Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 11

1. Section 6.1: Problems 10, 15(a), 16(a).

For problems 15(a) and 16(a), just count multiplications/divisions.

2. Derive a linear system of equation corresponding to the following boundary value problem

\begin{matrix}u′′(x) + u′(x) = f(x), x ∈ (0, 1) \\ u(0) = α, u(1) = β \end{matrix}

with uniformly spaced grids 0 = x_{0} < x_{1} < ⋯ < x_{N} = 1, x_{i} − x_{i−1} = h = 1/N, using second order finite difference method. That is, given α, β, and f_{i}, i = 1, 2, ⋯, N − 1, try to derive a linear system of equations to solve for u_{i}, i = 1, 2, ⋯, N − 1. You don’t have to solve the linear system this week (but soon will).

3. Consider an N^{2} × N^{2} matrix A with a_{ij} = 0 except for i − j = 0, ±1, ±2 (only five diagonals have nonzero entries). Estimate total number of multiplication needed for Gaussian elimination (without pivoting) on A. Give the leading order of the number of multiplication as KN^{p}. Find K and p.

4. Do the same for an N^{2} × N^{2} matrix B with b_{ij} = 0 except for |i − j| ≤ N (only 2N + 1 diagonals have nonzero entries).

5. Do the same for an N^{2} × N^{2} matrix C with c_{ij} = 0 except for i − j = 0, ±1 and ±N (only 5 separated diagonals have nonzero entries).

6. Section 6.2: Problems 1(a), 3(a), 5(a), 31 (all in paper-and-pencil instead of programming).