On in Polynomial Time Approximable Real Numbers and Analytic Functions

Abstract

The set ℓ _Pol of complex numbers ζ , which can be approximated in a polynomial time cζ ·n ^k depending from the precision 2⁻ⁿ, is a field. The operations of the field can be approximated in a time<c · n ² and better relative to the time one needs to approximate the operands ζ ₁ ,ζ ₂ with precision 2⁻ⁿ. In the case of addition and multiplication it can be done for |ζ ₁ |, |ζ ₂ |<a with a c depending from a and the constants in the running time of the polynomial time approximation of ζ 1,ζ 2, but in the case of division the constant depends additionally from |ζ | . The same holds for the ring of the polynomials ℓ _Pol [x ] and the field of the rational functions f ∈ ℓ _Pol (x ). The constant factor in the polynomial time bound of the approximation of f (ζ ) depend additionally from the distance of ζ from the singularities of f . The field ℓ _Pol is algebraically closed. In the case that the polynomials have constant coefficients there exist algorithms to approximate the roots of the polynomials in a linear time with the Newton method if the roots are simple and an isolation of the roots is precomputed [Sch82]. But this precomputation needs much time [ESY06]. The best known algorithms to solve this problem need a time O (m ⁵(τ +logm )²), m the degree of the polynomial and τ the length of the binary representation of the coefficients. The problem with not constant coefficients has been attacked on base of bitstreams in [Eig08], [ESY06]. In [Eig08] on can find an excellent overview of the state of art. In the case the coefficients are not fixed constants but given by approximations the situation is much more complicated because the precomputation depends from the precision of the approximation. It is clear that in general the approximation of a root of a polynomial with coefficients, which cannot be approximated in a time O (n ^k ), cannot be done in this time. Ker-I Ko discusses the approximation problems on base of the unary representation of the numbers [Ko91]. Polynomial time algorithms relative to dyadic and unary representation are obviously not the same. The motivation of this paper is the question how far the mentioned problems can be solved by global polynomial time algorithms. This are algorithms with a running time <c·n ^k with the constant c only depending from the constants of the polynomial time approximations of the arguments and the domain of the function to be computed. We prove that there exist analytic functions, which are computable on all finite domains in this sense and that analytic functions for which this is true can be represented by a power series in z.