% Exercise 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear
x=2.5e-3
y=2.9e3
z=x/y
format long
z
format short
% answer
% x =
%    0.0025
% y =
%        2900
% z =
%   8.6207e-07
% z =
%     8.620689655172414e-07
%
input('press return to continue: ')
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% Exercise 2
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clear
t=0:0.2:5;
y=exp(-sin(t));
plot(t,y,'r')
hold on
z=exp(-cos(t));
plot(t,z,'b+')
set(gca,'FontName','Helvetica','FontSize',16)
title('two functions (y and z)')
xlabel('time')
ylabel('function value')
hold off
input('press return to continue: ')
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% Exercise 3
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clear
[root1,root2]=quadratic(1,-3,2);
disp('roots of quadratic are: ')
disp([root1 root2])

[root1,root2]=quadratic(1,3,3);
disp('roots of quadratic are: ')
disp([root1 root2])


% roots of quadratic are: 
%     2     1
%
% roots of quadratic are: 
%  -1.5000 + 0.8660i  -1.5000 - 0.8660i
input('press return to continue: ')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%     quadratic.m
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function [root1,root2]=quadratic(a,b,c)
%
%   a,b,c are input arguments
%   root1, root2 are returned by the function
%   and contain the roots of the quadratic
%   a x^2 + b x + c
%
    temp = sqrt(b^2-4*a*c );
    root1 = (-b + temp)/(2*a);
    root2 = (-b - temp)/(2*a);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exercise 4
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%
% plot a circle, radius 3, centre (1,2)
%
theta=0:0.01:2*pi;
radius=3;
xcen=1;
ycen=2;
x=xcen+radius*cos(theta);
y=ycen+radius*sin(theta);
plot(x,y)
axis equal % preserves correct shape
%%%%%%%%%%%%%%%%%
%
% (a) plot the ellipse x^2/4+y^2/9=1
%
theta=0:0.01:2*pi;
x=2*cos(theta);
y=3*sin(theta);
plot(x,y)
axis equal

% (b) filled ellipse

patch(x,y,'r')
hold on
plot(x,y,'r')   % makes boundary of ellipse same colour as interior

% (c) a square centred on the origin, side 1

x=[0.5 0.5 -0.5 -0.5 0.5];
y=[-0.5 0.5 0.5 -0.5 -0.5];
plot(x,y)
axis equal

% (d) regular hexagon, centred on the origin

cx=1/2;
cy=sqrt(3)/2;
x=[cx 0 -cx -1 -cx 0 cx 1 cx];
y=[cy cy cy 0 -cy -cy -cy 0 cy];
plot(x,y)
axis equal
axis([-1.2 1.2 -1.2 1.2]) %leaves clear area around the figure


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exercise 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear
t=[0:0.1:3];                    % a vector, elements 0, 0.1, 0.2, ...

constant = pi                    
temp1 = size(t)                 % temp1 gives the size of the vector t
temp2 = ones(temp1)             % temp2 is a vector of ones, with the same
                                % dimension as t
temp3 = pi*temp2                % the vector has all elements with value pi
plot(t,temp3)

%  in two lines:
%  t=[0:0.1:3];
%  plot(t,pi*ones(size(t)))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exercise 6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear

f1 = factorial(5)
f2 = factorial(1)
f3 = factorial(0)
f4 = factorial(-1)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    factorial.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function y = factorial(n)
%
% function returns y = n!
% for integer n >= 0
%
if n< 0 
 disp ( ' error in factorial.m: n is less than zero')
 y = [];
elseif n == 0
  y = 1;
else
  y = 1;
  for j = 1:n               % loop can run from 2 i.e. for j=2:n
    y = j*y;
  end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that in fact there is a factorial function
built in to MATLAB that is about 3 times faster than
the code given above (it uses the in-built routine
prod.m)

It is also possible to code the factorial function
recursively, but this is about 10 times slower
than the version given above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exercise 7
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% fibonacci.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function f = fibonacci(n)
%
% function returns the nth number in the Fibonacci sequence
%
if n == 0
  f = 0;
else
  y(1) = 1;
  y(2) = 1;
  for j=3:n
    y(j) = y(j-1)+y(j-2);   
  end
  f = y(n);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Using the format long switch we find:

fibonacci(39)/fibonacci(38) = 1.61803398874990
fibonacci(40)/fibonacci(39) = 1.61803398874989
fibonacci(41)/fibonacci(40) = 1.61803398874989


(1+sqrt(5))/2 = 1.61803398874989

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