Class:- Numerical Analysis

Program used: Matlab

Write Matlab program to calculate Pi and plot darts with the targets as shown in the sample graph above. Sample Matlab code for plotting is given below: axis([-1, 1,-1, 1] ); axis manual equal set(gca,'xTick',[-1 0 1]); set(gca,'yTick', [-1 0 1]); fill(x, y, [0.8, 0.8, 0.8]); %fill a polygon with color space (RGB) 0-1 plot(x, y, 'ko', 'MarkerEdgeColor','k','MarkerFaceColor','r', 'MarkerSize',5); %'ko' for black circles

Approximation of a a is a mathematical constant that is the ratio of any circle's circumfer-ence to its diameter. 7 is approximately equal to 3.14 in the usual deci-mal notation. Many formulas in mathematics, science, and engineering in-volve , which makes it one of the most important mathematical constants. (3.14159265...) is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator. Consequently, its decimal representation never ends or repeats. (from wikipedia.com) The following describes a method of approximating the value of a by throwing darts. Suppose that a dart player throws a number of darts at a circular target inscribed into a square and all the darts land inside the square. Figure 1 shows an imaginary distribution of darts on the target. Assume that the chances to hit anywhere inside the square are equal. If a dart lands inside the circle, it is counted as a hit, otherwise a miss. It is observed that the chance (probability) to hit inside the circle is proportional to the ratio of the area of the circle to the area of the square. A mathematical formula is used to express the relationship: R2 i = (2R)2 = where H is the number of hits and T is the total number of darts thrown. Then it can be expressed as: T = 4 The steps to approximate the value of a: 1. Generate a pair of x and y coordinates of darts that marks the lo- cation of a dart hit on the target using a random number generator function called rand(1). It generate a decimal number between 0 and 1 (exclusive) Simulation of Pi Figure 1: Distribution of Darts on a Target 2. Calculate the distance between the darts and the origin of the coordi- nate system with the formula below: d=V x2 + y2 3. Assume that the radius R of the circle target is 1. If d is less than 1, count as a hit, otherwise a miss. 4. Count the number of hits and the total number of darts thrown. 5. Calculate y using the preceding formula a = 4