Recurrence relation with python.

Question

How to find the terminating value of the continued fractions $$ S=3-\cfrac2{3-\cfrac2{3-\cfrac2{\ddots}}} $$ by writing a recurrence relation in Python? (Start from any guess value other than 1.)

Answer 1

Write your equation as $$S=3-\frac{2}S$$ Notice that $S=1$ is a solution, however I think it's unstable. Then just start with any number $S_0$ not equal to $1$. Then repeat $$S_{n}=3-\frac2{S_{n-1}}$$ until $|S_n-S_{n-1}|<\varepsilon$.

Additional: Following the comment from @RobArthan, let's see what's happening if you are close to either of the two solutions $S=1$ or $S=2$.

Let's choose $S_n=1+\alpha$, where $|\alpha|\ll1$. Then $$S_{n+1}-1=3-\frac 2{1+\alpha}-1=\frac{2\alpha}{1+\alpha}\approx2\alpha$$

So starting from any point in the vicinity of $1$ the next iteration will be further away (about a factor of $2$ further than the initial condition).

How about $2$? We repeat the same steps: $S_n=2+\alpha$ $$S_{n+1}-2=3-\frac{2}{2+\alpha}-2=\frac{\alpha}{2+\alpha}\approx\frac\alpha2$$ So starting close to $2$, in the next step you are getting twice as close as before. Therefore $2$ is a stable solution

Answer 2

For a suitable function $f$, we can iterate an estimate $S$ to $f(S)$ with a for loop, terminated either when the change in $S$ is small or after a large number of iterations. Fewer iterations are needed if $f$ is Newton-Raphson inspired than if you just use $f(S):=3-2/S$. In particular, $S=3-2/S\implies S^2-3S+2=0$, so you could choose $f(S)=S-\frac{S^2-3S+2}{2S-3}=\frac{S^2-2}{2S-3}$.

Of course, there's no need to iterate anyway, as clearly $S=3-2/S\implies S\in\{1,\,2\}$. Mathematically, there are two interesting questions: which value of $S$ if either is mandated by the definition of $S$ (is it even well-defined?), and which choice of $f$ gives stable convergence to such a value from a wide range of nearby estimates of $S$?

We must define $S$ as the limit of a sequence. The obvious choice is $S_0:=3,\,S_{n+1}:=3-\frac{2}{S_n}$. You can easily prove by induction that $S_n\in(2,\,3]$, so $S=2$; $S\ne1$. However, you'll find an estimate close to either $1$ or $2$ leads to stable behaviour with the above Newton-Raphson choice of iteration. (This can be proven by considering the first few derivatives of $f$.)

Answer 3

We can easily show that your continued fraction is equal to $1$ or $2$. In fact: $$S=3-\dfrac2S\implies S^2-3S+2=0\iff S=1$$

Here I will post a very useful algorithm that I always use when I have to operate with continued fraction:

from decimal import Decimal
from fractions import Fraction

class CFraction(list):

    def __init__(self, value, maxterms=15, cutoff=1e-10):
        if isinstance(value, (int, float, Decimal)):
            value = Decimal(value)
            remainder = int(value)
            self.append(remainder)

            while len(self) < maxterms:
                value -= remainder
                if value > cutoff:
                    value = Decimal(1) / value
                    remainder = int(value)
                    self.append(remainder)
                else:
                    break
        elif isinstance(value, (list, tuple)):
            self.extend(value)
        else:
            raise ValueError("CFraction requires number or list")

    def fraction(self, terms=None):
        "Convert to a Fraction."

        if terms is None or terms >= len(self):
            terms = len(self) - 1

        frac = Fraction(1, self[terms])
        for t in reversed(self[1:terms]):
            frac = 1 / (frac + t)

        frac += self[0]
        return frac

    def __float__(self):
        return float(self.fraction())

    def __str__(self):
        return "[%s]" % ", ".join([str(x) for x in self])

if __name__ == "__main__":
    from math import e, pi, sqrt

    numbers = {
        "phi": (1 + sqrt(5)) / 2,
        "pi": pi,
        "e": e,
    }

    print "Continued fractions of well-known numbers"
    for name, value in numbers.items():
        print "   %-8s  %r" % (name, CFraction(value))

    for name, value in numbers.items():
        print
        print "Approximations to", name
        cf = CFraction(value)
        for t in xrange(len(cf)):
            print "   ", cf.fraction(t)

    print
    print "Some irrational square roots"
    for n in 2, 3, 5, 6, 7, 8:
        print "   ", "sqrt(%d)  %r" % (n, CFraction(sqrt(n)))

    print
    print "Decimals from 0.1 to 0.9"
    for n in xrange(1, 10):
        cf = CFraction(n / 10.0)
        print "   ", float(cf), cf

As you can note, it can be used to print the continued fraction for all the square roots, irrational number and also general continued fraction as yours.