CS 151L – Spring 2015 Programming Assignment 2 Due: Wednesday, February 4, 2015 at 11:59 PM (Email in two separate MatLab scripts called “cs151sp15assn2a.m” and “cs151sp15assn2b.m”. Each of these scripts must be well-commented so the graders can follow and grade your work.) 1. Compute an approximation of to arbitrary precision by dividing the perimeter of an n-sided regular polygon inscribed in a unit circle by the radius, r = 1. s r=1 Write a MatLab script that does the following: a) Create a 1 by 6 matrix, or row vector, representing the number of sides of a polygon where n is equal to 4, 8, 16, 32, 64 and 128. b) Compute another row vector, theta (), whose elements correspond to those of the array, n. You will need to calculate the value of theta based on the number of sides of the polygon and knowing that a complete circle encompasses 360 degrees. (Do not do the calculations using a calculator; you should be using MATLAB operations to determine these values.) c) Compute a row vector, s, representing the length of one side of the polygon, whose elements correspond to those of the array, n. Note that you now have calculated the angle within each triangle and that two of the sides of the triangle are equal to 1. d) Compute the row vector, p, representing the perimeter of each polygon. e) For a circle, the ratio of the circumference to the diameter is c d and c d ). Therefore, is approximately perimeter/(2*r). Using this information, compute pi_approx, for each polygon. 1 f) Compute the row vector, percent_error, whose elements represent the percent error in pi_approx for each polygon. The percent error is defined as (note that this value should always be positive): approximate exact 100 exact g) Plot the percent error (vertical axis) versus n (horizontal axis) using the plot command plot(a, b) where a is the horizontal value and b is the vertical value. 2. Compute an approximation of by using random numbers. Assume a circle of radius ½ unit is inscribed within a square with sides of length a = 1 unit. The area of the circle is given by: Acircle r 2 0.5 2 Therefore, , while the area of the unit square is given by Asquare a 1 1 . 2 4 2 A Acircle / 4 , and 4 circle . We can approximate probabilistically by A Asquare 1 4 square recognizing that if we randomly throw darts at the unit square region, the number of “hits” within the circle to “hits” within the square is approximately in the ratio of A n ncircle A circle . Therefore nsquare Asquare 4 circle 4 circle . Asquare nsquare y a=1 y=1 . . r = 1/2 . . a=1 . y=0 . x x=0 x=1 Step-by-Step Solution a) Create the variable n as a 1 by 1 matrix, or scalar, where the user must input the value using the input command. b) Create two 1 by n row vectors x and y, each containing n random numbers between 0 and 1. (Use the MATLAB function rand().) 2 c) Compute a 1 by n matrix, dist, containing the distance of each point (xi, yi) from the center of the circle (Note that the center of the circle is located at (0.5, 0.5) on the coordinate system in the diagram). You will need to use the distance formula ( x 0.5 y 0.5 ) to create this variable (you can use the sqrt() function or raise the values to the ½ power.) d) Compute the row vector, in_out, which contains a “1” if each point is within the circle, and a “0” if each point is not within the circle. (Hint: you can use the MATLAB function floor() and multiply the distance by 2 – the radius of the circle is only ½; therefore, if the distance of one of the points is 0.44, then it will be in the circle and if the distance of one of the points is 0.51, then it is outside the circle. There are always other methods to perform this calculation.) 2 2 Acircle 4 ncircle . A n square square e) Determine an approximation of , pi_approx, by the formula 4 (Hint: you can use the MATLAB function sum() and remember that the value in the denominator is the number of hits in the square, not the number of hits outside the circle!) f) Determine the percent error, percent_error, in the approximation of (this value should always be positive). g) Now, re-run the script using n = 1e2, 1e3, etc. (i.e. 100, 1000, etc.) In your m-file, answer the following question: what must be the order of magnitude of n to routinely get to within 1 percent accuracy? 3

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