Advanced MATLAB
programming
Morris Law
Jan 19, 2013
1
Outline
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Advance programming in MATLAB
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MATLAB toolboxes
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Error analysis using Hilbert matrix
Solving non-linear equations
Function approximation using Taylor's expansion
Solving ordinary differential equations
Simulink
Image processing
MATLAB GUI
MATLAB for fun
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Error analysis using Hilbert matrix
 Hilbert matrix is an NxN matrix with element (1/i+j-1).
It is a typical example of ill conditioned matrix.
 hilb(5) in MATLAB will give 5x5 Hilbert matrix
1
1
2
 13
1
4
1
5
1
2
1
3
1
4
1
3
1
4
1
5
1
4
1
5
1
6
1
5
1
6
1
7
1
6
1
7
1
8

1
6
1
7

1
8
1
9
1
5
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Error analysis using Hilbert matrix
Try the following MATLAB code (hilbtest.m)
n = 5;
H = hilb(n);
x = [1:n]’;
b = H * x;
xx = H \ b ;
error = x – xx
condH = cond(H)
Change n to 10, 15, 20 to test the error
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Solving non-linear equations
 The following non-linear equations are considered.
y  sin( x) *sin( x)  2 x  x
2
y  e  2 x  cos( x)
x
2
y  1/ (1  x )  0.5
2
 Above equations are stored in testfun.m.
 Consider one equation for each test.
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Solving non-linear equations
 You may write your own solver using bisection method.
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Refer to nonlineq.m
 Or you may solve it simply for matlab function fzero.
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Refer to solvefun.m in the following
function x0 = solvefunc % solve nonlinear function
x = [-5:0.02:5];
y = testfun(x);
hold on;
plot(x,y);
plot(x,zeros(size(x)),'r');
hold off
x0=fzero('testfun',1); % solve with initial guess 1
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Function approximations using
Taylor’s expansion
 The Taylor series of f(x)
xa
( x  a)2
''
f ( x)  f (a )  f (a )
 f (a)

1!
2!
'
( x  a)n
 f (a)

n!
n
 sin(x) can be approximated by expanding in Taylor
series with f(x)=sin(x) at a=0
x3 x5
sin( x)  x   
3! 5!
2 n 1
x
 (1)n

(2n  1)!
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Function approximations using
Taylor’s expansion
 The MATLAB code for sinappx.m can be written as,
%approximate sin by taylor polynomial
function y=sinappx(x)
term=x; y=term; x2=x*x; n=1;
%initialization
while abs(term)>eps
%have we summed enough yet?
term=-term*x2/((n+1)*(n+2));
%update term
y=y+term;
%update sum
n=n+2;
%update n
end
 Similarly you can modify the above to write cosappx.m
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Solving ordinary differential equations
 Ordinary differential equation can be solved
in MATLAB using functions like ode23 or
ode45.
 For first order ODE, simply prepare a function
y’ = f(x) and name it as yprime.m, then solve
by [t,y]=ode23(@yprime, tspan,y0)
 Plot the graph to show the solution.
 For higher order ODE, rewrite it into a system
of first order ODE and solve similarly.
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Solving ordinary differential equations
 To solve y’ - y – t = 0, the matlab code
yprime.m should be like,
function yp = yprime(y,t)
yp = y + t;
 Solve it by
 tspan=[0,10];
 y0 = 0;
 [t,y] = ode23(@yprime,tspan,y0)
 Plot the graph t vs y to show the solution
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plot(t,y)
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Solving ordinary differential equations
 Another example to solve a 2nd order ODE, y'' + y' + y
+ t = 0, the matlab code yprime1.m should be like,
function yp=yprime1(y,t)
yp(1) = y(2);
yp(2) = -y(2) - y(1) -t;
 Solve it by
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tspan=[0,10];
y0 = 0;
[t,y] = ode23(@yprime1,tspan,y0)
 Plot the graph t vs the first and second column of y to
show the solution y and y’
plot(t,y(*,1),y(*,2))
 Refer to forfun/orbit.m to solve an orbit trajectory
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using ode23.
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MATLAB Toolboxes
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Toolboxes add more functions and feature into MATLAB. You may also write own toolboxes.
OEE501 has another MATLAB classroom license with the following toolboxes,
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Simulink
Control System Toolbox
Image Processing Toolbox
Signal Processing Toolbox
Simulink control design Toolbox
Statistics Toolbox
Science Faculty has subscribed the following toolboxes,
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Simulink
Control System Toolbox
Image Processing Toolbox
MATLAB Compiler
Neural Network Toolbox
Optimization Toolbox
Partial Differential Equation Toolbox
Signal Processing Toolbox
Simulink Control Design
Spline Toolbox
Statistics Toolbox
Symbolic Math Toolbox
System Identification Toolbox
Wavelet Toolbox
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Simulink Toolbox
• Provide a simulation environment for common discrete/continuous system
• Invoke by typing ‘simulink’ in command windows
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Image Processing Toolbox
 Functions specialised for image processing
such as imread, imshow, imadjust, imhist,
histeq, ……
 Support almost all image format input and
output.
 RGB vs index vs BW images
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Examples
 Counting grain from microscopic images
%
% Demo for functions in image processing toolbox
% imgdemo.m
%
I=imread('rice.png');
%J=imread('demo.jpg');
%I=rgb2gray(J);
imshow(I)
background = imopen(I,strel('disk',15));
figure, surf(double(background(1:8:end,1:8:end))),zlim([0 255]);
set(gca,'ydir','reverse');
I2 = I - background;
imshow(I2)
I3 = imadjust(I2);
imshow(I3);
level = graythresh(I3);
bw = im2bw(I3,level);
bw = bwareaopen(bw, 50);
imshow(bw);
cc = bwconncomp(bw, 4);
grain = false(size(bw));
grain(cc.PixelIdxList{8}) = true;
figure;imshow(grain)
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MATLAB GUIDE
 Development Environment for Graphical User
Interface
 Invoke with ‘guide’ in command window
 Plenty of user interface like button, textbox,
scrollbar to develop screen interfaces.
 Callback functions can be written for each
graphics object.
 Save and load the GUI as figures
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MATLAB for fun
 Check machine constants
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machine.m
 Plot circle using parametric equation
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circle.m, drawpattern.m
 Plot graph by loading data from file
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dataplot.m, grid.plt
 tic-tac-toe game written in MATLAB
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play.m
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Thank you!
For enquiry: send e-mail to
morris@hkbu.edu.hk
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