Accurate And Efficient Evaluation Of Laguerre Series And Chebyshev Tensor Product Surface

Since round-off error is generated by incompletely representation of real numbers in computers,some large-scale and time-consuming numerical calculations produce inaccu-rate results.To solve this problem,we propose an accurate and efficient evaluation for some Laguerre series and Chebyshev Tensor Product surface.Our algorithm is mainly based on Error-free Transformations(EFTs)to improve floating-numbers calculations.The main work of this paper have the following two aspects:1.We present a compensated algorithm to evaluate Laguerre series accurately.Based on EFTs to record round-off error in each step of Clenshaw algorithm,we can eval-uate all errors.This compensated algorithm in stable,and produce accurate results even in ill-conditioned problems.Besides,we analyze the relative error bound of Clenshaw algorithm and compensated Clenshaw algorithm,and the bound of lat-ter method illustrate that its result is as accurate as if computed by double-double classic Clenshaw scheme.In addition,we analyze the dynamic error bound of Clen-shaw and compensated Clenshaw scheme through evaluating error bound in each step.We can obtain a more adequate bound.Numerical experiments verify the compensated algorithm is accurate and efficient,and illustrate that dynamic error is more approximate actual error than theoretical bound.2.A Chebyshev tensor product surface is widely used in image analysis and numeri-cal approximation.This article illustrates an accurate evaluation for the surface in form of Chebyshev tensor product(CompCTP).This algorithm is based on EFTs,we use compensated Clenshaw scheme in nested Clenshaw method(CTP)to record round-off errors in each loop,and approximate the actual error of CTP.Next,we analyze relative error bound of CTP and compensated algorithm.Our error analysis shows that the error bound is u+O(u~2)×cond(P,x,y)in contrast to classic scheme u×cond(P,x,y),where u is working precision and cond(P,x,y)is a condition number of bivariate polynomial,which means that the accuracy of the computed result is similar to that produced by classical approach with twice working preci-sion.Moreover,we propose an dynamic error of CTP and CompCTP,and present two algorithms to evaluate dynamic error bound,respectively.Beside,numerical experiments show that the proposed algorithm is stable and efficient,and illustrate that dynamic error bound is more approximate the real error than theoretical bound.