6 01 Fall Semester 2007 Assignment 4 Issued Tuesday Sept 25 1 MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6 01 Introduction to EECS I Fall Semester 2007 Assignment 4 Issued Tuesday Sept 25 To do this week in Tuesday software lab 1 Start writing code and test cases for the numbered questions in the software lab Paste all your code including your test cases into the box provided in the Software Lab Part 4 1 problem on the on line Tutor This will not be graded before the start of lab on Thursday 1 Read the lecture notes 2 Do the on line Tutor problems for week 4 that are due on Thursday Part 4 2 3 Read through the entire description of Thursday s lab in Thursday robot lab 1 Answer the numbered questions in the robot lab and demonstrate them to your LA 2 Do the nanoquiz it will be based on the material in the lecture notes and the on line Tutor problems due on Thursday before the start of lecture next Tuesday 1 Do the lab writeup providing written answers including code and test cases for every numbered question in this handout 6 01 Fall Semester 2007 Assignment 4 Issued Tuesday Sept 25 2 On Athena or the lab laptops make sure you execute athrun 6 01 update so that you can get the Desktop 6 01 lab4 directory which has the files mentioned in this handout You need the file polynomial pyc for the software lab You need the file diffeq py for the software lab You need the file soar graph py for the software lab During software lab if you are using your own laptop download the files from the course Web site on the Calendar page Make sure you are using Python version 2 5 as we are giving you a compiled version of the polynomial class Tuesday Software Lab 1 Solving second order difference equations For this lab you will be writing a Python program that solves arbitrary second order homogeneous linear difference equations analytically by computing natural frequencies Your program should take as inputs the initial values x 0 and x 1 as well as coefficients a1 and a2 for the difference equation in the form y n a1 y n 1 a2 y n 2 Your program should print out the solution to the difference equation and also return a procedure that when called with n returns y n For example if the difference equation is y n 2y n 1 2y n 2 and the initial conditions are y 0 0 y 1 1 your program should print something equivalent to y n 2 7755575 e 017 0 5 j 1 1 j n 2 7755575 e 017 0 5 j 1 1 j n magnitude of natural frequencies 1 4142135623730951 1 4 142 13 56 23 73 09 51 and your program should also return a function which can be used to evaluate y n for any n NOTE you can assume that the natural frequencies are distinct but your program should indicate an error when the two natural frequencies are identical The special case of what to do with repeated natural frequencies is studied in more advanced courses Polynomial manipulation To save time you should use our implementation of the polynomial manipulation program from the recent post lab exercises and edit diffeq py to create your second order difference equation solver Please be sure diffeq py and polynomial pyc are in the same directory as your difference equation solver Executing diffeq py in IDLE will import our version of the polynomial manipulation routines implemented in the class Polynomial You can see the class interface by typing help Polynomial at the IDLE command line Note that the Polynomial class has functions to add multiply print and find the roots of polynomials The following code fragment 6 01 Fall Semester 2007 Assignment 4 Issued Tuesday Sept 25 3 p Polynomial 1 6 9 p roots 3 0 j 3 0 j generates an instance of the polynomial x2 6x 9 0 and then computes the polynomial s roots Our implementation only computes roots correctly for polynomials up to third order cubics Checkpoint 11 45 You should understand the polynomial class A note on complex numbers For this problem your procedure will sometimes need to compute zn where z is a complex number Python understands complex numbers to some extent but you have to write them in the form a bj where a is the real part and b is the imaginary part Note that Python uses the convention that j 1 As an EECS student you will have to learn to accept the fact that the variable i must be reserved for electrical current You can t raise a negative real number to a fractional power but you can do so with a complex number So for example the Python command 4 0 5 will produce an error but the Python command 4 0 j 0 5 produces 1 2246063538223773e 016 2j Why the very small real part Note use y 0 5 to compute the square root of a number The python sqrt function does not understand complex numbers You can get the real and imaginary parts of a complex number x with x real and x imag also abs x will return its magnitude It will be important to ensure that the function returned by your difference equation solver only returns real not complex values Be sure you understand why mathematically the output should always be real if the coefficients and initial values are real 6 01 Fall Semester 2007 Assignment 4 Issued Tuesday Sept 25 4 Demonstrate that your program works Question 1 Describe how your program works Question 2 Test your program by solving the difference equation y n 2y n 1 2y n 2 with the initial conditions y 0 0 y 1 1 Question 3 Demonstrate that your program returns the correct function How did you verify your program was correct Checkpoint 12 30 PM You should have a working difference equation solving program You can also test your program by plotting the values the program generates Here are the steps 1 Start SoaR just type SoaR in the terminal it s best to quit Idle before you do this 2 Click on the editor window to get an Idle window from inside SoaR 3 Open the file soar graph py which has the following commands from diffeq import g1 GraphingWindow 400 400 0 20 100 100 diffeq f solveDiffEq 0 1 2 2 g1 graphDiscrete f blue The first in the above sequence of commands generates a a graphing window that is 400 by 400 pixels with x coordinates ranging from 0 to 20 and y coordinates from 100 to 100 and gives the graph the window title diffeq The second command in the sequence calls a difference equation solver that returns a procedure f The third command causes a blue graph to be generated in the graphing window The function graphDiscrete will call the function f with integer arguments from 0 to 20 and plot …
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