Numerical Analysis I, Fall 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 03

Nov 13, 2009.

1. Find the Newton divided difference f[0, 1, 2, 3, 4] where f(x) = x^{4} + πx^{3} − x^{2} + 1. 2.

s(x) = \{ \begin{matrix}x^{3} + 2x^{2} − x + 1 & −1 ≤ x ≤ 0 \\ αx^{3} + βx^{2} + γx + δ & 0 ≤ x ≤ 1 \end{matrix}

Find ALL values of (α, β, γ, δ) such that s(x) is a cubic spline on [−1, 1].

3. Define the Chebyshev polynomial T_{n}(x) on [−1, 1] and find its roots.

4. State (need not prove) the error formula for polynomial interpolation. Give an error bound in sup-norm for polynomial interpolation of e^{x} on [−1, 1] using roots of T_{5}(x) as interpolating points.

5. Does

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} tan^{3}(\frac{iπ}{2n})

converge? If so, what is the limit? If not, what is the leading order term in terms of n?

Numerical Analysis I, Fall 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 03

Nov 13, 2009.

1. Find the Newton divided difference f[0, 1, 2, 3, 4] where f(x) = x^{4} + πx^{3} − x^{2} + 1. 2.

s(x) = \{ \begin{matrix}x^{3} + 2x^{2} − x + 1 & −1 ≤ x ≤ 0 \\ αx^{3} + βx^{2} + γx + δ & 0 ≤ x ≤ 1 \end{matrix}

Find ALL values of (α, β, γ, δ) such that s(x) is a cubic spline on [−1, 1].

3. Define the Chebyshev polynomial T_{n}(x) on [−1, 1] and find its roots.

4. State (need not prove) the error formula for polynomial interpolation. Give an error bound in sup-norm for polynomial interpolation of e^{x} on [−1, 1] using roots of T_{5}(x) as interpolating points.

5. Does

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} tan^{3}(\frac{iπ}{2n})

converge? If so, what is the limit? If not, what is the leading order term in terms of n?