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

Quiz 06

Jan 05, 2010.

1. Let A, B be two real n × n matrices. Show that ∥AB∥ ≤ ∥A∥∥B∥. Here the matrix norm is not specified (it could be generated from L^{2}, L^{∞} or other norms in ℝ^{n}). You can prove it for one particular norm, or give a proof that works for any norm.

2. Show that cond(A) ≥ 1 for any A.

3. Describe Jacobi iteration and Gauss-Siedel iteration.

4. Let A = ( \begin{matrix}3 & 1 & 0 \\ 1 & 3 & 0 \\ −1 & 1 & 3 \end{matrix} ) . Give a convergent iteration method for solving Ax = b. Explain why it converges.

5. Let B = ( \begin{matrix}0.5 & 1 & 0 \\ 1 & 0.5 & 0 \\ −1 & 1 & 0.5 \end{matrix} ) . Give a convergent iteration method for solving Bx = b. Explain why it converges.

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

Quiz 06

Jan 05, 2010.

1. Let A, B be two real n × n matrices. Show that ∥AB∥ ≤ ∥A∥∥B∥. Here the matrix norm is not specified (it could be generated from L^{2}, L^{∞} or other norms in ℝ^{n}). You can prove it for one particular norm, or give a proof that works for any norm.

2. Show that cond(A) ≥ 1 for any A.

3. Describe Jacobi iteration and Gauss-Siedel iteration.

4. Let A = ( \begin{matrix}3 & 1 & 0 \\ 1 & 3 & 0 \\ −1 & 1 & 3 \end{matrix} ) . Give a convergent iteration method for solving Ax = b. Explain why it converges.

5. Let B = ( \begin{matrix}0.5 & 1 & 0 \\ 1 & 0.5 & 0 \\ −1 & 1 & 0.5 \end{matrix} ) . Give a convergent iteration method for solving Bx = b. Explain why it converges.