課程名稱︰計算數學導論
課程性質︰
課程教師︰薛克民
開課學院：理學院
開課系所︰數學系
考試時間︰Part A:11/13 8:10~10:00 Part B:11/16 13:20~15:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Part A
You are free to use whatever computer program or softwares you like durig
the test.this test sould be done on your own, without any collaboration from
others. A short essay should be turned in at the end of the test in either
the printed or electronic from that describe the answer of the problems for
the test. It is not okay(i.e., you will not get any credit for it) if you
only present the results, but do not provide any information on how they were
obtained.
1. (50 points) Let f(x) = max{0,1-x}.
a) (10 points) Plot the function f on the interval [-4,4].
b) (20 points) Find interpolating polynomials p of degrees 8 and 32 to
f on [-4,4], using the equally spaced nodes. For each of
the cases, plot the error │f(x) － p(x)│at 128 equally
spaced points. What can you say about the convergence of
the interpolating polynomial as n－>∞.
c) (20 points) Redo the problem in b) using the Chebyshev nodes, which
is defined by
xi = (a+b)/2 + (b-a)cos[(2i+1)π/(2n+2)]/2, (0<=i<=n)
on an arbitrary interval [a,b].
2. (50 points) It can be shown that the exact solution for the interval-value
problem of the Burger's equation Ut+U*Ux = 0 with initial data
U(x,0) = U0(x) is U(x,t) = U0(x-ut) for time t<ts, where ts is
the time for the occurrence fo the singularity in the solution.
Here Ut and Ux mean the partial derivatives of U(x,t) with
respect to t and x, respective.
a) (10 points) When U0(x) = sin(2πx), plot the initial condition U(x,0) on
the spatial interval [-1,1].
b) (20 points) Find the solution U(x,t) at time t = 0.1 at 100 equally
spaced nodes on [-1,1], sin[2π(x-0.1u)] = 0, at 100 spatial
points.
c) (20 points) Redo the problem in b) when time t = 0.2.
Part B
Close books and notes.
1. (40 points) Assume that the equation f(x) = 0 has the root r with
multiplicity m = 2.
a) (20 points) If Newton's method converges to this root, show that the rate
of convergence of the method is linear.
b) (20 points) If we modify the Newton's method to be of the form
Xn+1 = Xn - mf(Xn)/f'(Xn),
show that we obtain quardratic convergence again, when the
method convergernce to this root.
2. (20 points) Show that the exact solution of the recurrence equation
Qn = QnQn-1
with intial condition Q0 and Q1 is
Qn = ({[(Q0^β)(Q1^(-1))]^(α^n)}*{[(Q0^(-α))Q1]^(β^n)})^(-1/√5)
where α = (1+√5)/2 and β = (1-√5)/2.
3. (20 points) Find a polynomial p that takes these values: p(1) = 3, p(2) = 1,
p(0) = -5. You may use any method you wish. Next, find a new
polynomial q that takes those same three values and q(3) = 7.
4. (20 points) Consider the points P0(x0,y0), P1(x0+h,y0), and Pb(x0-αh,y0) as
shown in Fig.1, 0<α<1. The problem is to find a second order
approximation of the partial derivative Ux of U(x,y) at the
points P0 in terms of U0 = U(P0), U1 = U(P1), and Ub = U(Pb).
αh h
‧───‧─────‧
Pb P0 P1
Figure 1: Computing partial derivative Ux of U(x,y) at point P0
near the boundary.
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