Topics:

• Organizational details, introduction to the lab at UNGAR 426.
• What is scientific computing?
• An Ancient calculation of sqrt(2): Sumerian tablet YBC7289
• Simulated flow trajectoris during the Deepwater Horizon Oil Spill. NOAA
• Heron's Original paper Newton's Original paper Raphson's Original paper Newton's Method

Assignment:

1. Read the original paper of Heron (see the link above). Carry out one more iteration of Heron's algorithm by hand (using paper and pencil, that is). Make sure to use fractions only. What is the error bound after this second iteration? Write the precise steps in the Heron procedure to make it an algorithm that a computer can understand. After some web search, write a paragraph about the life of Heron.

2. Using the modern version of Newton-Raphson method described in the lectures, find an approximation to a root of Newton's cubic polynomial y^3 - 2y -5 = 0 starting with the initial guesses y_0 = 2.2, and also y_0 = 1.5. How many iterations do you have to make to get Newton's finding 2.09455148? (See the Newton's original paper above.) How good is this answer as an approximate root? To to this problem, you first need to determine which function to iterate; then enter this function into Phaser as a Custom Equation (MAP). Does Newton's polynomial have any other real root? (Hint: Using a function plotter, plot the graph of the cubic polynomial.)

3. Enter the difference equation x1 -> x1 - 0.21*(x1*x1 - 2) into PHASER. Determine the initial conditions whose iterates converge to sqrt(2). Compare the rate of convergence of this method of computing sqrt(2) with that of Newton's method. To make a fair comparison, use the same initial condition, say 2.1, in both iterations.

4. Using Newtons's method, with starting value x = 0, try to find a root of the function f(x) = x^3 - 2x + 2. Does the iteration converge?

5. Compare two of the genomic sequences we examined in class, with accession numbers 'AF176722' and 'AF315498'. These are the mitochondrial D-loop sequences of the Chimp Schweinfurthii and Chimp Vellerosus respectively. To compare the sequences, use the tool blast2seq, which can be found here: http://www.ncbi.nlm.nih.gov/blast/bl2seq/wblast2.cgi [www.ncbi.nlm.nih.gov] [www.ncbi.nlm.nih.gov [www.ncbi.nlm.nih.gov]] (and is the first google match when searching for 'blast2seq'). You could download (cut/paste) the sequences from the genbank database (just search for 'genbank' on google), but it is even easier to use the accession numbers. Please report the number of substitutions and insertion/deletions (indels) to transform one sequence to the other. These can be found by looking at the alignment on the bottom of the result page from blast2seq. What is the ratio of substitutions/indels? Which one do you think is more probable to occur by chance in a genomic sequence with time, a substitution or an indel? Why?