Approximating the \(p\)th root by composite rational functions (Q2029808)

It is shown that a rational function \(r\) of the form \(r(x)=r_k(x,r_{k-1}(x,r_{k-2}(\dots (x,r_1(x,1)))))\) approximates the function \(x^{1/p}\) on the interval \([0,1]\) with superalgebraic accuracy close to \(p\)th root exponential convergence. This convergence is doubly exponential with respect to the number of degree of freedom. The error is \(O(\exp(-c_1\exp(c_2d)))\) for some constants \(c_1,c_2>0\) and \(d\) is the number of parameters expressing the rational function, \(d=\sum_{i=1}^k m_i+l_i+1\) if \(r_i\) is of the type \((m_i,l_i), i=1,2,\dots,k\).