Numerical Analysis I, Fall 2011 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 13

Assigned Dec 09, 2011.

1. Section 7.4: Problems 2(c), 4, 7.

2. Construct the matrices A, B corresponding to the equations v′′(x) = f and u(x) + u′(x) = g, respectively. Verify numerically that the condition numbers depend on the size of the matrix, or on the grid size as κ(A) = O(1/h^{2}) and κ(B) = O(1/h), respectively.

3. Let A be the matrix resulted from discretizing

(∂_{x}^{2} + ∂_{y}^{2})u(x, y) = f(x, y), (x, y) ∈ (0, 1)^{2} u = 0, on the boundary of (0, 1)^{2}

(1)

with uniformly spaced grids 0 = x_{0} < x_{1} < ⋯ < x_{N} = 1, 0 = y_{0} < y_{1} < ⋯ < y_{N} = 1, x_{i} − x_{i−1} = y_{j} − y_{j−1} = h = 1/N, using second order centered finite difference method.

Write a pseudo-code for solving Au = f using Jacobi, Gauss-Siedel and SOR iteration, respectively (first study your class note for the 1D case and try to generalize it to this 2D case).