Working with the set of real numbers (was: Finding size of Variable)

[Ian Kelly]

On Mon, Mar 3, 2014 at 11:35 PM, Chris Angelico <rosuav at gmail.com> wrote:
> In constant space, that will produce the sum of two infinite sequences
> of digits. (And it's constant time, too, except when it gets a stream
> of nines. Adding three thirds together will produce an infinite loop
> as it waits to see if there'll be anything that triggers an infinite
> cascade of carries.) Now, if there's a way to do that for square
> rooting a number, then the CF notation has a distinct benefit over the
> decimal expansion used here. As far as I know, there's no simple way,
> in constant space and/or time, to progressively yield more digits of a
> number's square root, working in decimal.

The code for that looks like this:

def cf_sqrt(n):
    """Yield the terms of the square root of n as a continued fraction."""
   m = 0
    d = 1
    a = a0 = floor_sqrt(n)
    while True:
        yield a
        next_m = d * a - m
        next_d = (n - next_m * next_m) // d
        if next_d == 0:
            break
        next_a = (a0 + next_m) // next_d
        m, d, a = next_m, next_d, next_a


def floor_sqrt(n):
    """Return the integer part of the square root of n."""
    n = int(n)
    if n == 0: return 0
    lower = 2 ** int(math.log(n, 2) // 2)
    upper = lower * 2
    while upper - lower > 1:
        mid = (upper + lower) // 2
        if n < mid * mid:
            upper = mid
        else:
            lower = mid
    return lower


The floor_sqrt function is merely doing a simple binary search and
could probably be optimized, but then it's only called once during
initialization anyway.  The meat of the loop, as you can see, is just
a constant amount of integer arithmetic.  If it were desired to halt
once the continued fraction starts to repeat, that would just be a
matter of checking whether the triple (m, d, a) has been seen already.

Going back to your example of adding generated digits though, I don't
know how to add two continued fractions together without evaluating
them.