課程名稱︰計算數學導論
課程性質︰必修
課程教師︰王偉仲
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/11/10
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
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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