May 30, 2005 Matlab Tutorial 1. Arrays and Matrices Row Vector - - PDF document

May 30, 2005 Matlab Tutorial

• 1. Arrays and Matrices
• Row Vector

r = [1, 2, 3, 4] or r =

• 1

2 3 4

• r =
• 1

2 3 4

• Column Vector

c = [1; 2; 3; 4] c =     1 2 3 4    

• Matrix

m = [1, 2, 3; 4, 5, 6; 7, 8, 9] m =   1 2 3 4 5 6 7 8 9  

• Transposes

ctrans = c′ ctrans =

• 1

2 3 4

• mtrans = m′

mtrans=

• Exercises

x1=zeros(3,1) x1= x2=zeros(1,3) x2= x3=zeros(3) x3= y1=ones(1,3) y1= 1

y2=ones(3,2) y2= y3=ones(3) y3= I=eye(3) I=

• 2. Operators
• Matrix operators: +, -, *, /, ˆ,\
• Array (element-by-element) operators: +, -, .*, ./, .ˆ
• Example

A = B = 1 2 3 4

• C = A ∗ B;

D = A. ∗ B C = D =

• Exercises

K =     1 2 3 4    , H =     1 2 3     and compute if you can the following and check your answers using MATLAB 1) 2 − 3 ∗ 2 2) (2 − 3) ∗ 2 3) K + 2 2

4) 2 − K 5) K − H 6) K ∗ H 7) K. ∗ H 8) K./H 9) K/2

• 3. .m-files
• Write programs or subprograms
• script - defined variables are global on the workspace
• function - defined variables are local (helps keep code easy to read and more portable)
• Programs (script m-file)

clear all; %clear all variables in MATLAB memory x=0:pi/12:2*pi; % x goes from 0 to 2π with the stepsize

π 12

f=sin(x);

• Programs (function m-file)

function output =exfunc1(input) 3

x=input;%save input as x f=sin(x);

• utput=f;

⋆ help function

• Exercises

1) Create a script m-file to calculate 2sin(x) for x ∈ [−2π, 2π] with the stepsize=.01 2) Create a function m-file to calculate f=bsin(x) for x ∈ [−2π, 2π] with the stepsize=.01 where the input is b and the output is f. Run the program for b=2,3,5.

• 4. Programming Syntax
• The ’for’ loop

N=1000; M=50; for i=1:N for j=1:M x(i,j)=i + j2; end end ⋆ help for

• The ’if’ statement

i=1; j=1; if i==j x=1; else x=100; 4

end

• Other Boolean Operators: ˜ (NOT); <=; >=; < ; >

⋆ help if

• 5. Statistical Tool Box

v=[1; 2; 3; 4]

• mean(v);
• var(v);
• std(v);
• 6. Graphical Results (Plotting)

x=0:pi/12:2*pi; % x goes from 0 to 2π with the stepsize

π 12

f=sin(x); g=cos(x);

• Plot the function against the values

plot(x,f,’r’);% plot sin(x) with a red line xlabel(’x’); % labeling the x-axis ylabel(’f’); % labeling the y-axis title(’Plot of sin(x)’); % title of the graph

• Plot multiple functions on the same figure

plot(x,f,’- -’,x,g,’r’);% plot sin(x) with a dashed line and cos(x) with a red line xlabel(’x’); % labeling the x-axis ylabel(’y’); % labeling the y-axis title(’Plot of sin(x) and cos(x)’); % title of the graph 5

legend(’sin(x)’,’cos(x)’ ); % give a legend to match the plots ⋆ help plot or run demos

• 7. ODE Solvers

The main command: [t,y]=ode23(@fun,[t0 tf],y0,options,a1,a2,...)

• Input
• Name of a function (@fun)
• Timespan ([t0 tf])
• Initial conditions (y0)
• Options for ODE solver (options=(’AbsTol’,1e-4,’RelTol’,1e-4))
• Values to pass to the function
• Output
• The variable timestep values of t
• Values of solution at corresponding values of t (y(t))

The function where ODE is written out: function dy=fun(t,y,flag,a1,a2,...,aN)

• Function dy evaluates y′ = f(t, y(t))
• Input
• Values of t and y to evaluate y′
• flag is the place holder for options
• a1,...,aN are extra values to pass into the function besides t, y

Higher Order of ODE

• Rewrite ODE as system of first order ODE: y′ = Ay + b

6

• Pass values in A and b as a1,...,aN

Matlab Codes tspan=0:.01:5; %define tspan from 0 to 20 y0=1; [t,y]=ode23(@odefun,tspan,y0); plot(t,y);% plot the results function dy=odefun(t,y) dy=t*y; Other ODE solvers: ode45, ode113, ode23s, ode15s ⋆ help ode45; help ode23, etc.; run demos

• Exercises: Write a program to solve ¨

y − 3 ˙ y − 4y = 0, ˙ y(0) = 2, y(0) = 1 from [0,10]. Plot y(t).

• 8. Optimization Methods (fminsearch)
• Find a local minimizer X of the function FUN
• FUN accepts input X and returns a scalar value evaluated at X
• X can be a scalar, vector, or matrix
• Matlab commands: X = fminsearch(@FUN, X0); % find a value X

to minimize the funtion FUN with the initial guess X0 function f=FUN(x) f = cos(x) ∗ exp(2 ∗ x2);% function that we want to minimize 7

⋆ help fminsearch; demos

• Exercises: Using fminsearch, find a minimizer of x2
• 9. Data Saving and Loading

Matlab command: save(’filename.dat’,’variablename’);%saving %variablename to filename.dat s=importdata(’filename.dat’); %save the contents of filename in s ⋆ help load m=[5;6;7;8]; save(’data.dat’,’m’);%saving m to file data.dat Exercises: load up the file data.dat into variable z 8