The golden section search method, an iterative numerical method for finding the minimum or maximum of a unimodal function, involves dividing an interval into two subintervals with the ratio of the lengths of the subintervals equal to the golden ratio. This method is closely related to the Fibonacci sequence, the golden ratio, the golden rectangle, and the golden angle.
Golden Section Search Method: An Optimal Search Algorithm
The golden section search method is a technique used to find the minimum of a unimodal function. It is an iterative algorithm that converges to the minimum point with quadratic convergence rate. The golden section search method is based on the principle of the golden ratio, which is a special number approximately equal to 1.618.
Algorithm
The golden section search method starts with an interval [a, b] that contains the minimum point. The interval is then divided into two subintervals, [a, c] and [c, b], such that the ratio of the lengths of the subintervals is equal to the golden ratio. The point c is given by:
c = a + (b - a) * (1 - sqrt(5)/2)
The function is then evaluated at the points c and d, where d is given by:
d = a + (b - a) * sqrt(5)/2
If the function value at c is less than the function value at d, then the interval [a, c] is retained, and the interval [c, b] is discarded. Otherwise, the interval [a, c] is discarded, and the interval [c, b] is retained.
This process is repeated until the interval [a, b] is sufficiently small. The minimum point is then estimated as the midpoint of the interval [a, b].
Convergence
The golden section search method converges to the minimum point with a quadratic convergence rate. This means that the error in the estimated minimum point is reduced by a factor of approximately 1.618 with each iteration.
Advantages
The golden section search method has several advantages:
- It is a simple and easy to implement algorithm.
- It converges to the minimum point with a quadratic convergence rate.
- It does not require any derivatives of the function.
Disadvantages
The golden section search method also has some disadvantages:
- It can be slow for functions that are not unimodal.
- It can be sensitive to the initial interval [a, b].
Applications
The golden section search method is used in a variety of applications, including:
- Optimization of functions
- Root finding
- Numerical integration
- Data fitting
Example
The following table shows the steps of the golden section search method for finding the minimum of the function f(x) = x^2 – 1.
| Iteration | Interval [a, b] | Point c | Point d | Function value at c | Function value at d | Retained interval |
|---|---|---|---|---|---|---|
| 1 | [-1, 2] | 0.382 | 1.618 | -0.618 | -0.091 | [0.382, 2] |
| 2 | [0.382, 2] | 1.091 | 1.618 | -0.182 | -0.091 | [1.091, 2] |
| 3 | [1.091, 2] | 1.409 | 1.618 | -0.009 | -0.091 | [1.409, 2] |
| 4 | [1.409, 2] | 1.554 | 1.618 | -0.001 | -0.091 | [1.554, 2] |
| 5 | [1.554, 2] | 1.611 | 1.618 | 0.000 | -0.091 | [1.611, 2] |
The minimum point is estimated as the midpoint of the interval [1.611, 2], which is approximately 1.809.
Question 1:
What is the Golden Section Search Method?
Answer:
The Golden Section Search Method is an iterative optimization technique used to locate the minimum or maximum of a unimodal function. It is based on the Fibonacci sequence, where each subsequent term is the sum of the two preceding terms.
Question 2:
How does the Golden Section Search Method work?
Answer:
The Golden Section Search Method starts by creating an initial interval containing the potential minimum or maximum. It then iteratively divides the interval into two subintervals with a ratio of approximately 1.618:1, known as the golden ratio. The function is evaluated at the endpoints of the subintervals.
Question 3:
Why is the Golden Section Search Method considered efficient?
Answer:
The Golden Section Search Method is efficient because it converges quickly to the minimum or maximum with a relatively small number of iterations. This is due to the use of the golden ratio, which provides a near-optimal way to divide the interval at each iteration.
Hey there, nerds and math enthusiasts! Thanks a bunch for sticking with me on this wild exploration of the golden section search method. I hope it’s given you a better understanding of this fascinating mathematical tool. Don’t forget to check back later for more mind-boggling adventures into the world of numbers. Until then, keep those golden spirals and Fibonacci sequences dancing in your head!