Function: bestappr
Section: number_theoretical
C-Name: bestappr
Prototype: GDG
Help: bestappr(x,{B}): return a rational approximation to x, whose
 denominator is limited by B, if present. This function applies to reals,
 intmods, p-adics, and rationals of course. Otherwise it applies recursively
 to all components.

Doc: using variants of the extended Euclidean algorithm, returns a rational
 approximation $a/b$ to $x$, whose denominator is limited
 by $B$, if present. If $B$ is omitted, returns the best approximation
 affordable given the input accuracy; if you are looking for true rational
 numbers, presumably approximated to sufficient accuracy, you should first
 try that option. Otherwise, $B$ must be a positive real scalar (impose
 $0 < b \leq B$).

 \item If $x$ is a \typ{REAL} or a \typ{FRAC}, this function uses continued
 fractions.
 \bprog
 ? bestappr(Pi, 100)
 %1 = 22/7
 ? bestappr(0.1428571428571428571428571429)
 %2 = 1/7
 ? bestappr([Pi, sqrt(2) + 'x], 10^3)
 %3 = [355/113, x + 1393/985]
 @eprog
 By definition, $a/b$ is the best rational approximation to $x$ if
 $|b x - a| < |v x - u|$ for all integers $(u,v)$ with $0 < v \leq B$.
 (Which implies that $n/d$ is a convergent of the continued fraction of $x$.)

 \item If $x$ is a \typ{INTMOD} modulo $N$ or a \typ{PADIC} of precision $N =
 p^{k}$, this function performs rational modular reconstruction modulo $N$. The
 routine then returns the unique rational number $a/b$ in coprime integers
 $|a| < N/2B$ and $b\leq B$ which is congruent to $x$ modulo $N$. Omitting
 $B$ amounts to choosing it of the order of $\sqrt{N/2}$. If rational
 reconstruction is not possible (no suitable $a/b$ exists), returns $[]$.
 \bprog
 ? bestappr(Mod(18526731858, 11^10))
 %1 = 1/7
 ? bestappr(Mod(18526731858, 11^20))
 %2 = []
 ? bestappr(3 + 5 + 3*5^2 + 5^3 + 3*5^4 + 5^5 + 3*5^6 + O(5^7))
 %2 = -1/3
 @eprog\noindent In most concrete uses, $B$ is a prime power and we performed
 Hensel lifting to obtain $x$.

 The function applies recursively to components of complex objects
 (polynomials, vectors, \dots). If rational reconstruction fails for even a
 single entry, returns $[]$.
