Whittaker method. It is a method of numerical mathematics (therefore iterative), used for the numerical solution of linear equations. It is said to be a quasi- Newton method because of its similarities to it. The Newton-Raphson method has the advantage that its convergence is quadratic, therefore it is much faster than the linear one obtained with other methods such as the Bisection method .
Summary
[ hide ]
- 1 Description of the method
- 1 Disadvantages
- 2 Algorithm
- 1 In pseudocode
- 3 See also
- 4 Bibliography
Description of the method
Whittaker’s method solves the problem present in Newton’s method since only the derivative of the function is evaluated for the first point and this value is made constant for the remaining iterations.
x_ (n + 1) = x_n – (f (x_n)) / m , where:
m = f ‘(x_0) , while Newton’s method approximates the solution in the form:
x_ (n + 1) = x_n – (f (x_n)) / (f ‘(x_n))
Disadvantages
The disadvantage of this method is that it is necessary to evaluate the derivative of the function as many times as iterations are performed, which is very costly from a computational point of view, and it is possible that the derivative will cancel or change its sign, which can cause the appearance of a horizontal tangent or an approximation very far from the root where the function could not even be defined.
The success of the Whittaker method lies in the selection of a good X0 that does not nullify m and ensures rapid convergence.
This method does not have a quadratic convergence like Newton’s but is always faster than the linear convergence of the Bisection method .
Algorithm
In pseudocode
- Enter X0
- Enter TE
- xprevious = x
- repeat
- x = xanterior – f (xanterior) / f ‘(xanterior)
- Error = | x – xprevious |
- xprevious = x
- until Error <TE
- The searched root is x and its maximum absolute error is Error
- End up
