Topics:

• Organizational details, introduction to the labs at UNGAR 311 and 426.
• What is scientific computing?
• An Ancient calculation of sqrt(2): Sumerian tablet YBC7289
• 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. 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.

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 guess y_0 = 2. How many iterations do you have to make to get Newton's finding 2.09455148? Does Newton's polynomial have any other real root?

3. Enter the difference equation x1 -> x1 - 0.25*(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.

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 sequences we examined in class, with accession numbers 'AY079510' and 'AF050738' respectively. These are the mitochondrial D-loop sequences of the Western Lowland Goriila and the Eastern Lowland Gorilla. 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] (and is the first google match when searching for 'blast2seq'). You can download (cut/paste) the sequences from the ncbi database (just search for 'ncbi' on google). 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 easy to happen by chance in a genomic sequence, a substitution or an indel? Why?

6. Extra Credit: Try to read Newton's original paper linked above. For extra credit you can write an algorithm for Newton's original procedure.