## Pi

The other day I ran across the following continued fraction for π.

Source: L. J. Lange, An Elegant Continued Fraction for π, The American Mathematical Monthly, Vol. 106, No. 5 (May, 1999), pp. 456-458.

While the continued fraction itself is interesting, I thought I’d use this an example of how to typeset and compute continued fractions.

## Typesetting

I imagine there are LaTeX packages that make typesetting continued fractions easier, but the following brute force code worked fine for me:

\pi = 3 + \cfrac{1^2}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cdots}}}}

This relies on the `amsmath`

package for the `\cfrac`

command.

## Computing

Continued fractions of the form

can be computed via the following recurrence. Define *A*_{-1} = 1, *A*_{0} = *a*_{0}, *B*_{-1} = 0, and *B*_{0} = 1. Then for *k* ≥ 1 define *A*_{k+1} and *B*_{k+1} by

Then the *n*th convergent the continued fraction is *C*_{n} = *A*_{n} / *B*_{n}.

The following Python code creates the *a* and *b* coefficients for the continued fraction for π above then uses a loop that could be used to evaluate any continued fraction.

N = 20 a = [3] + ([6]*N) b = [(2*k+1)**2 for k in range(0,N)] A = [0]*(N+1) B = [0]*(N+1) A[-1] = 1 A[ 0] = a[0] B[-1] = 0 B[ 0] = 1 for n in range(1, N+1): A[n] = a[n]*A[n-1] + b[n-1]*A[n-2] B[n] = a[n]*B[n-1] + b[n-1]*B[n-2] print( n, A[n], B[n], A[n]/B[n] )

Python uses -1 as a shortcut to the last index of a list. I tack *A*_{-1} and *B*_{-1} on to the end of the A and B arrays to make the Python code match the math notation. This is either clever or a dirty hack, depending on your perspective.

## Back to pi

You may notice that these approximations for π are not particularly good. It’s a trade-off for having a simple pattern to the coefficients. The continued fraction for π that has all *b*‘s equal to 1 has a complicated set of *a*‘s with no discernible pattern: 3, 7, 15, 1, 292, 1, 1, etc. However, that continued fraction produces very good approximations. If you replace the first three lines of the code above with that below, you’ll see that four iterations produces an approximation to π good to 10 decimal places.

N = 4 a = [3, 7, 15, 1, 292] b = [1]*N

Just curious. How do you generate your image dealing with math expressions.

I used http://rogercortesi.com/eqn/index.php

I played with continued fractions of pi a while ago “Perl 6 Pi and continued fractions”. I didn’t try and come up with generative code like that, opting instead for just reversing the input and calling reduce on that.

I decided to convert your code to Perl 6.

my \a = 3, |( 6 xx * );

my \b = ->{ (2 * $++ + 1)² } … *;

my (@A,@B);

@A = a[0], ->{ my \n = ++$; a[n]*@A[n-1] + b[n-1]*(@A[n-2]//1) } … *;

@B = 1, ->{ my \n = ++$; a[n]*@B[n-1] + b[n-1]*(@B[n-2]//0) } … *;

my \N = 20;

for 0..N -> \n {

put ( n, @A[n], @B[n], @A[n]/@B[n] );

}

It gets a bit simpler if I combine the two arrays together, and remove the

`b`

array.constant pi-continued = ( 3,7,15,1,292,1,1,1 );

my \AB = gather {

my \iter = pi-continued.iterator;

take my ($a1,$b1) = (iter.pull-one,1);

my ($a2,$b2) = (1,0);

until IterationEnd =:= my \a = iter.pull-one {

my $a = a*$a1 + $a2;

my $b = a*$b1 + $b2;

NEXT {

($a2,$b2) = ($a1,$b1);

($a1,$b1) = take ($a,$b);

}

}

}

for 0..* Z AB -> (\n, (\A,\B)) {

put (n,A,B,FatRat.new: A,B)

}

What I find interesting is that this is not faster than just:

my \one = FatRat.new(1,1);

say pi-continued.reverse.reduce: -> \a, \b { one / a + b }

( The FatRat is there so that it will do something useful on the list of 250 that I tried it with )