Saturday, April 25, 2009

Numerical Methods - The Newton-Raphson Method to Solve Mechanical Design Problems Part I

Machine designers have to deal with several number of equations in their design projects. There are a number of ways to solve for roots of algebraic and transcendental equations. Sometimes, the roots could be obtained by direct methods. However, there are many more that could not. The classical equation such as f(x) = e-x - x cannot be solved analytically. For this case, the only alternative is an approximate solution technique.

There are several methods available to solve the root finding problem such as "bracketing methods" and "Open methods".

The bracketing methods require 2 initial guesses for the root. These guesses must "bracket" the root. The numerical methods using bracketing methods consist of the following techniques:
  • The Bisection Method: The idea of this technique is incremental search that related to the sign change. Sometimes, this technique is called binary chopping, or Bolzano's method.
  • The False-Position Method: It's the improved technique from the bisection methods. It replaces the curve f(x) by a straight line and gives a false-position of the root. The false-position method is also called the linear interpolation method.
However, my favorite technique in root finding method is not the bracketing methods but the open methods.

The bracketing methods are said to be convergent. In contrast, the open methods are based on formulas that require a single starting value of x. They sometimes diverge. However, when the open methods converge they usually do so much more quickly than the bracketing methods.

The followings are the root finding techniques using open methods.
  • Simple One-Point Iteration
  • The Newton-Raphson Method: The most widely used of all root-locating formulas. The Newton-Raphson method uses the slope (first derivative) of the function to find the root. It's my favorite one.
  • The Secant Method: Instead of using the first derivative of the function to find the slope as in The Newton-Raphson Method, the first derivative in secant method can be approximated by a finite divided difference.
In the next post, let's see how we can use Microsoft Excel VBA again to solve the root finding problems using the Newton-Raphson Method. I'll show you the simple VBA code that you can copy & paste into your excel file and use it as the module to calculate for the root of your equation.

No comments: