1. Scientific Computing with MATLAB
Morris Law
Apr 11, 2018

2. Outline Advance programming in MATLAB MATLAB toolboxes MATLAB for fun
Error analysis using Hilbert matrix
Solving non-linear equations
Function approximation using Taylor's expansion
Solving ordinary differential equations
MATLAB toolboxes
Simulink
Image processing
MATLAB GUI
MATLAB for fun

3. 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

4. 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

5. Solving non-linear equations
The following non-linear equations are considered.
Above equations are stored in testfun.m.
Consider one equation for each test.

6. Solving non-linear equations
You may write your own solver using bisection method.
Refer to nonlineq.m
Or you may solve it simply for matlab function fzero.
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

7. Function approximations using Taylor’s expansion
The Taylor series of f(x)
sin(x) can be approximated by expanding in Taylor series with f(x)=sin(x) at a=0

8. 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

9. 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 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.

10. 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] =
Plot the graph t vs y to show the solution
plot(t,y)

11. 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
tspan=[0,10];
y0 = 0;
[t,y] =
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 using ode23.

12. MATLAB Toolboxes
Toolboxes add more functions and feature into MATLAB. You may also write own toolboxes.
OEE501 has another MATLAB classroom license with the following toolboxes,
Simulink
Control System Toolbox
Image Processing Toolbox
Signal Processing Toolbox
Simulink control design Toolbox
Statistics Toolbox
Science Faculty has subscribed the following toolboxes,
MATLAB Compiler
Neural Network Toolbox
Optimization Toolbox
Partial Differential Equation Toolbox
Simulink Control Design
Spline Toolbox
Symbolic Math Toolbox
System Identification Toolbox
Wavelet Toolbox

13. Simulink
Simulink is a great toolbox for building Mathematical Method and performing Simulation
Plenty of block diagrams mimics useful functions, data generations and outputs
Invoke using >> simulink

14. 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

15. 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)

16. 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

17. MATLAB for fun Check machine constants machine.m
Plot circle using parametric equation
circle.m, drawpattern.m
Plot graph by loading data from file
dataplot.m, grid.plt
tic-tac-toe game written in MATLAB
play.m
MATLAB Mobile
Download from App Store and Play Store

18. For enquiry: send e-mail to morris@hkbu.edu.hk
Thank you! Please give me some comment on the questionnaire :
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