2
votes
1answer
82 views

About continued fractions as best rational approximations

I'm reading this notes about continued fractions: http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf I had no problems understanding everything there, except one thing that has ...
2
votes
1answer
45 views

Why does this pattern fail (sometimes) for the continued fraction convergents of $\sqrt{2}$?

This is connected to my post on the continued fraction convergents of pi. Motivated by Calvin Lin's comment whether a similar pattern exists for other constants, I checked $\sqrt{2}$. Its convergents ...
10
votes
1answer
383 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
1
vote
1answer
171 views

How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?

I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products: $A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
1
vote
2answers
181 views

Continued Fractions Approximation

I have come across continued fractions approximation but I am unsure what the steps are. For example How would you express the following rational function in continued-fraction form: $${x^2+3x+2 ...
5
votes
1answer
951 views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
3
votes
0answers
75 views

Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))

I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
3
votes
0answers
190 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
7
votes
3answers
695 views

Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...