## Look And Say, Revisited

### March 28, 2011

Our solution uses two techniques of mathematics: Horner’s rule and bisection. First, here are the coefficients of the polynomial that computes Conway’s constant:

`(define cs '(-6 3 -6 12 -4 7 -7 1 0 5 -2 -4 -12 2 7`

12 -7 -10 -4 3 9 -7 0 -8 14 -3 9 2 -3 -10 -2 -6 1

10 -3 1 7 -7 7 -12 -5 8 6 10 -8 -8 -7 -3 9 1 6 6 -2

-3 -10 -2 3 5 2 -1 -1 -1 -1 -1 1 2 2 -1 -2 -1 0 1))

Horner's rule allows us to easily calculate the value of the polynomial at any given point:

`(define (horner x cs)`

(let loop ((z 0) (cs (reverse cs)))

(if (null? cs) z

(loop (+ (* x z) (car cs)) (cdr cs)))))

Then we can use bisection to find a root. Here, *lo* evaluates to a negative value and *hi* evaluates to a positive value, regardless whether *lo* < *hi* or *hi* < *lo*:

`(define (bisect lo hi eps cs)`

(let loop ((lo lo) (hi hi))

(if (< (abs (- hi lo)) eps)

(exact->inexact (/ (+ lo hi) 2))

(let* ((mid (/ (+ lo hi) 2))

(fmid (horner mid cs)))

(if (negative? fmid)

(loop mid hi)

(loop lo mid))))))

Conway’s polynomial evaluates to -6 at *x*=0, -6 at *x*=1, and 1256202183622743302568 at *x*=2, so we know there is a root between 1 and 2. That makes sense; it looks like the sequence increases by about 30% at each step. Thus, we calculate Conway’s constant as follows:

`> (bisect 1 2 1e-15 cs)`

1.3035772690342964

You can see Conway’s constant to a greater degree of accuracy at A014715, and you can run the program at http://programmingpraxis.codepad.org/3ia1g1r8.

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My Haskell solution (see http://bonsaicode.wordpress.com/2011/03/28/programming-praxis-look-and-say-revisited/ for a version with comments):

[...] today’s Programming Praxis exercise, our goal is to calculate Conway’s constant. Let’s get [...]

Cheating with Sage:

I’ll come up with a real solution soon.

Your polynomial appears to have some typos, and doesn’t match what is shown at Mathworld. For example, see the x^61 term.

Fixed. Thanks.

As it turns out, I should have used

`find_root(f, 1, 2)`

in Sageinstead. As for my own code, I went for Newton’s Method. It requires knowing

the derivative of your function (simple enough to calculate for polynomials),

so I wrote a second Horner’s Rule to build it. The nice thing about Newton’s

Method is that if the function is reasonably well-behaved in the neighborhood

of interest, it converges quadratically to the root.

Come to think of it, I didn’t actually use Horner’s Rule to construct my

polynomials, just the naive definition. Apologies on misnaming my functions!

A version that actually uses Horner’s Rule:

My c++ solution :)

Factor solution. Interesting that if epsilon is set to 1e-8, Newton’s method doesn’t converge to that precision. Instead it alternately cycles between 1.303577269034296 and 1.303577269034297 until the iteration count breaks the cycle. If the precision is set to 1e-7, it converges rapidly (5 iterations) to 1.303577269034296.

( scratchpad ) conway-constant .

1.303577269034296

( scratchpad )