课程名称︰计算数学导论 课程性质︰必修 课程教师︰王伟仲 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010/11/10 考试时限（分钟）：110 是否需发放奖励金：是 （如未明确表示，则不予发放） Problem 1 [14%] Let f(x) = exp(x)-exp(-2x) and α be a small positive number. (a) What numerical difficulty you may have while evaluating f(α). (b) How can you overcome the difficulty to improve the accuracy of f(α)? Problem 2 [8%] Which of the following methods can converge to the number 7^(1/5) faster? Why? [p_(n-1)]^5 - 7 (a) p_n = p_(n-1) - ————————— 5[p_(n-1)]^4 [p_(n-1)]^5 - 7 (b) p_n = p_(n-1) - —————————— 12 Problem 3 [8%] Give examples or sketch graphics to explain why we do not prefer using (a) |f(x_n)|<tolerance or (b) |x_n - x_(n-1)|<tolerance as the stopping criterion of Newton's method. Problem 4 [5%] Suppose we have the following five points (on the coordinates) that x1=(1,π/2), x2=(1,-π/2), x3=(1,π/4), x4=(1,-π/4), and x5=(1,0). Can you find a interpolation polynomial passing these five points which is like a half-circle? Explain you idea. Problem 5 [15%] Derive in detail the linear system you need to solve for finding the natrual spline s that interpolates the point (0,1), (1,e), (2,e^2), and (3,e^3). Problem 6 [15%] (a)Use Taylor's Theorem to derive the following approximation of the first derivative: u'(x) = -3u(x)/2h + 2u(x+h)/h - u(x+2h)/2h. (b) What is the order of this approximation? Problem 7 [15%] 1 (a)Use the composite Simpson's rule to approximate the integral ∫ x^3dx with an absolute error less than 2×10^(-5). 0 Problem 8 [20%] (a)Construct the 20×20 matrix A whose main diagonal elements are 1,2,...,20 and the first lower off-diagonal are 219,218,...,201. (b)Construct a 20×1 random vector b. (c)Solve the linear system Ax=b by a MATLAB built-in function. (d)Write a code to slove Ax=b by using a for-loop. --

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