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MATLAB Tutorial ECE 002 Professor S. Ahmadi

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**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.

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**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.

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**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;

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**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

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**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)]

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**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)

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**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

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**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 ≈

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**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.

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**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

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**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

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**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;

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**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.

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**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).

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**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

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