Standard and Corrected Numerical Differentiation Formulae

Year 2019, Volume: 2 Issue: 2, 135 - 153, 27.06.2019

François Dubeau

https://doi.org/10.33434/cams.512796

Abstract

Standard numerical differentiation rules that might be established by the method of undetermined coefficients are revisited. Best truncation error bounds are established by a direct method and by the method of integration by parts "backwards". A new method to increase the order of the truncation error using a primitive is presented. This approach leads to corrected numerical differentiation rules. Differentiation formulae and numerical tests are presented.

Keywords

Absolutely continuous function , Method of undetermined coefficients , Numerical differentiation rules , Peano kernel , Taylor's expansion

References

• [1] P. J. Davis, Interpolation and Approximation, Dover, N.Y., 1975.
• [2] A. S. Househoulder, Principles of Numerical Analysis, McGraw Hill, Columbus, N.Y., 1953.
• [3] D. Kincaid, W. Cheney, Numerical Analysis, Brooks/Cole Pub. Co., Cal., 1991.
• [4] A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, N.Y., 1965.
• [5] N. Macon, A. Spitzbart, Inverses of Vandermonde matrices, Amer. Math. Monthly, 65(2) (1958), 95-100. http://dx.doi.org/10.2307/2308881
• [6] E. Asplund, L. Bungart, A First Course in Integration, Holt, Rinehart and Winston, N.Y., 1966.
• [7] L. L. Schumaker, Spline Functions Basic Theory, Wiley, N.Y., 1981.
• [8] G. Peano, Resto nelle formule di quadratura espresso con un integrale definito, Atti. Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., Serie 5, 22(I) (1913), 562-569.
• [9] R. von Mises, U¨ ber allgemeine quadraturformeln, J. Reine Angew. Math., 174 (1935), 56-67; reprinted in Selected Papers of Richard von Mises, Vol. 1, 559-574, American Mathematical Society, Providence, R.I., 1963.
• [10] A. Ghizzetti, A. Ossicini, Quadrature Formulae, Academic Press, N.Y., 1970.
• [11] F. Dubeau, Revisited optimal error bounds for interpolatory integration rules, Adv. Numer. Anal., 2016 (2016), Article ID 3170595, 8 pages, http://dx.doi.org/10.1155/2016/3170595.
• [12] F. Dubeau, The method of undetermined coefficients: general approach and optimal error bounds, J. Math. Anal., 5(4) (2014), 1-11.
• [13] J. S. C. Prentice, Truncation and roundoff errors in three-point approximations of first and second derivatives, Appl. Math. Comput., 217 (2011), 4576-4581. http://dx.doi.org/10.1016/j.amc.2010.11.008
• [14] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd Ed., SIAM, Philadelphia, PN, 2002. http://dx.doi.org/10.1137/1.9780898718027