Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

A.5. SECOND MIDTERM, FALL 2004 151
P2 (10 pnts) Complete this statement of the polynomial interpolation error theorem: Let p be the
polynomial of degree at most n interpolating function f at the n+1 distinct nodes x 0 , x 1 , . . . , x n
on [a, b]. Let f (n+1) be continuous. Then for each x ∈ [a, b] there is some ξ ∈ [a, b] such that
f(x) − p(x) = ? ?
n∏
(?) .
P3 (10 pnts) State the conditions which define the function S(x) as a natural cubic spline on the
interval [a, b].
[Problems] Answer no more than five of the following six.
P1 (16 pnts) How large must n be to interpolate the function e x to within 0.001 on the interval
[0, 2] with the polynomial interpolant at n equally spaced nodes?
P2 (16 pnts) Let φ(h) be some computable approximation to the quantity L such that
i=0
φ(h) = L + a 5 h 5 + a 10 h 10 + a 15 h 15 + . . .
Combine evaluations of φ(·) to devise a O ( h 10) approximation to L.
P3 (16 pnts) Derive the constants A and B which make the quadrature rule
∫ 1
−1
( √ )
(√ )
15
15
f(x) dx ≈ Af − + Bf(0) + Af
5
5
exact for polynomials of degree ≤ 2. Use this quadrature rule to approximate
π =
∫ 1
−1
2
1 + x 2 dx
P4 (16 pnts) Find the polynomial of degree ≤ 3 interpolating the data
x −1 0 1 2
y −2 8 0 4
(Hint: using Lagrange Polynomials will probably take too long.)
P5 (16 pnts) Find constants α, β such that the following is a quadratic spline on [0, 4].
⎧
⎪⎨ x 2 − 3x + 4 : 0 ≤ x < 1
Q(x) = α(x − 1)
⎪⎩
2 + β(x − 1) + 2 : 1 ≤ x < 3
x 2 + x − 4 : 3 ≤ x ≤ 4
P6 (16 pnts) Consider the following approximation:
f ′ (x) ≈ f(x + h) − 5f(x) + 4f(x + h 2 )
3h
• Derive this approximation via Taylor’s Theorem.
• Assuming that f(x) has bounded derivatives, give the accuracy of the above approximation.
Your answer should be something like O ( h ?) .
• Let f(x) = x 2 . Approximate f ′ (0) with this approximation, using h = 1 3 .