Remember: You must hand in your matlab codes along with the mathematics (if needed), discussion and results.

engineering

Description

Notes:

• Remember: You must hand in your matlab codes along with the mathematics (if needed), discussion and results. Also, as indicated in class, these codes must be commented in accordance with the examples appearing on the website.

1. Apply Newton’s method for n > 1 dimensions to find an approximate solution of the (“simultaneous”) pair of equations

x 3 2 /3 + x2 − x1 exp(−x 2 1 /2) = 0, x 2 1 + x 4 1 /4 = x 3 2 + 3x2. Start from (x1, x2) = (2, 2).

2. Suppose you will solve x = g(x) using a fixed-point method. Suppose you know that |g 0 (x)| ≤ 0.3 for all x. Starting from x0 = 0, you obtain x1 = 2.0. How many steps would be required to guarantee an error no greater that 10−5 in your estimate of the solution? That is, if the true solution is denoted by ¯x, what is the minimum value of n such that you can guarantee |xn − x¯| ≤ 10−5 ?

You may choose to stop based on the number of iterations, or on a more complex condition, but your solution approximation should be reasonably close to a true solution.