% Written by Jiwon Seo (jwseo@stanford.edu, jiwons@gmail.com)
% for EE 265 class (2/22/2008)

% Press F5 to run this code.  
% F5 is one of the most useful shortcuts in Matlab.

function SolveByBisection()
% This code is for solving a general 'f(x) == 0' equation
% using the Bisection method, which is usually taught in the
% first class of Numerical Analysis.
%
% After learning many sophisticated algorithms in the Numerical 
% Analysis class during my undergrad in Aerospace Engineering 
% at KAIST (Korea Advanced Institute of Science and Technology),
% this simple Bisection algorithm has been my favorite for
% about 10 years in many practical situations.
%
% This algorithm is simple and slow but numerically stable, 
% so you can always find the solution if you selected 
% search interval correctly.  Slowness is not a problem
% in most cases because computers are so fast these days
% and you can get a solution within a second anyway. :)

clc;        % Clear screen.
close all;  % Close all existing figures.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Define your search interval.
searchFrom = 5;
searchTo = 30;

% This value determines the resolution of your solution.
delta = 0.00001;  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Plot y = f(x) graph in your search interval and check
% zero crossings, which helps you to figure out if your 
% interval is adequately chosen.
%
% If there's no zero crossing in your search interval,
% your final solution will be incorrect.  You may want 
% to stop running by pressing Ctrl+C if this function is 
% running more than several seconds.  
% Multiple zero crossings should also be avoided, but you 
% can still find all solutions one by one. :)

x = linspace(searchFrom, searchTo, 1000);
y = f(x);
plot(x, y);   hold on;
x = [searchFrom, searchTo];    y = [0, 0];
plot(x, y, 'r');
xlim([searchFrom, searchTo]);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Solve the equation.  You don't need to change this part.
% Just change the function definition at the end of this file.
while abs(searchFrom - searchTo) > delta
    middlePoint = (searchFrom + searchTo)/2;

    f_searchFrom = f(searchFrom);
    f_middlePoint = f(middlePoint);

    if (f_searchFrom * f_middlePoint) <= 0
        searchTo = middlePoint;
    else 
        searchFrom = middlePoint;
    end
end

format long;
answer = searchFrom
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Many engineers don't know how to make multiple functions
% within one 'M file', so their folders usually have 
% a lot of small M files with single function. -_-;
%
% Matlab does provide a smarter way to help structured
% programming and you should know about this if you're
% educated at Stanford!! (B university in Bay Area does
% not necessarily teach this. :P)
%
% The following function is valid within this M file only,
% which is what we exactly want.
% However, your main function should start with a proper
% function definition as this file, otherwise you can't 
% define multiple functions in one M file.  
% That's the key point!

function y = f(x)
% Equation should be in the form of 'f(x) == 0'.
% The output 'y' has to be real value.

% The input 'x' could be a vector, so you should use
% .^ operation instead of ^ operation.
y = 0.456 * (x-15).^3 + 0.786*(x-10) - 3.685;