CS 221 Fall 2011 Problem Set 4

CS 221 Fall 2011 Problem Set 3

Out: Tuesday, November 8
Due: 23:59:59 Wednesday, November 16

Problem 1: Implementing the Fixed-Point Root-Finding Method

Implement a function fixpoint() that takes three arguments:

A function handle (the function for which the fixed point is to be found);
An initial guess; and
A bound on the estimated relative error (as defined in lecture).

The function returns a single value, x, such that |f(x)-x|/f(x) is less than the given bound.

Your function should include checks to detect when it is diverging (Hint: conditions for convergence are given in the slides from lecture; the computation is diverging if successive estimates are getting further apart.)

Save the function in a file "fixpoint.m", and include it in the zip archive you turn in.

Problem 2: Solving Problems

Solve the following problems using (i) the root-finding methods of MATLAB (you may use whichever method applies); (ii) the fixpoint() function you implemented above; and (iii) Excel. For each method, show the value of the root calculated with that method, and the value of the function evaluated at that point. When you have a choice, use an error bound of 10^-4.

You are designing a water tank for a village in Africa. The tank is spherical, with a radius R meters. The volume of water it holds when filled to height h is given by:
V = πh²(3R-h)/3
If the radius of the tank is 4 meters, to what height must it be filled to hold 30 m³ of water?
Repeat for a tank with radius 3 meters and desired volume of 20 m³?
When calculating the payment on a mortgage, the relationship between the principal P (i.e., the loan amount), the monthly payment m, the term n of the loan in months, and the annual interest rate r, is given by the annuity equation:
```
             P r
 m =     -----------
       12(1 - 1/(1+r/12)^n)
```
What duration (n) is required for $200,000 loan at 3.7% (use 0.037 as the parameter value) to get a monthly payment of $1200?

Submission Instructions

To submit your solutions, follow these steps:

Create a folder titled Last_First_HW4 (replacing "Last" and "First" with your actual names) on the Desktop.
Move the files you are supposed to turn in into that folder. Nothing else should be in the folder.
Create a zip file containing the folder and its contents. (The main course web page has instructions for creating a zip archive.)
Important: Verify that the contents of the zip archive are correct. It should contain the directory (folder) and the three files, and nothing else. (Do this by double-clicking the zip file.)
Upload the zip archive via the portal, as
Important: save the folder and its contents to a flash drive or elsewhere, for two reasons: (i) in case something goes wrong with your submission; and (ii) we will sometimes build on your solutions in later problems.
Very Important: save the number the submission system gives after you upload your submission. It is the only acceptable proof that you uploaded your file.