Computable analysis: an introduction
Computable analysis: an introduction
E2 -computability of e, π and Other Famous Constants
Electronic Notes in Theoretical Computer Science (ENTCS)
| Hi-index | 0.00 |
Let us call an approximator of a complex number α any sequence γ0,γ1,γ2,... of rational complex numbers such that $$|\gamma_t-\alpha|\le \frac{1}{t+1},\ \ t=0,1,2,\ldots$$ Denoting by ℕ the set of the natural numbers, we shall call a representation of α any 6-tuple of functions f1,f2,f3,f4,f5,f6 from ℕ into ℕ such that the sequence γ0,γ1,γ2,... defined by $$\gamma_t=\frac{f_1(t)-f_2(t)}{f_3(t)+1}+\frac{f_4(t)-f_5(t)}{f_6(t)+1}i,\ \ t=0,1,2,\ldots\,$$ is an approximator of α. For any representations of the members of a finite sequence of complex numbers, the concatenation of these representations will be called a representation of the sequence in question (thus the representations of the sequence have a length equal to 6 times the length of the sequence itself). By adapting a proof given by P. C. Rosenbloom we prove the following refinement of the fundamental theorem of algebra: for any positive integer N there is a 6-tuple of computable operators belonging to the second Grzegorczyk class and transforming any representation of any sequence α0,α1,...,αN−−1 of N complex numbers into the components of some representation of some root of the corresponding polynomial P(z)=zN+αN−−1zN−−1+⋯+α1z+α0.