15-212-X : Homework Assignment 1

Due Wed Sep 11, 10:00 am (electronically); papers at recitation.

Maximum Points: 50 (+10 extra credit)


Problem 1: Binomial numbers (35 pts)

Binomial numbers play an important role in combinatorics because they answer the question of how many subsets can be formed by picking r elements out of a set of n ("n choose r" or "n over r"). There are several ways to calculate binomials. One way is by using factorials, the other one by reading them out of Pascal's triangle. One of the aims of this problem is to show the equivalence between both definitions.

First, the factorial function on natural numbers is defined mathematically as follows:

0! = 1
n! = n * (n-1)! for n > 0

The factorial function is the basis for the first definition of binomial numbers. BN(n,r) denotes the binomial number for (n,r) or "n choose r". It is defined as follows:

BN(n,r) = n!
r! (n-r)!
for 0 <= r <= n

A way to calculate binomial numbers which does not require factorials is to use Pascal's triangle. The triangle has a 1 at its top, every other number in the Pascal triangle is defined as the sum of the numbers to the right and to the left above. If there isn't any number, it counts as zero.

We denote a binomial number defined by the Pascal triangle as PT(n,r), where n stands for the row in the figure above, and r counts from left to right, starting at 0.

Question 1.1 (5 pts)

Write an ML program to calculate PT(n,r). Your function should be named pascal and must have type int * int -> int . Calculate the following values PT(5,3) and PT(7,1) .

Question 1.2 (10 pts)

Show by induction that pascal is a correct implementation of the binomial as defined using factorials. Carefully state and check the boundary conditions! You don't need to type your proof in (writing it by hand and handing it in at recitation is usually faster and easier).

Binomial numbers have other somehow surprising properties:

Question 1.3 (10 pts)

Consider the sum BN(n,0) + .... + BN(n,n) for an arbitrary n. This sum can be written in closed form. Find the closed form, and prove it correct. To find the closed form, write an ML program sumAll which is of type int -> int. Experiment with the program and guess the closed form. Then prove by induction that your claim is correct.

The next problem is very similar to the previous one.

Question 1.4 (10 pts)

Consider the sum BN(n,0) - BN(n,1) + .... + (-1)^n * BN(n,n) again for an arbitrary n. This sum can be written in closed form. Find the closed form, and prove it correct. To find the closed form, write an ML program sumNeg which is of type int -> int. Experiment with the program and guess the closed form. Then prove by induction that your claim is correct.

Question 1.5 (extra credit, 10 pts)

Give a straightforward implementation of a function binomial which uses factorials. This implementation is likely to lead to rather frequent overflow exceptions. Test it to find some values of n and r where the computation requires intermediate results which exceed the size allowed by the machine representation of integers. Compare this to the function pascal. Consider how you might define a function to compute binomials based on multiplication which works for a larger set of natural numbers. [Of course, eventually the result will be too large to be representable with machine integers and no further improvement is possible without another implementation of integers.]

Problem 2: Numerical Integration (15 pts)

The second problem treats some computations with real numbers and functions over them. We will discuss and implement a method for numerical integration of functions of one argument. Numerical integration means, that we look for a good approximation of the integral, but not for a "closed form". Naturally, we cannot expect to guess the solution with the first approximation, therefore we must calculate a sequence of approximation values, each an improvement of the previous one. The idea behind this approximation is very simple: Consider the following graph:
and an equi-distant partition of the x-axis. What happens if we calculate the area of the rectangles in the graph below (colored blue)?
Note: The height of the blue rectangles is always defined by the function value at the left border of the interval.

We expect to obtain an approximation value which is close to the integral. It should be clear from the graph above, that the approximation value might be slightly above or below the real value. It is one of the easier theorems of Calculus that with finer partitions we can hope for a more refined approximation result. You can see this, in the next picture where we make one refinement step by splitting every partition on the x-axis in two equal parts. The sum of the red rectangles determines now the approximation value. Note that the light blue areas have been counted for in the previous step, but not any more.

Typically in search for a good approximation value, we continue this kind of splitting in the same fashion. The nth approximation value is hence determined by the sum of 2n rectangles. To decide now about good and bad approximations we must define a criteria for deciding when to stop with the partition refinement. Unfortunately, your program cannot access the "real" value of the integral, so we must define the criteria on the basis of the previous approximation value and the current value. We accept an approximation value if the absolute value of the difference between previous and current value is less then a given (positive) epsilon.

Question 2.1 (10 pts)

Write an SML function

integrate : real -> (real -> real) -> (real * real) -> (int * real)

where integrate epsilon f (a,b) approximates the integral of f between a and b up to epsilon. It returns a a pair, where the first component yields the current refinement level (n, not 2n), and the second the approximation value, fulfilling the condition from above. Integrate
  1. the sine function between 0 and pi
  2. the exponential function between 0 and 1
  3. the function f(x) = cos(2*pi*sin(x)) between 0 and pi
and check which integral converges fastest for epsilon = 10-6 (that is, 0.000001).

Hint: Use the context browser to access information about the mathematical functions from the Math structure.

Question 2.2 (5pts)

Now fix epsilon = 10-6 (= 0.000001) and write a function

integrate0 : (real -> real) -> (real -> real)

where integrate0 f is the function g(x) which represents the value of the integral of f between 0 and x, approximated with bound epsilon.

Handin instructions