Find The Minimizer Of a Function Using The Bisection Method

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  1. securityhope

    securityhope Administrator Staff Member

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    The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method.

    Minimization with the Bisection Method

    Assume that a single-variable continuous function has a unique minimum (and, thus, a unique minimizer) in a closed interval [a, b]. If this function is differentiable in [a, b], then calculus can be used easily to find the minimizer. However, if the function has a cusp or a kink, then it’s not differentiable at that point, and numerical methods are needed instead. For example, consider the following function.

    cusped-function.png
    Within the interval [1, 3], this function has a unique minimizer near x = 1.5, but it also has a cusp at x = 2. Thus, it is not differentiable at x = 2. A simple, stable, but slow numerical method to find the minimizer is the bisection method.

    https://chemicalstatistician.wordpr...-the-golden-ratio-for-numerical-optimization/
     

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