Study of linear information for classes of polynomial equations

Summary We study linear sequential (adaptive) information for approximating zeros of polynomials of unbounded degree and establish a theorem on constrained approximation of smooth functions by polynomials.For a positive we seek a pointx ^* such that|x ^* – _p | , where _p is a zero of a real polynomialp in the interval a, b]. We assume thatp belongs to the classF ₁ of polynomials of bounded arbitrary seminorm and having a root in a, b] or to the classF ₂ of polynomials which are nonpositive ata, nonnegative atb and have exactly one simple zero in a, b]. The information onp consists ofn sequential (adaptive) evaluations of arbitrary linear functionals. The pointx ^* is constructed by means of an algorithm which is an arbitrary mapping depending on the information onp. We show that there exists no information and no algorithm for computingx ^* for everyp fromF ₁, no matter how large the value ofn is. This is a stronger result than that obtained by us for smooth functions.For the classF ₂ we can find a pointx ^* for arbitraryp and. Anoptimal algorithm, i.e., an algorithm with the smallest error, is thebisection of the smallest known interval containing the root ofp. We also exhibitoptimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is minimal. It turns out that in the class of nonsequential (parallel) information, i.e., when the functionals are given simultaneously, optimal information consists of the evaluations of a polynomial atn-equidistant points in a, b]. In the class of sequential continuous information, optimal information consists of evaluations of a polynomial atn points generated by thebisection method. To prove this result we establish a theorem on constrained approximation of smooth functions by polynomials. More precisely, we prove that a smooth function can be arbitrarily well uniformly approximated by a polynomial which satisfies constrains given byn arbitrary continuous linear functionals.Our results indicate that the problem of finding an -approximation to a real zero of a real polynomial (of unknown degree) is essentially of the same difficulty as the problem of finding an -approximation to a zero of an infinitely differentiable function.