A Highly Accurate Technique for Interpolations Using Very High-Order Polynomials, and Its Applications to Some Ill-Posed Linear Problems

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 expanded by the monomials is the simplest one, which is easy to handle mathematically. For higher accuracy, one always attempts to use a higher-order polynomial 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 + . . . + ānx/R0, 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, R0 = R0/(b−a) [where a and b are the end-points of the interval for x]. The range of values for R0 for numerical stability is identified, and one can overcome the difficulty due to Runge, as well as increase the accuracy and stability in the numerical interpolation by very-high-order polynomials, for these values of R0. Numerical results which validate the current theory are presented for (i) the first and higher-order derivatives of noisy numerical data [such as determining da/dN in fatigue mechanics], (ii) the solution of the Abel integral equation under noisy data, and (iii) the numerical determination of an inverse Laplace transform under noisy data. These results confirm the validity of the present approach for very high-order polynomial interpolation.