Bird-Like Function

I. Mathematical Expression:

$$ f(x)=\frac{2x^4+x^2+2}{x^4+1}$$

where:

\(\bullet\) \(-4 \leq x \leq 4\)

\( \rhd\) For Minimization Mode:

\(\bullet\) It has one global minima: \(f_{min}(x^*)=2\)

\(\bullet\) \(x^*=2\)

\( \rhd\) For Maximization Mode:

\(\bullet\) It has two global maxima: \(f_{max}(x^*)=2.5\)

\(\bullet\) \(x^*=\{-1,1\}\)

II. Citation Policy:

If you publish material based on databases obtained from this repository, then, in your acknowledgments, please note the assistance you received by using this repository. This will help others to obtain the same data sets and replicate your experiments. We suggest the following pseudo-APA reference format for referring to this repository:

Ali R. Al-Roomi (2015). Unconstrained Single-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/unconstrained]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2015,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Single-Objective Benchmark Functions Repository}},
year = {2015},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/unconstrained}
}

III. 2D-Plot:

IV. MATLAB M-File:

% Bird-Like Function
% Range of initial points: -4 <= x <= 4
% One global minima: x=2 and two global maxima: x={-1,1}
% fmin(x)=2 and fmax(x)=2.5
% Coded by: Ali R. Alroomi | Last Update: 17 May 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=-4;
xmax=4;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=(2*x(i)^4+x(i)^2+2)/(x(i)^4+1);
end
 
plot(x,f,'r','LineWidth',2);grid;set(gca,'FontSize',12);axis([-4 4 -0.5 3.5]);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');

Click here to download m-file

V. References:

[1] SageMath, "Calculus Tutorial," [Accessed July 08, 2015]. [Online]. Available: http://www.sagemath.org/calctut/inflimits.html
[2] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications