Where are “Special Numbers” mentioned in Concrete Maths used?

Question

I was glancing through the contents of Concrete Maths online. I had at least heard most of the functions and tricks mentioned but there is a whole section on Special Numbers. These numbers include Stirling Numbers, Eulerian Numbers, Harmonic Numbers so on. Now I have never encountered any of these weird numbers. How do they aid in computational problems? Where are they generally used?

Answer 1

Most of these numbers count certain kinds of discrete structures (for instance, Stirling Numbers count Subsets and Cycles). Such structures, and hence these sequences, implicitly arise in the analysis of algorithms.

There is an extensive list at OEIS that lists almost all sequences that appear in Concrete Math. A short summary from that list:

• Golomb's Sequence
• Binomial Coefficients
• Rencontres Numbers
• Stirling Numbers
• Eulerian Numbers
• Hyperfactorials
• Genocchi Numbers

You can browse the OEIS pages for the respective sequences to get detailed information about the "properties" of these sequences (though not exactly applications, if that's what you're most interested in).

Also, if you want to see real-life uses of these sequences in analysis of algorithms, flip through the index of Knuth's Art of Computer Programming, and you'll find many references to "applications" of these sequences. John D. Cook already mentioned applications of Fibonacci & Harmonic numbers; here are some more examples:

Stirling Cycle Numbers arise in the analysis of the standard algorithm that finds the maximum element of an array (TAOCP Sec. 1.2.10): How many times must the current maximum value be updated when finding the maximum value? It turns out that the probability that the maximum will need to be updated k times when finding a maximum in an array of n elements is p[n][k] = StirlingCycle[n, k+1]/n!. From this, we can derive that on the average, approximately Log(n) updates will be necessary.

Genocchi Numbers arise in connection with counting the number of BDDs that are "thin" (TAOCP 7.1.4 Exercise 174).

Answer 2

Harmonic Numbers appear almost everywhere! Musical Harmonies, analysis of Quicksort... Stirling Numbers (first and second kind) arise in a variety of combinatorics and partitioning problems. Eulerian Numbers also occur several places, most notably in permutations and coefficients of polylogarithm functions.

Answer 3

A lot of the numbers you mentioned are used in the analysis of algorithms. You may not have these numbers in your code, but you'll need them if you want to estimate how long it will take for your code to run. You might see them in your code too. Some of these numbers are related to combinatorics, counting how many ways something can happen.

Sometimes it's not enough to know how many possibilities there are because you need to enumerate over the possibilities. Volume 4 of Knuth's TAOCP, in progress, gives the algorithms you need.

Here's an example of using Fibonacci numbers as part of a numerical integration problem.

Harmonic numbers are a discrete analog of logarithms and so they come up in difference equations just like logs come up in differential equations. Here's an example of physical applications of harmonic means, related to harmonic numbers. See the book Gamma for many examples of harmonic numbers in action, especially the chapter "It's a harmonic world."

Answer 4

These special numbers can help out in computational problems in many ways. For example:

• You want to find out when your program to compute the GCD of 2 numbers is going to take the longest amount of time: Try 2 consecutive Fibonacci Numbers.

• You want to have a rough estimate of the factorial of a large number, but your factorial program is taking too long: Use Stirling's Approximation.

• You're testing for prime numbers, but for some numbers you always get the wrong answer: It could be you're using Fermat's Prime test, in which case the Carmicheal numbers are your culprits.

The most common general case I can think of is in looping. Most of the time you specify a loop using a (start;stop;step) type of syntax, in which case it may be possible to reduce the execution time by using properties of the numbers involved.

For example, summing up all the numbers from 1 to n when n is large in a loop is definitely slower than using the identity sum = n*(n + 1)/2.

There are a large number of examples like these. Many of them are in cryptography, where the security of information systems sometimes depends on tricks like these. They can also help you with performance issues, memory issues, because when you know the formula, you may find a faster/more efficient way to compute other things -- things that you actually care about.

For more information, check out wikipedia, or simply try out Project Euler. You'll start finding patterns pretty fast.

Answer 5

Not necessarily a magic number from the reference you mentioned, but nonetheless --

0x5f3759df

-- the notorious magic number used to calculate inverse square root of a number by giving a good first estimate to Newton's Approximation of Roots, often attributed to the work of John Carmack - more info here.

Not programming related, huh? :)

Answer 6

Is this directly programming related? Surely related, but I don't know how closely.

Special numbers, such as e, pi, etc., come up all over the place. I don't think that anyone would argue about these two. The Golden_ratio also appears with amazing frequency, in everything from art to other special numbers themselves (look at the ratio between successive Fibonacci numbers.)

Various sequences and families of numbers also appear in many places in mathematics and therefore, in programming too. A beautiful place to look is the Encyclopedia of integer sequences.

I'll suggest this is an experience thing. For example, when I took linear algebra, many, many years ago, I learned about the eigenvalues and eigenvectors of a matrix. I'll admit that I did not at all appreciate the significance of eigenvalues/eigenvectors until I saw them in use in a variety of places. In statistics, in terms of what they tell you about uncertainty of an estimate from a covariance matrix, the size and shape of a confidence ellipse, in terms of principal component analysis, or the long term state of a Markov process. In numerical methods, where they tell you about convergence of a method, be it in optimization or an ODE solver. In mechanical engineering, where you see them as principal stresses and strains.

Answer 7

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