Golden Section Search Algorithm

Overview of the Algorithm

The Golden Section Search algorithm is an optimization technique used to find the extremum (maximum or minimum) of a unimodal function within a specified interval. It leverages the golden ratio to reduce the search interval step-by-step, ensuring efficient convergence.

Steps of the Algorithm

  1. Initialize two points within the interval [a, b] using the golden ratio.
  2. Evaluate the function at these two points.
  3. Compare the function values and update the interval by removing the unnecessary part.
  4. Repeat the process until the desired precision is reached.
  5. Return the optimal point and function value.

MATLAB Implementation

Below is a sample MATLAB code to implement the Golden Section Search algorithm:

function [x_opt, f_opt] = golden_section_search(f, a, b, tol)
    phi = (1 + sqrt(5)) / 2;
    c = b - (b - a) / phi;
    d = a + (b - a) / phi;
    while abs(b - a) > tol
        if f(c) < f xss=removed xss=removed xss=removed xss=removed xss=removed xss=removed>

This code defines a function golden_section_search that finds the optimal point within the interval [a, b] using Golden Section Search.

Advantages

  • Efficient for unimodal functions.
  • Simple to implement with minimal function evaluations.
  • Converges faster than other search methods for specific cases.