Numerical Analysis I, Fall 2020 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 06

Goal: Study both theoretical and practical aspects of Fixed Point Iteration and Newton’s method for nonlinear system of equations.

1. Section 10.1:

Suppose that D ⊂ R^{n} and G : D ↦ D is sufficiently smooth.

Denote by the matrix norm ∥|A∥|_{2} = sup_{x≠0} \frac{∥Ax∥_{2}}{∥x∥_{2}} for any n × n real matrix A, where

∥x∥_{2} is the standard ℓ_{2} norm for x ∈ R^{n}.

(a) Give a sufficient condition for uniqueness of the fixed point x = G(x) in terms of ∥|G′∥|_{2}.

(b) Give a sufficient condition for convergence of the fixed point iteration x^{k+1} = G(x^{k}) and its error estimate in terms of ∥|G′∥|_{2}.

Hint: Go through and mimic the proofs of Theorem 2.3, 2.4 and Corollary 2.5.

2. Section 10.1: Problems 3(a,b), 4(a,b).

3. Section 10.1:

As in the scalar case, where knowing the approximate value of f′(x_{∗}) would help to design a fixed point iteration for solving the equation f(x) = 0 by finding a suitable equivalent form x = g(x) by introducing a free parameter α. Try apply this technique to the following system of equations

\begin{aligned}1x_{1} + 2x_{2} + 0.03 ∗ sin(x_{1} + x_{2}) & = 4 \\ 5x_{1} + 6x_{2} + 0.07 ∗ cos(x_{1} − x_{2}) & = 8 \end{aligned}

and find a convergent fixed point iteration. If the equivalent form is not obvious to you, to use the trick as in the scalar case, but take α to be a 2 × 2 matrix.

If you already did it on Tuesday’s recitation, change the numbers and practice again yourself.

4. Section 10.2: Problems 7(a,b).