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
考試時間︰2006年11月16日 13:20-15:10
是否需發放獎勵金：是
試題：
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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