Homework $-$ Lab. 1

These exercices should be done by Monday February 3rd at 9.00 am, (I gave you an extra weekend to waste on them), email the results from your runs of numbers 2, 3 and 4 to me at sph11@cornell.edu, finding out how to save the results as a text file and how to email a text file are part of the assignment!

1. Using the web

• Using Bearaccess connect to the 494/694 home page at
  http://www.biom.cornell.edu/btry494/. Look for the reference to Charles Van Loan's matlab programs.
• FTP to his ftp site and download, in text format the files Float.m, Represent.m, Convert.m, Pretty.m from the folder Chapter 1 of his SciCompFiles into a local directory that you should create either on c: or on a diskette (a:).
• Download the leafstat.dat file (as text file) and the file lesson1.m from the site: http://www.math.utk.edu/~gross/matlabfiles/quant.lifesci.matlab.html

2. Simulating floating point arithmetic

Using Van Loan's functions, show how floating point errors occur, find an example of three numbers whose variance computed with the textbook algorithm has negative variance when the computations are done with 3 decimal places only.

3. A monte carlo experiment to find $\pi$

If you throw uniform random darts within a square board, the ratio of the number of throws within a circle of diameter the side of the square to the total number of throws will be $\frac{\pi}{4}$, use this method to compute an approximation to $\pi$.

Take an average over 500 simulations of 1000 throws.

Compute the variance of your estimate.

You can compare with a very good representation of $\pi$ you can consult through the command vpa(pi).

4. A computer experiment to show how $\phi^n$ diverges

As we saw in class, the golden ratio $\phi=\frac{\sqrt{5}-1}{2}$ obeys the recurrence relation

$$\phi^{n+1}=\phi^{n-1}-\phi^n$$

Compute the first 150 terms of this recurrence, and compare to the first 150 powers of $\phi$ (from 0 to 150), plot the terms when the series starts to diverge by more than 1.

5. Testing data entry and a little statistics

Try doing lesson 1 from the matlab lessons:
http://www.math.utk.edu/~gross/matlabfiles/quant.lifesci.matlab.html