Steffensen's Method - Advantages and Drawbacks

Advantages and Drawbacks

The main advantage of Steffensen's method is that it has quadratic convergence like Newton's method – that is, both methods find roots to an equation just as 'quickly'. In this case quickly means that for both methods, the number of correct digits in the answer doubles with each step. But the formula for Newton's method requires a separate function for the derivative; Steffensen's method does not. So Steffensen's method can be programmed for a generic function, as long as that function meets the constraints mentioned above.

The price for the quick convergence is the double function evaluation: both and must be calculated, which might be time-consuming if is a complicated function. For comparison, the secant method needs only one function evaluation per step, so with two function evaluations the secant method can do two steps, and two steps of the secant method increase the number of correct digits by a factor of 1.6 . The equally time-consuming single step of Steffensen's (or Newton's) method increases the correct digits by a factor of 2 – only slightly better.

Similar to Newton's method and most other quadratically convergent algorithms, the crucial weakness in Steffensen's method is the choice of the starting value . If the value of is not 'close enough' to the actual solution, the method may fail and the sequence of values may either flip flop between two extremes, or diverge to infinity (possibly both!).

Read more about this topic:  Steffensen's Method

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