The Verification Of Algebraic Interpolation On Experiment Point Set

The rapid development of science and technology has increased the requirements for the accuracy of calculation results.There are a lot of problems,such as raw data errors,limited precision representation of real numbers,and error accumulation,which have made calculation inaccuracies everywhere.In high-risk applications including aircraft design,satellite orbit determination,and rocket launch,it is necessary to know the credible upper bound of the numerical calculation results.For these key issues,the accumulation of small calculation errors may lead to qualitative changes of calculation results.Further,it may lead to major accidents.The issues about how to ensure that the errors in the calculation process are controllable and the results are true and credible,are needed to be solved in scientific computing.In engineering calculations,point sets are mostly obtained from the results of experiments.Therefore the coordinates of points inevitably have errors.The point sets whose coordinates are within a certain range are called experimental point sets.Since approximate algebraic interpolation of experimental point sets can reflect the needs of engineering practice,it has attracted much attention from many domestic and overseas scholars.This paper uses Rump interval algorithm and Kantorovich theorem to design trusted verification algorithms for approximate algebraic interpolation.The main research contents are as follows:(1)Design an error-controllable algorithm for multivariate polynomial interpolation on the experimental point set.Given the experimental point set,the designed algorithm outputs a low-order polynomial,the admissible point set of the given experimental point set and its credible error bound.The numerical part of the algorithm calculates an order ideal and an admissible point set,and the polynomial corresponding to the order ideal approximately vanishes on the admissible point set.The verification part of the algorithm transforms the verification of the solution of a multivariable polynomial system into the verification of the solutions to several univariate equations.Using Kantorovich's theorem and Rump's interval theorem,the algorithm calculates the verified error bound for the set of admissible points which is computed by the numerical part.The algorithm guarantees that there exists a set of admissible points within the verified error bound of the output admissible point set,and the computed low-order polynomials precisely vanishes on this set of admissible points.(2)Design an error controllable algorithm for the barycentric rational interpolation over univariate experimental point set.This algorithm uses the generalized Vandermonde matrix and the barycentric rational interpolation,combines the interval algorithm toolbox,Kantorovich theorem and Rump's interval theorem,in order to compute the barycentric representation of rational interpolant whose weight vector is an interval vector.Besides,this algorithm outputs an admissible interval point set with verified error bound.The algorithm guarantees that there exists an admissible point set within the computed an admissibleinterval point set,and there exists a barycentric rational interpolant within the computed barycentric rational interval interpolant.This barycentric rational interpolant exactly satisfies the interpolation condition on this admissible point set.