Research On Accurate Evaluation And Its Running Error Analysis Of A Polynomial In Bernstein Form

Recently, as high performance computer rapidly develops, large scale scientific andengineering calculation has been applied to many areas. Since the round-off error of float-ing point operations is inevitable and inherent to computer operations, the round-off errorwould probably accumulate ceaselessly, especially under the condition of intensive, largescale, long time computing or extremely ill-conditioned problem, which will lead to un-acceptable distortion of the numerical results. Hence, the research of high-precision andreliable numerical algorithm has already turned into and will continue to be an importantproblem in the field of Computational Mathematics. Computer aided geometric design(CAGD) has played a significant role in modern industry and science. As the polynomialin Bernstein form is fundamental and crucial in CAGD, our research on high precisionalgorithm and its running error analysis of function value and k-th derivative value forthis kind of polynomial is of great significance.This article takes advantage of error-free transformation to analyze the running er-ror of the high precision algorithm for the evaluation of a polynomial in Bernstein form.Furthermore, we have analyzed the high-precision algorithm of k-th derivative value of apolynomial in Bernstein form and its running error. The main content is as follows:Asforthehigh-precisionevaluationofpolynomials, scientistsandengineersnotonlywant to obtain the numerical results, but also care more about the error bound of the nu-merical results to ensure the reliability of the obtained results. Through the running erroranalysis of the high-precision compensated algorithm of a polynomial in Bernstein form,we discover that when evaluating the polynomial at a fixed point, the error of the resultis related to a Bernstein polynomial which takes the absolute values of round-off errors inthe computation process as parameters; then the running error bound theorem is provedand the corresponding compensated algorithm with running error bound is presented. Nu-merical experiments show the algorithm proposed in this paper gains higher precision andbetter validity than previous algorithms.Concerning the high-precision computation of k-th derivative value of a polynomialin Bernstein form, by means of the error-free transformation technique, the compensatedalgorithm of the difference operation for accurate computing the coefficients of the k- th derivative of a polynomial in Bernstein form is put forward; then the high-precisioncompensated algorithm of computing k-th derivative value of a polynomial in Bernsteinform is designed and its global error bound theorem is proved. For the sake of higherreliability of the numerical results, the running error analysis of the algorithm is carriedout and a more accurate error bound is gained. The running error bound theorem and thealgorithmwithrunningerrorboundareshownandthenumericalexperimentsdemonstratethe validity of the algorithms.