Exercises to Hand In

1.

Use Newton's Method to solve the simultaneous equations $x^{2}-y^{2}=-1,\,y^{2}-x=3$ with initial guess x=y=-1. What is the actual (i.e. exact) solution? How many interations of Newton's method does it take to get the correct answer to 16 places?

2.

Find a local minimum for the function $f\,(x,y)=x^{3}+y^{3}-3x-3y+5$.(First, look at some surface and contour plots to find an estimate for where the minimum occurs -- don't forget that x³ is entered as x.^3 in setting up the plot -- next, use Newton's method to find the critical point accurately.)

3.

(Challenging) Find one maximum and one minimum (to 16 places) for the function

$$z=.02\sin(x)\sin(y)-.03\sin(2x)\sin(y)+.04\sin(x)\sin(2y)+.08\sin(2x)\sin(2y)$$

(This represents a Fourier approximation to a certain vibrating membrane.) You should draw some plots to find good starting approximations for using Newton's Method to find the critical points -- contour plots are probably easiest to use for this. Here is an m-file for use in plotting this function -- note the use of a broken string (with a +) to make the function more readable:

$$\begin{tabular}[c] {c}\texttt{membrane.m}\\ \hline \multicol... ...t l\vert}{\texttt{z=eval(vectorize(f));}}\\ \hline\end{tabular}$$