangent of the curve y = f(x) in the point  (x₁,y₁).  The tangent crosses the x-axis at the point
  x₂ = x₁ - y₁ / y₁'
• now x₂ is (hopefully) nearer the solution than x₁.
  Calculate   x₃ = x₂ - y₂ / y₂'  and so on until the solution is as accurate as you desire.
• stop with an error message if the x-values do not converge towards a solution.

Let your program write out   x₁, y_1,   x₂, y₂,   x₃, y₃,   and so on.   Try with different values for the initial guess, x₁, and see if it converges towards the solution.