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

Homework Assignment for Week 14

1. Section 6.6: Problems 3(a,c), 7(a,c), 14(a,b).

Remark: These are paper and pencil assignments. You are asked to construct the matrices L and D. Do so by performing Gauss elimination with paper and pencil and collect l_{ij} and d_{i} in the process. Then in problem 7, you can either use matlab backslash or a few loops to solve for the corresponding linear systems.

2. Section 6.6: Problems 17, 20, 21, 25, 32.

3. Prove Corollary 6.29.

4. Construct the matrix for 2D and 3D Laplace equation Δu = f on [0, 1]^{2} and [0, 1]^{3} with Dirichlet boundary condition (u prescribed on the boundary) using the partition 0 = x_{0} < x_{1} < ⋯ < x_{N} = 1, 0 = y_{0} < y_{1} < ⋯ < y_{N} = 1, 0 = z_{0} < z_{1} < ⋯ < z_{N} = 1 with uniform mesh size x_{i} − x_{i−1} = y_{j} − y_{j−1} = z_{k} − z_{k−1} = h = \frac{1}{N}. The unknowns are u_{ij} or u_{ijk} with 1 ≤ i, j, k ≤ N − 1, respectively. Give the leading order operation KN^{p} for for LU decomposition and forward, backward substitution in 2D and 3D, respectively.

5. Implement your own tridiagonal solver using indexing of the entries as in problem 27.