A highly accurate technique for interpolations using very high-order polynomials, and its applications to some ill-posed linear problems

Abstract: Since the works of Newton and Lagrange, interpolation had been a mature technique in the numerical mathematics. Among the many interpolation methods, global or piecewise, the polynomial interpolation p(x) = a0+a1x+...+ anx n expanded by the monomials is the simplest one, which is easy to handle math-ematically. For higher accuracy, one always attempts to use a higher-order polyno-mial as an interpolant. But, Runge gave a counterexample, demonstrating that the polynomial interpolation problem may be ill-posed. Very high-order polynomial interpolation is very hard to realize by numerical computations. In this paper we propose a new polynomial interpolation by p(x) = ā0 + ā1x/R0 +...+ ānxn/Rn0, where R0 is a characteristic length used as a parameter, and chosen by the user. The resulting linear equations system to solve the coefficients āα is well-conditioned, if a suitable R0 is chosen. We define a non-dimensional parameter, R∗0 = R0/(b−a) [where a and b are the end-points of the interval for x]. The range of values for R∗0 for numerical stability is identified, and one can overcome the difficulty due t