College Math Teaching

February 17, 2014

Aitken acceleration: another “easy” example for the students

In a previous post I showed some spreadsheet data to demonstrate the Aitken acceleration process. Here, I’ll go through an example where a sequence converges linearly: let p_n = L +a^n + b^n where 0 < a < b < 1 . We use the form q_n = \frac{p_{n+2}p_{n} -(p_{n+1})^2}{p_{n+2} - 2p_{n+1} + p_n} (I am too lazy to use the traditional “p-hat” notation). First, note that the denominator works out to a^n(1-a)^2 +b^n(1-b)^2

The numerator is a tiny bit more work: the L^2 terms cancel and as far as the rest:
L(a^{n+2} + b^{n+2}) + L(a^n + b^n) +(a^{n+2}+b^{n+2})(a^n + b^2)-2L(a^{n+1}+b^{n+1})-(a^{n+1}+b^{n+1})^2
which simplifies to a term involving L and one that doesn’t. Here is the term involving L :

L(a^{n+2}-2a^{n+1}  + a^n + b^{n+2} -2b^{n+1} +b^n) = L(a^n(1-a)^2 +b^n(1-b)^2)

which, of course, is just L times the denominator.

Now the terms not involving L: (a^{n+2}+b^{n+2})(a^n+b^n) - (a^{n+1} + b^{n+1})^2 = b^na^n(b^2+a^2-2ab) = b^na^n(b-a)^2

So our fraction is merely

\frac{L((a^n(1-a)^2 +b^n(1-b)^2)) + b^na^n(b-a)^2}{a^n(1-a)^2 +b^n(1-b)^2} = L + \frac{b^na^n(b-a)^2}{a^n(1-a)^2 +b^n(1-b)^2}

This can be rearranged to L + \frac{(b-a)^2}{\frac{(1-a)^2}{b^n} + \frac{(1-b)^2}{a^n}}

Clearly as n goes to infinity, the error goes to zero very quickly. It might be instructive to look at the ratio of the errors for p_n and q_n :

This ratio is
(a^n + b^n)\frac{a^n(1-a)^2 + b^n(1-b)^2}{a^nb^n(b-a)^2} =(a^n +b^n)(\frac{1}{b^n}(\frac{1-a}{b-a})^2 + \frac{1}{a^n}(\frac{1-b}{b-a})^2)

Note that in the right hand factor: both squared factors are fixed and the coefficients go to infinity as n goes to infinity. If one multiplies out, one obtains:

((\frac{a}{b})^n +1)(\frac{1-a}{b-a})^2 +  ((\frac{b}{a})^n +1)(\frac{1-b}{b-a})^2 . In the limit, the first term decreases to (\frac{1-a}{b-a})^2 and the second goes to infinity.

Hence the errors in the accelerated sequence are smaller.


1 Comment »

  1. […] To see an abstract example where where , go to the next post in this series. […]

    Pingback by Demonstrating Aitken’s sequence acceleration | College Math Teaching — February 17, 2014 @ 3:06 am

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