MATLAB Tutorial

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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 (arg1arg2arg3)/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
• Yaverage3(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?')
• VarAverageaverage3(VarA, VarB, VarC)
• num2str converts a number to a string.
• resultThe 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 i0
• 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, well use a trapezoid shape.
• Area of trapezoid ½ (b1 b2) h

f(b)
f(a)
base 2 (b1)
h
base 2 (b2)
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Numerical Integration

• Trapezoidal Rule (cont)

• 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)

• 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

• 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) f(a) 2 f(x1) f(b)
  

f(b)
f(x1)
f(a)
Area of Trapezoid 1
Area of Trapezoid 2
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, well 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 x1
  x2
  b
• ? f(x) dx ? f(x) dx ? f(x) dx ? f(x)
  dx
• a a
  x1
  xn-1
• (b-a)/2n f(x) f(x1) f(x1)
  f(x2) f(xn-1) f(b)
• (b-a)/2n f(x) 2 f(x1) 2
  f(x2) 2 f(xn-1) f(b)
• b
• ? f(x) dx ?x/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 , lets integrate
  it from e,2e
• 2e 2e
• ? 1/x dx ln (x) ln (2e) ln
  (e) ln (2) 0.6931
• e
  e
• Matlab Equivalent
• function ITrapez(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 i1n-1
• x(i) a ih
• s s2 feval (f,x(i))
• end
• s s feval(f,b)
• I sh/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 2.7182818 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 IFastTrap(f, a, b, n)
• Same as the previous Trapezoidal function
  example, but using more
• efficient MATLAB vectorized operations.
• h(b-a)/n Increment value
• sfeval(f, a) Starting value
• in1n-1
• xpointsainh Defining the x-points
• ypointsfeval(vectorize(f),xpoints) Get
  corresponding y-points
• sig2sum(ypoints) Summing up values in
  ypoints, and mult. by 2
• sssigfeval(f,b) Evaluating last term
• Ish/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.
• foninline('log) Defines the function we
  wish to integrate.
• aexp(1) Starting point.
• b2a Ending point.
• n1000 Number of intervals.
• tic Start counter.
• OutValueTrapez (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
• N1,000
• N100,000.
• 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, 2p.
• Define the integral
• x(i)
• Int (i) ? sin (t) dt
• 0
• Calculate Int(i) for all values, i 1, , 101
• Plot y(i) and Int(i) versus x(i), for i 1, , 101