CSCI 1120 (Low-Level Computing), Spring 2020:
Homework 4

Credit:

20 points.

Reading

Be sure you have read, or at least skimmed, the assigned readings for classes through 9/18.

Programming Problems

Do the following programming problems. You will end up with at least one code file per problem. Submit your program source (and any other needed files) by sending mail to my TMail address with each file as an attachment. Please use a subject line that mentions the course and the assignment (e.g., “csci 1120 hw 4” or “LL hw 4”). You can develop your programs on any system that provides the needed functionality, but I will test them on one of the department's Linux machines, so you should probably make sure they work in that environment before turning them in.

1. (10 points) NOTE that this problem is meant as an opportunity to get some practice with while loops in C, so you will only get full credit if you use at least one.

   Newton's method for computing the square root of a non-negative number $x$ starts with an initial guess $r_0$ and then repeatedly refines it using the formula

   $$r_n = (r_{n-1} + (x / r_{n-1})) / 2$$
   Repetition continues until the absolute value of $(r_n)^2$ - $x$ is less than some specified threshold value. An easy if not necessarily optimal initial guess is just $x$. So for example the calculation starts like this for $x = 2$:
   

   $$r_0 = 2$$ (1)

   $$r_1 = (r_0 + 2/r_0) / 2$$ (2)

   $$= (2 + 2/2) / 2$$ (3)

   $$= 1.5$$ (4)

   $$r_2 = (r_1 + 2/r_1) / 2$$ (5)

   $$= (1.5 + 2/1.5) / 2$$ (6)

   $$= 1.1417 \;\; \mathrm{(approximately)}$$ (7)

   
   

   Write a C program that implements this algorithm and compares its results to those obtained with the library function sqrt(). Have the program prompt for $x$, the threshold value, and a maximum number of iterations; do the above-described computation; and print the result, the actual number of iterations, the square root of $x$ as computed using library function sqrt(), and the difference between the value you compute and the one you get from sqrt(). Also have the program print an error message if the input is invalid (non-numeric or negative -- but note that zero is okay).

   Here are some sample executions (assuming you call your program newton and compile with make)
   

   [bmassing@diasw04]$ ./newton
   enter values for input, threshold, maximum iterations
   2 .0001 10
   square root of 2:
   with newton's method (threshold 0.0001):  1.41422 (3 iterations)
   using library function:  1.41421
   difference:  2.1239e-06
   
   [bmassing@diasw04]$ ./newton
   enter values for input, threshold, maximum iterations
   2 .000001 10
   square root of 2:
   with newton's method (threshold 1e-06):  1.41421 (4 iterations)
   using library function:  1.41421
   difference:  1.59472e-12
   

   Hints:

   • For this problem I recommend that you resist any impulse to split up your program into several functions; I think it's simplest and clearest to just put all the needed computation in main().
   • Remember that to use the library function sqrt() you need not only the appropriate #include line (as documented in its man page) but also the compile flag -lm (also documented in the man page). (The suggested Makefile includes this flag.)
   • You may find the library function fabs() useful.

2. (10 points) NOTE that this problem is meant as an opportunity to get some practice with for loops in C, so you will only get full credit if you use at least one.

   C, like many programming languages, has a library function (rand()) that can be used to generate a “random” sequence of numbers (quotes because it's not truly random -- more in video lecture). Many languages have a similar function that generates “random” numbers in some specified range, useful if for example you're trying to simulate rolling a 6-sided die. C doesn't have such a function, but you can get the same effect using rand() and a little additional code. rand() itself generates a number between 0 and the library-defined constant value RAND_MAX, so to get a value in a smaller range you have to somehow map the larger range to the smaller one. The somewhat obvious way to do this is by computing a remainder (e.g., to map to two possible values, assign even values to 0 and odd values to 1). (I'll call this the “remainder method”.) But with some implementations of rand() this gives results that aren't very good. The conventional wisdom is therefore to instead try to do a more-direct map (e.g., to map to two possible values, assign values from 0 through RAND_MAX/2 to 0 and the remaining values to 1). (I'll call this the “quotient method”.)

   Your mission for this problem is to complete a C program that, given a number of samples N and a number of “bins” B generates a sequence of N “random” numbers, uses both methods (remainder and quotient) to map each generated number to a number between 0 and B-1 inclusive, and counts for each method how many elements of the sequence fall into each bin (e.g., for each method bin 0 is how many elements of the sequence map to 0), and prints the result, as in the sample output below. To help you (I hope!) I'm providing a starter program, link below, which you should use as your starting point. Code at the bottom of the program shows how to apply both methods to something returned by rand(). The remainder method is straightforward; the quotient method is less so, but see the footnote¹for an explanation.

   One other thing to know about rand() is that by default it always starts with the same value (and produces the same sequence). To make it start with a different value, you can call srand() with an integer “seed”, so your program should prompt for one of those too.

   Sample execution (assuming you call your program rands and compile with make):
   

   [bmassing@diasw04]$ ./rands
   seed?
   5
   how many samples?
   1000
   how many bins?
   6
   counts using remainder method:
   (0) 154
   (1) 188
   (2) 171
   (3) 161
   (4) 155
   (5) 171
   counts using quotient method:
   (0) 172
   (1) 175
   (2) 183
   (3) 150
   (4) 168
   (5) 152
   

   If you feel a bit more ambitious you could also have the program print maximum and minimum counts and the difference between them, as a crude measure of how uniform the distribution is:
   

   [bmassing@diasw04]$ ./rands
   seed?
   5
   how many samples?
   1000
   how many bins?
   6
   counts using remainder method:
   (0) 154
   (1) 188
   (2) 171
   (3) 161
   (4) 155
   (5) 171
   min = 154, max = 188, difference 34
   counts using quotient method:
   (0) 172
   (1) 175
   (2) 183
   (3) 150
   (4) 168
   (5) 152
   min = 150, max = 183, difference 33
   

   (You will get an extra-credit point for doing this.)

   Here is a starter program that prompts for the seed, generates a few “random” numbers, and illustrates the two methods of mapping to a specified range: rands.c.

   Of course, your program should check to make sure all the inputs are positive integers. (Yes, error checking is a pain, but it's an incentive to get better at copy-and-paste?)

   Hints:

   • If you find the problem description confusing (many students do, alas), maybe an example will clarify a bit:

     Suppose input specifies 100 samples and 4 bins.

     The program should generate a sequence of 100 “random” numbers.

     For the remainder method, bin 0 will be a count of elements $n$ of this sequence for which $n / 4$ gives a remainder of 0, bin 1 will be a count of elements $n$ of this sequence for which $n / 4$ gives a remainder of 1, and so forth.

     For the quotient method, bin 0 will be a count of elements $n$ of this sequence that fall into the first 1/4 of the range 0 through RAND_MAX, bin 1 will be a count of elements $n$ of this sequence that fall into the second 1/4 of the range 0 through RAND_MAX, and so forth.

     If you still find this confusing, please ask for help! I think all the ACM tutors have done this problem and they can likely explain to you, or I'm happy to give it another try myself.

Pledge

Include the Honor Code pledge or just the word “pledged”, plus at least one of the following about collaboration and help (as many as apply).²Text in italics is explanatory or something for you to fill in. For programming assignments, this should go in the body of the e-mail or in a plain-text file pledge.txt (no word-processor files please).

• This assignment is entirely my own work. (Here, “entirely my own work” means that it's your own work except for anything you got from the assignment itself -- some programming assignments include “starter code”, for example -- or from the course Web site. In particular, for programming assignments you can copy freely from anything on the “sample programs page”.)
• I worked with names of other students on this assignment.
• I got help with this assignment from source of help -- ACM tutoring, another student in the course, the instructor, etc. (Here, “help” means significant help, beyond a little assistance with tools or compiler errors.)
• I got help from outside source -- a book other than the textbook (give title and author), a Web site (give its URL), etc.. (Here too, you only need to mention significant help -- you don't need to tell me that you looked up an error message on the Web, but if you found an algorithm or a code sketch, tell me about that.)
• I provided help to names of students on this assignment. (And here too, you only need to tell me about significant help.)

Essay

Include a brief essay (a sentence or two is fine, though you can write as much as you like) telling me what if anything you think you learned from the assignment, and what if anything you found found interesting, difficult, or otherwise noteworthy. For programming assignments, it should go in the body of the e-mail or in a plain-text file essay.txt (no word-processor files please).

Footnotes

... footnote¹

If you're curious about the admittedly cryptic formula: Suppose n is the value returned by rand(). We could convert it to a bin number in two steps: First, we scale it to a floating-point number in the range from 0 up to but not including 1, thus:

$$x = \mathit{n} / (\mathtt{RAND\_MAX} + 1)$$

(We divide by RAND_MAX+1 so the largest possible value maps to something still slightly less than 1. Note also that in the code we have to be sure this addition is done using floating point, since otherwise it could overflow; we can do this by writing the 1 as 1.0.) Then we scale this range of floating-point values to our desired range (still in floating point) by multiplying by B, which gives from 0 up to but not including B. Finally, we convert back to an int with a cast, which drops the fractional part, and leaves us with an index between 0 and B-1 inclusive.

... apply).²

Credit where credit is due: I based the wording of this list on a posting to a SIGCSE mailing list. SIGCSE is the ACM's Special Interest Group on CS Education.