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Problem 6 At the end of a car race, we hare the (x.y) position of car A and car B Car A Car B t x (meters) y (meters) if I (meters) z (meters) 10:00:54 -3.8 5 10:00:54 -4 5 10:00:55 -1.8 4 10:00:55 -3.75 4 10:00:56 -0.97 3 10:00:56 -3 3 10:00:59 0.67 1 10:00:59 2.25 1 10:01:00 1.1 1 10:01:00 5 1 Use an interpolation method to find: (a) Which car aon the race? The end-line is the y-azis at I = 0. (b) Which anas the speed of the winner at the end-line. (c) Will the outcomes of a) and b) change if at 10 : 00 : 54 the acceleration of car A is 4m/s and car B is 6m/s29 Problem 7 Use the Jacobi and Gauss-Seidel Methods to solve the sparse system within six correct decimal places (residual error in infinity norm) for n = 100 and n = 1000. The correct solution is [1, 1]. Report the namber of steps needed. The system is 3 1 I1 2 3 1 I2 1 : = : - 3 -1 In-1 1 3 Tn 2 Problem 8 Use the SOR method to solve the sparse system within sir correct decimal places (residual error in infinity norm) for n = 100 and n = 1000. The correct solution is [1. -1. 1. - 1, - -1]. Report the number of steps needed and the optimal w parameter. The system is 2 1 I1 1 1 2 1 I2 : : 1 2 1 x, 1 2 r, -1 Problem 9 The Legendre polynomials Pn are defined by the recurrence formula (2n + = =(n+1)P=+1(=)+RPA-1(x) n= 1.2.3. where Po(x) = 1 and P1(x) = x. Find all the roots of Pn. n = 2.3.4, 5 in the interval [-1,1 1].

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Numerical Analysis
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