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
考試日期（年月日）︰1月8日，2007
考試時限（分鐘）：8:10-10:00am(110分鐘)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Part A : Computer Programming
This test should be done on your own, without any collaboration from others.
A short essay should be turned in at the end of the test that describes the
answers 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. (25 points) The Bessel function of order n (a nonnegative interger) is
defined by the equation
π
Jn(x) = (∫ cos(x*sin(t)-n*t) dt)/π, n = 0,1,2...
0
Calculate J2(1) by applying the Romberg algorithm to the
integral so that the result is in agreement with the approximate
solution 0.1149034849319 obtained using the Matlab command:
bessel(2,1).
2. (25 points) Consider the definite integral
x
I(x) = ∫ (1+t^3)^(1/2) dt.
0
(a) Use your own Runge-Kutta code compute I(x) when x = 1.
(b) Compute I(1) again, but this time you are free to use whatever method
you like, except the Runge-Kutta method as in (a).
3. (25 points) For n = 100, solve this tridiagonal system of equations
2*x(1) - x(2) = 1,
-x(j-1) + 2*x(j) = j, j = 2,3,...,n-1,
-x(n-1) + 2*x(n) = n.
using your own tridiagonal solver.
4. (25 points) Solve
(d^2)y dy dy
─── + ─ + y^2 - 2*t = 0, with y(0) = 0, ─(0) = 0.1,
d(t^2) dt dt
on t at [0,3] by any nimerical method with approprate time
step △t. Make a graph of your solution y with respect to time
t.