How to Perform Monte Carlo Integration in MATLAB | MATLAB Tutorial

How to do a Monte Carlo Integral in MATLAB! Covers the theory behind the numerical method and integration while presenting a program to execute montecarlo integration in MATLAB!

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CHAPTERS
0:00 Goal
0:07 Monte Carlo Theory
3:11 Setting up Code
5:24 Generate and Assign Points
9:35 Clean the Output
11:51 Plotting the Solution
12:26 Integral Calculation
13:18 MATLAB's integral()
13:51 Percent Error
14:25 Applying Script to Other Integrals

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--------------------------FREE CODE FROM VIDEO ------------------------------------------------------------------------------------
% MonteCarlo - Integral Calculation
clc, clearvars, close all, format compact

% Equation to integrate over. Integral of this from a to b.
f = @(x) exp(x).*x.^2.*sqrt(exp(x));

% Parameters
N = 10000; % number of points to generate randomly
a = 2; % integral lower x limit - manual
b = 4; % integral upper x limit - manual
M = 1.4*max(f(linspace(a,b))); % upper y bound = c*max y_value from equation

% Generate Dots
for i = 1:N

% generate random pt
x_val = rand(1)*(b-a) + a;
y_val = rand(1)*M;

% compare random against the curve
fx = f(x_val);

% logic statement
if y_val REPLACE_WITH_LESS_THAN_SYMBOL fx % under the curve - blue
under(i,1) = x_val;
under(i,2) = y_val;
else % above the curve - red
above(i,1) = x_val;
above(i,2) = y_val;
end
end

% Filter out zeros
under2(:,1) = nonzeros(under(:,1));
under2(:,2) = nonzeros(under(:,2));
above2(:,1) = nonzeros(above(:,1));
above2(:,2) = nonzeros(above(:,2));

% Plotting
plot(above2(:,1),above2(:,2),'ro','MarkerFaceColor','r')
hold on
plot(under2(:,1),under2(:,2),'bo','MarkerFaceColor','b')
title('Monte Carlo Integration'), xlabel('x'), ylabel('y')
legend('above','under')

% Integral Calcs
MonteCarlo_Integral = length(under2) / N * (M*(b-a))
MATLAB_Integral = integral(f,a,b)
PercentError = abs(MATLAB_Integral - MonteCarlo_Integral)/MATLAB_Integral * 100


-------------------------VECTORIZED VERSION OF CODE-------------------------------------------------------------
% Monte Carlo Integration
% Vectorized Version - shorter and faster than what is shown in the video
clc, clearvars, close all, format compact

% This method optimizes performance over simplicity
% Dots are not colored red and blue

tic
f = @(x) 10 + 5*sin(5*x);

% Parameters
N = 10000;
a = 2; b = 4;
M = 1.4*max(f(linspace(a,b)));

% Generate Random Points
x = rand(1,N)*(b-a) + a;
y_val = rand(1,N)*M;

% Perform Integral Calculation
fx = f(x);
PercentUnderCurve = sum(y_val REPLACE_WITH_LESS_THAN_SYMBOL fx) / N;
Monte_Integral = PercentUnderCurve * M * (b-a)
Matlab_Integral = integral(f,a,b)
PercentError = abs(Monte_Integral-Matlab_Integral)/Matlab_Integral*100
toc

% Plot
plot(x,y_val','.')
hold on
plot(x,fx,'.')
----------------------------------------------------------------------------------------------------------------------

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