1. MATLAB Tutorial
ECE 002
Professor S. Ahmadi

2. Tutorial Outline Functions using M-files.
Numerical Integration using the Trapezoidal Rule.
Example #1: Numerical Integration using Standard Programming (Slow).
Example #2: Numerical Integration using Vectorized MATLAB operations (Fast).
Student Lab Exercises.

3. What Are Function M-Files
Function M-files are user-defined subroutines that can be invoked in order to perform specific functions.
Input arguments are used to feed information to the function.
Output arguments store the results.
EX: a = sin(d).
All variables within the function are local, except outputs. NOTE: Local variables are variables that are only used within the function. They are not saved after the M-file executes.

4. Example of a Function M-File
function answer = average3(arg1,arg2, arg3)
% A simple example about how MATLAB deals with user
% created functions in M-files.
% average3 is a user-defined function. It could be named anything.
% average3 takes the average of the three input parameters: arg1,
% arg2, arg3
% The output is stored in the variable answer and returned to the
% user.
% This M file must be saved as average3.m
answer = (arg1+arg2+arg3)/3;

5. How to call a function in matlab
X = sin(pi);
% call the interior function of Matlab, sin
Y = input(“How are you?”)
% call the interior function of Matlab, input
% input is a function that requests information from
% the user during runtime
Y=average3(1, 2, 3);.
% average3 is a user-defined function.
% The average value of 1,2 and 3 will be stored in Y

6. Example: Calling a Function from within the Main program.
% Example1: This is a main program example to illustrate how functions are called.
% input is a function that requests information from
% the user during runtime.
VarA = input('What is the first number?');
VarB = input('What is the second number?');
VarC = input('What is the third number?');
VarAverage=average3(VarA, VarB, VarC);
% num2str converts a number to a string.
result=[‘The average value was’, num2str(VarAverage)]

7. Numerical Integration
General Approximation Made With Num. Integration:
b n
∫ f(x) dx ≈ ∑ ci f(xi)
a i=0
Trapezoidal rule:
Integration finds the “Area” under
a curve, between two points (a and b).
To approximate the area under a curve,
use a familiar shape whose area we
already know how to calculate easily.
In this case, we’ll use a trapezoid shape.
Area of trapezoid = ½ (b1 + b2) h x y
f(b)
f(a)
base 2 (b1) a b h
base 2 (b2)

8. Numerical Integration
Trapezoidal Rule (con’t): x y
Area under curve ≈ Area of trapezoid
under the curve
Area of trapezoid = ½ h [ b1 + b2 ]
Area under curve ≈ ½ (b-a) [ f(a) + f(b) ]
Therefore, Trapezoidal Rule: b
∫ f(x) dx ≈ ½ (b-a) [ f(a) + f(b) ] a
f(b)
f(a) a b
How can we get a better approximation of the area under the curve?
Answer: Use More Trapezoids

9. Numerical Integration
More trapezoids give a better approximation of the area under curve x y
Add area of both trapezoids together,
more precise approx. of area under curve
Area of trapezoid = ½ h [ b1 + b2 ]
Area under curve ≈
½ (x1-a) [ f(a) + f(x1) ] + ½ (b-x1) [ f(x1) + f(b) ]
Simplify:
½ ((b-a)/2) [ f(a) + f(x1) + f(x1) + f(b) ]
(b-a)/4* [ f(a) + 2 * f(x1) + f(b) ]
f(b)
f(x1)
f(a)
Area of Trapezoid 1
Area of Trapezoid 2 a b x1
Area under curve ≈

10. Numerical Integration
Using more trapezoids to approximate area under the curve is called: Composite Trapezoidal Rule
The more trapezoids, the better, so instead of two trapezoids, we’ll use n trapezoids.
The greater the value of n, the better our approximation of the area will be.

11. Composite Trapezoidal Rule
Divide interval [a,b] into n equally spaced subintervals (Add area of the n trapezoids)
b x x b
∫ f(x) dx ≈ ∫ f(x) dx + ∫ f(x) dx + … + ∫ f(x) dx
a a x xn-1
≈ (b-a)/2n [ f(a) + f(x1) + f(x1) + f(x2) +…+ f(xn-1) + f(b) ]
≈ (b-a)/2n [ f(a) + 2 f(x1) + 2 f(x2) +…+ 2 f(xn-1) + f(b) ] b
∫ f(x) dx ≈ Δx/2 [ y0 + 2y1 + 2y2 + … + 2yn-1 + yn ] a
Composite Trapezoidal Rule

12. Example 1 on Numerical Integration
Implementing Composite Trapezoidal Rule in Matlab
Example Curve: f(x) = 1/x , let’s integrate it from [e,2e] :
2e e
∫ 1/x dx = ln (x) | = ln (2e) – ln (e) = ln (2) =
e e
Matlab Equivalent
function I=Trapez(f, a, b, n) % take f, add n trapezoids,from a to b
% Integration using composite trapezoid rule
h = (b-a)/n ; % increment
s = feval(f,a) ; % starting value
for i=1:n-1
x(i) = a + i*h ;
s = s+2 * feval (f,x(i)) ;
end
s = s + feval(f,b) ;
I = s*h/2 ;
Area under curve: 1/x, from [e,2e]
Inline(‘1/x’)
In our case, input to the function will be:
f = inline (’1/x’)
a = e =
b = 2e

13. Example 2 on Numerical Integration: Using MATLAB Vectorized Operations
This function carries out the same function as the previous example, but by using MATLAB Vectorized operations, it runs much faster.
function I=FastTrap(f, a, b, n)
% Same as the previous Trapezoidal function example, but using more
% efficient MATLAB vectorized operations.
h=(b-a)/n; % Increment value
s=feval(f, a); % Starting value
in=1:n-1;
xpoints=a+in*h; % Defining the x-points
ypoints=feval(vectorize(f),xpoints); % Get corresponding y-points
sig=2*sum(ypoints); % Summing up values in ypoints, and mult. by 2
s=s+sig+feval(f,b); % Evaluating last term
I=s*h/2;

14. Example 3: Integrating Trapezoidal/FastTrap Function into Main Program
Main program to test numerical integration function, and to measure difference in speed between the two previous functions.
% Example 3: Main program to run the numerical integration function,
% using the user-created Trapezoidal/FastTrap methods.
fon=inline('log‘); % Defines the function we wish to integrate.
a=exp(1); % Starting point.
b=2*a; % Ending point.
n=1000; % Number of intervals.
tic % Start counter.
OutValue=Trapez (fon, a, b, n) % Calling Trapezoidal function.
toc % Stop counter, and print out counters value.
% Try replacing the Trapez function with the vectorized FastTrap
% function, and notice the difference in speeds between the two.

15. Example 3: Continuation
Try two different values of N
N=1,000
N=100,000.
For both N values, test the code using both functions for the Trapezoidal method: The normal Trapez Method, and the FastTrap Method.
Compare the difference in execution time between the standard way (Trapez), and the vectorized approach (FastTrap).

16. Additional MATLAB Exercise
Function: y(i) = sin [ x(i) ]
Where x(i) is “defined” by 101 uniformly spaced points in [0, 2π].
Define the integral:
x(i)
Int (i) = ∫ sin (t) dt
Calculate Int(i) for all values, i = 1, … , 101
Plot y(i) and Int(i) versus x(i), for i =1, …, 101