Mathematics 524 Test #3 Name:

This due in class the Tuesday after break.

1. Find the minimum and the minimizer of the function f(x, y) = x + 2y + 3xy + \frac{4}{xy^{2}} on the set

x, y > 0.

2. Let g_{1}(x, y) = x + 2y − 4 and g_{2}(x, y) = 2x + 4y − 7 and set

g = [\begin{matrix}g_{1}(x, y) \\ g_{2}(x, y) \end{matrix}] .

Then take enough steps in Broyden’s for solving g = 0 starting with x^{(0)} = [\begin{matrix}0 \\ 0 \end{matrix}] and D_{0} = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}]

to verify that it is converging to what it should converge to.

3. Problem 17 on page 128 of the text. Write up your answer carefully.

4. In the last problem (#1 page 128) you showed

|x^{(k+1)} − r| ≤ \frac{M}{2m}|x^{(k)} − r|.

Explain and show by example why this means that the number of correct decimal places doubles in each step of Newton’s method.