課程名稱︰計算數學導論
課程性質︰數學系必修
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︰2007年01月11日
考試時限：110分鐘，13:20-15:10
是否需發放獎勵金：是
試題 :
Close books and notes.
1.(25 points) Derive a formula of the form.
h
∫ f(x) dx = w0*f(0) + w1*f(h) + w2*f'(0) + w3*f'(h)
0
that is exact for polynomials of the highest degree possible.
2. (10 points) True or false of the following two questions:
(a) Gaussian elimination without pivoting fails only when the matrix
ill-conditioned or singular.
(b) In numerical solution of an ordinary differential equation, the global
error grows only if the equation is unstable.
3. (15 points) Suppose we wish to use Gaussian elimination to consruct an
upper triangular matrix from the following matrix A:
┌ 4 -8 1 ┐
A = │ 6 5 7 │.
└ 0 -10 -3 ┘
What will the initial pivot element be if
(a) No pivoting is used ?
(b) Partial pivoting is used ?
(c) Complete pivoting is used ?
4. (25 points) Describe a numerical method in algorithmic details for solving
an n×n linear system of equation Ax = z, where xεR^n, zεR^n,
and the matrix AεR^(n×n) is nonsingular and is of the form:
┌ b c d ┐
│ a b c │
│ a b c │
│ . . . │
A = │ . . . │.
│ . . . │
│ a b c │
│ a b c│
└ d a b┘
for non-trivial real constants: a, b, c, and d.
5. (25 points) The centered difference approximation
dy y(k+1) - y(k-1)
─ ≒ ────────,
dt 2*Δt
leads to the two-step leapfrog method of the form
y(k+1) = y(k-1) + 2*Δt*f(t(k),y(k))
dy
for solving the ordinary differential equation ─ = f(t,y),
dt
where y(k) ≒ y(t*(k)) and Δt is the time step. Determine the
order of accuracy of the leapfrog method.
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