One of the most famous methods for solving non-linear equations is the Newton-Raphson method. The Newton-Raphson method is a kind of open method which employs Taylor series for estimation the position of the root. For arbitrary function f(x), the Taylor series around a stsrting point can be written as follows:
where is a point between x_{i} and x_{i+1}. If we neglect the second order term, the linear approximation of f(x) around x_{i} is as follows:
As it is shown in Fig. 1 the new guess of the root can be found from following equation:
or
Fig. 1. The Newton-Raphson method
The convergence speed of the Newton-Raphson method is very high in the most of cases. Using Taylor series, it can be shown that the error is approximately proportional to the square of the previous error. In other words the Newton-Raphson method is qudraticly convergent. The algorithm of the Newton-Raphson method is very simple as follows:
Step 1: Choose x_{0} as a starting point. Step 2: Let Step 3: if |x-x_{0}|<e then let root = x, else x_{0} = x ; go to step 2. Step 4: End.
e: Acceptable approximated error. Roots of Equations
(Source Code in C++)
o One point Interpolation Method |