In particular, you will write a second version of sum-of-divisors.Using the techniques you learned in class, you should do a theoretical analysis of each version to produce predictions about how the time the computer takes to do a computation changes with the size of the computation. Next, you will measure the time the computer takes to do computations of varying sizes. (This is known as empirical testing.) Then you can compare your theoretical predictions with your empirical results. You will also write a procedure that uses sum-of-divisors, and do similar analyses.
In class, we'll talk about how to present your timing data both graphically and by using a table as you try to confirm your theoretical run-time predictions. For the project report, while we recommend you present the same data both graphically and in a table (for practice and to get feedback), you are only required to present the data in one of the two formats.
aprocedure which computes the ratio of abundant and deficient numbers, in class on Monday, March 5. You'll then have time to do the less rote parts of the labs during the lab period.
On this lab, you will work in groups of 2 two (with perhaps one group of three, should that prove necessary due to an odd number of students). You may wish to choose a partner who is both in your lab section and your lecture. You should not work with the same partner twice during the course.
We warn you that working in a group of two does not mean that your work load will be halved; in fact, some groups may find that the amount of hassle will effectively increase the work. Our intention is that you gain experience in working in teams, since teamwork is very common in today's workplace. You should have a clear understanding of each other's responsibilities, and be dependable and punctual in carrying them out. Also, be sure that neither partner monopolizes the work, but rather that you share it as equally as possible. Finally, we strongly encourage you to set up a group meeting with a lab instructor after your first or second day of working together in lab, so that we can assess how the project is going and, if need be, suggest strategies for more successfully working together.
As we said in the first project, we will assign a specific audience for each project report. This report might be called scientific in nature, since the emphasis should be less on the writing of the procedures, and more on the theoretical and empirical analysis that you do of those procedures. When you write the project report, you may assume that your audience is generally knowledgeable about computer science and Scheme, but not about what you did.
After a short description the procedures you wrote (including full copies of those procedures and a brief description of how you tested that they were accurate), your lab report should concentrate on answering the central questions for this project. In this case, you have two sets of procedures. In each set, there are two procedures which produce the same result when given the same data. Your job is to compare the amount of time the computer takes to run the procedures, using both theoretical analyses and empirical results.
In writing this report, you will want to include not only the procedures you used and the measurements you made, but a clear analysis of the asymptotic orders of growth (in big Theta-notation) for each of the two methods of doing each of the two computations. In particular, for each method, you will need to
You might also want to make some additional predictions that go beyond your measurements. Suppose you were interested in pursuing the abundant/deficient ratio study well beyond n=3200, perhaps to n=102400. How long would you expect each method to take? Would you want to experimentally verify this? A harder (optional) question is, how large an n would the computer be able to handle in a day? a week? a year?
Finally, there are three documents you should consult when writing up your project report:
One use for the sum-of-divisors procedure other than finding perfect numbers is to measure the relative frequency of abundant and deficient numbers. As mentioned in section 3.3, very few numbers are perfect. The remaining numbers either have a sum of divisors that is more than twice the number, in which case they are called abundant, or less than twice the number, in which case they are called deficient. The empirical question we'd like to ask is, are these two kinds of numbers equally common, or does one predominate? If so, by how much? The usual mathematical way to formulate this is to count how many numbers less than or equal to n are abundant and how many are deficient, and take the ratio. Then repeat this for larger and larger values of n. If the ratio approaches closer and closer to some limiting value as n increases, then this limiting ratio tells us the relative frequency of abundant and deficient numbers. For example, if the ratio approaches a limit of 1, then abundant and deficient numbers are equally frequent. On the other hand, a limiting value of 1/2 would mean that there are roughly twice as many deficient numbers as abundant numbers.Write a procedure that generates an iterative process for computing this ratio, given n. Most of the work can be done by a subsidiary procedure that keeps count of how many abundant numbers it has found and how many deficient numbers it has found as it scans the range from 1 to n. Once the entire range has been scanned, one count can be divided by the other. Test your procedure to be sure that it works.
Do the abundant/deficient ratios seem to approach some simple limiting value? If so, what?
(load "~mc27/public/time.scm")into your definition window and execute it. Now you should time how long it takes to find the sum of divisors of 100, 1000, 10000, 100000, and 1000000. You should do the same timing repeatedly, at least for the smaller numbers, to get some idea how precisely repeatable the timings are. To illustrate how you would do a timing, here is a sample timing of the sum-of-divisors of 100:
> (run-time sum-of-divisors 100)The time is given in seconds; e.g., in the above example, the evaluation took a little less than thousandth of a second. The time reported will not agree well with your own impression of the time taken, because it doesn't include time the computer spends doing other things, and also because for fast evaluations (such as the above) the evaluation is repeated until at least a second has elapsed and an average time for the repetitions is reported.
What is the ratio of times from each measurement to the next? (That is: What is the ratio of times going from 100 to 1000? From 1000 to 10000? And so forth.) How does this compare with your expectation? (Note: When answering questions like these in your project write-up, be sure to back up your answers with adequate arguments. In other words, give reasons, probably in the form of big Theta arguments, which explain your expectations.)