% Script 3 for Week 2 

% Example of incremental search in action.

f2=@(x) exp(x)-tan(x);
f1=@(x) x.^2-8;
incsearch(f1,-3,3)
incsearch(f1,ans(1,1),ans(1,2))

% Below are the contents of the varlist.m and testvarargin.m function files, used it to test our understanding of the varargin cell array.

%*******************************
function varlist(varargin)
fprintf('\n Number of arguments: %9d \n',nargin)
celldisp(varargin)
%******************************

varlist(ones(3),'some text',pi)

%******************************
function [x a b c]=testvarargin(x, varargin)
a=varargin{1};
b=varargin{2};
c=varargin{3};
fplot(@(t)a*sin(b*t+c),[-pi,pi])
end
%******************************

testvarargin(3, 4, 5, 6)

[x a b c]=testvarargin(3, 4, 5, 6)


% More on fixed-point iteration.

% This is a solution method producing a sequence of approximate solutions that MIGHT converge to a real solution.

% Suppose want to find x where exp(-x)-x = 0.

% So x=exp(-x)

% Let's try a quick plot so we can estimate the solution from that.

t=-1:.1:2;
plot(t,exp(-t),t,t)

% Now define a function whose fixed point is our desired solution.

g=@(x) exp(-x);
x=5
x=g(x)

% We want to know where g(x)=x.

% Form a sequence starting with some seed value: x_0=seed value.
% Iterate x=g(x) by hitting the up-arrow and return repeatedly.

% x_1=g(x_0)
% x_2=g(x_1)
% .
% .
% .
% x_{n+1}=g(x_n).

% Stop this procedure when ea=abs(  (x(i+1)-x(i))/x(i+1)  )
% is less than a designated target value (if it ever is less.)

% Often the sequence converges and when it does it converges to the  x value with x=g(x).

% If it doesn't YOU CAN OFTEN REFORMULATE THE ORIGINAL EXPRESSION and create an equivalent NEW g that does work. You will prefer a g with a smaller derivative (less than magnitude 1) near the fixed point.

% Now on to some 3-d graphing.

% meshgrid lets you avoid writing a little 
% loop to calculate z values in a surface plot.

% For a square domain you can use

[x,y] = meshgrid([-2:.2:2]);
Z = x.*exp(-x.^2-y.^2);
surf(x,y,Z)
colorbar

% For a more general rectangular domain you use


[x,y] = meshgrid([-2:.2:2],[-1:0.2:1]);
Z = x.*exp(-x.^2-y.^2);
surf(x,y,Z)
colorbar


X2=[-2:.2:2]; Y2=[-2:.2:2];

for i=1:length(X2)
for j=1:length(Y2)
Z2(i,j)=X2(i)*exp(-X2(i)^2-Y2(j)^2);
end
end

figure(7)
surf(X2,Y2,Z2)
colorbar


% Testing the Newton-Raphson script:

y = @(m) sqrt(9.81*m/0.25)*tanh(sqrt(9.81*0.25/m)*4)-36;
dy = @(m) 1/2*sqrt(9.81/(m*0.25))*tanh((9.81*0.25/m) ...
^(1/2)*4)-9.81/(2*m)*sech(sqrt(9.81*0.25/m)*4)^2;
newtraph(y,dy,140,0.00001)