## The Golden Section Search Method: Modifying the Bisection Method with the Golden Ratio for Numerical Optimization

#### Introduction

The first algorithm that I learned for root-finding in my undergraduate numerical analysis class (MACM 316 at Simon Fraser University) was the bisection method.  It’s very intuitive and easy to implement in any programming language (I was using MATLAB at the time).  The bisection method can be easily adapted for optimizing 1-dimensional functions with a slight but intuitive modification.  As there are numerous books and web sites on the bisection method, I will not dwell on it in my blog post.

Instead, I will explain a clever and elegant way to modify the bisection method with the golden ratio that results in faster computation; I learned this method while reading “A First Course in Statistical Programming with R” by John Braun and Duncan Murdoch.  Using a script in R to implement this special algorithm, I will illustrate how to minimize a non-differentiable function with the golden section search method.  In a later post (for the sake of brevity), I will use the same method to show that the minimizer of the sum of the absolute deviations from a univariate data set is the median.  The R functions and script for doing everything are in another post.

The Fibonacci spiral approximates the golden spiral, a logarithmic spiral whose growth factor is the golden ratio.

Source: Dicklyon via Wikimedia

## My Own R Function and Script for Simple Linear Regression – An Illustration with Exponential Decay of DDT in Trout

Here is the function that I wrote for doing simple linear regression, as alluded to in my blog post about simple linear regression on log-transformed data on the decay of DDT concentration in trout in Lake Michigan.  My goal was to replicate the 4 columns of the output from applying summary() to the output of lm().

To use this file and this script,

1) I saved this file as “simple linear regression.r”.

2) In the same folder, I saved a script called “DDT trout regression.r” that used this function to implement simple linear regression on the log-transformed DDT data.

3) I used setwd() to change the working directory to the folder containing the function and the script.

4) I made sure “DDT trout regression.r” used the source() function to call my user-defined function for simple linear regression.

5) I ran “DDT trout regression.r”.