Math 112: MATLAB Assignment 1
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NUMERICAL INTEGRATION * Due Date: October 9, 2008 |
Contents
Sample Program
The sample program below uses the left-endpoint rule, the right-endpoint rule and the trapezoid rule to approximate the definite integral of the function.
Matlab comments follow the percent sign (%)
a= 0; b= 1; N = 10; h=(b-a)/N; x=[a:h:b]; %creates a vector of n+1 evenly spaced points f=x.^2; IL=0; IR=0; IT=0; for k=1:N; %Note that the vector f has (N+1) elements IL=IL+f(k); IR=IR+f(k+1); IT=IT+(f(k)+f(k+1))/2; end; IL=IL*h; IR=IR*h; IT=IT*h; fprintf(' When N = %i, we find:\n',N); fprintf(' Left-endpoint approximation = %f.\n',IL); fprintf('Right-endpoint approximation = %f.\n',IR); fprintf(' Trapezoidal approximation = %f.\n',IT); % Output from this program:
When N = 10, we find: Left-endpoint approximation = 0.285000. Right-endpoint approximation = 0.385000. Trapezoidal approximation = 0.335000.
New Matab ideas
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A For Loop. Repeat the code between 'for' and 'end' once for each number between 1 and N
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the fprintf statement. This is a formatted printing statement, which uses almost identical syntax to the C programming language. It is used here to format the output for display. You should be able to use this part of the program without modifying it.
Parameters used in this program:
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a,b: the limits of integration
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x: the variable of integration
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f: the integrand
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N: the number of sub-intervals
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h: the width of each sub-interval
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IL & IR: the left and right endpoint approximation to the integral
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IT: the trapezoidal approximation
Your assignment
After running the above example, do the following to hand in: (Consult the Matlab TA if you have any questions.)
Approximate the integral of
Answer the following questions
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Run the code with N=10, N=100, and N=1000.
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For each approximation, when does the result agree with the exact value of the integral to 4 digits?
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How much better is the trapezoidal rule than the other two? Explain this result using the theory given in the textbook and in lecture.
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BONUS QUESTION Write a program to approximate the integral using Simpson's rule. How many points are needed to achieve the same accuracy as you found for the trapezoidal rule with 1000 points?