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Tue, 11 Dec 2007

More notes on power series
It seems I wasn't done thinking about this. I pointed out in yesterday's article that, having defined the cosine function as:

     coss = zipWith (*) (cycle [1,0,-1,0]) (map ((1/) . fact) [0..])
one has the choice to define the sine function analogously:

     sins = zipWith (*) (cycle [0,1,0,-1]) (map ((1/) . fact) [0..])
or in a totally different way, by reference to cosine:

     sins = (srt . (add one) . neg . sqr) coss
Here is a third way. Sine and cosine are solutions of the differential equation f = -f''. Since I now have enough infrastructure to get Haskell to solve differential equations, I can use this to define sine and cosine:

    solution_of_equation f0 f1 = func
        where func = int f0 (int f1 (neg func))
 
    sins = solution_of_equation 0 1
    coss = solution_of_equation 1 0
The constants f0 and f1 specify the initial conditions of the differential equation, values for f(0) and f'(0), respectively.

Well, that was fun.

One problem with the power series approach is that the answer you get is not usually in a recognizable form. If what you get out is

     [1.0,0.0,-0.5,0.0,0.0416666666666667,0.0,-0.00138888888888889,0.0,2.48015873015873e-05,0.0,...]
then you might recognize it as the cosine function. But last night I couldn't sleep because I was wondering about the equation f·f' = 1, so I got up and put it in, and out came:

     [1.0,1.0,-0.5,0.5,-0.625,0.875,-1.3125,2.0625,-3.3515625,5.5859375,-9.49609375,16.40234375,...]
Okay, now what? Is this something familiar? I'm wasn't sure. One thing that might help a bit is to get the program to disgorge rational numbers rather than floating-point numbers. But even that won't completely solve the problem.

One thing I was thinking about in the shower is doing Fourier analysis; this should at least identify the functions that are sinusoidal. Suppose that we know (or believe, or hope) that some power series a1x + a3x3 + ... actually has the form c1 sin x + c2 sin 2x + c3 sin 3x + ... . Then we can guess the values of the ci by solving a system of n equations of the form:

$$\sum_{i=1}^n i^kc_i = k!a_k\qquad{\hbox{($k$ from 1 to $n$)}}$$

And one ought to be able to do something analogous, and more general, by including the cosine terms as well. I haven't tried it, but it seems like it might work.

But what about more general cases? I have no idea. If you have the happy inspiration to square the mystery power series above, you get [1, 2, 0, 0, 0, ...], so it is √(2x+1), but what if you're not so lucky? I wasn't; I solved it by a variation of Gareth McCaughan's method of a few days ago: f·f' is the derivative of f2/2, so integrate both sides of f·f' = 1, getting f2/2 = x + C, and so f = √(2x + C). Only after I had solved the equation this way did I try squaring the power series, and see that it was simple.

I'll keep thinking.


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