Abstract
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer—the Infinity Computer—able to work numerically with finite, infinite, and infinitesimal numbers. It is proved that the Infinity Computer is able to calculate values of derivatives of a higher order for a wide class of functions represented by computer procedures. It is shown that the ability to compute derivatives of arbitrary order automatically and accurate to working precision is an intrinsic property of the Infinity Computer related to its way of functioning. Numerical examples illustrating the new concepts and numerical tools are given.
Similar content being viewed by others
References
Berz, M.: Automatic differentiation as nonarchimedean analysis. In: Computer Arithmetic and Enclosure Methods. pp. 439–450. Elsevier, Amsterdam (1992)
Bischof, C., Bücker, M.: Computing derivatives of computer programs. In: Modern Methods and Algorithms of Quantum Chemistry Proceedings NIC Series, vol. 3, 2 edn, pp. 315–327. John von Neumann Institute for Computing, Jülich (2000)
Chua L.O., Desoer C.A., Kuh E.S.: Linear and Non linear Circuits. MacGraw Hill, Singapore (1987)
Cohen J.S.: Computer Algebra and Symbolic Computation: Mathematical Methods. A K Peters, Ltd, Wellesley (1966)
Corliss, G., Faure, C., Griewank, A., Hascoet, L., Naumann, U. (eds): Automatic Differentiation of Algorithms: From Simulation to Optimization. Springer, New York (2002)
Daponte P., Grimaldi D., Molinaro A., Sergeyev Ya.D.: An algorithm for finding the zero-crossing of time signals with lipschitzian derivatives. Measurement 16, 37–49 (1995)
Lam H.Y.-F.: Analog and Digital Filters-Design and Realization. Prentice Hall Inc, New Jersey (1979)
Lyness J.N., Moler C.B.: Numerical differentiation of analytic functions. SIAM J. Numer. Anal. 4, 202–210 (1967)
Moin P.: Fundamentals of Engineering Numerical Analysis. Cambridge University Press, Cambridge (2001)
Muller J.M.: Elementary Functions: Algorithms and Implementation. Birkhäuser, Boston (2006)
Pardalos, P.M., Resende, M.G.C. (eds): Handbook of Applied Optimization. Oxford University Press, New York (2002)
Sergeyev Ya.D.: Arithmetic of Infinity. Edizioni Orizzonti Meridionali, CS (2003)
Sergeyev, Ya.D.: http://www.theinfinitycomputer.com (2004)
Sergeyev Ya.D.: A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4), 567–596 (2008)
Sergeyev, Ya.D.: Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. EU patent 1728149, (2009)
Sergeyev Ya.D.: Numerical computations and mathematical modelling with infinite and infinitesimal numbers. J. Appl. Math. Comput. 29, 177–195 (2009)
Sergeyev Ya.D.: Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlinear Anal. Ser. A Theory Methods Appl. 71(12), e1688–e1707 (2009)
Sergeyev Ya.D., Daponte P., Grimaldi D., Molinaro A.: Two methods for solving optimization problems arising in electronic measurements and electrical engineering. SIAM J. Optim. 10(1), 1–21 (1999)
Sergeyev, Ya.D., Kvasov, D.E.: Diagonal Global Optimization Methods. Fizmatlit, Moscow (2008, in Russian)
Strongin R.G., Sergeyev Ya.D.: Global Optimization and Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)
Walnut D.F.: An Introduction to Wavelet Analysis. Birkhäuser, Boston (2004)
Wilf H.S.: Generatingfunctionology, 3rd edn. A K Peters, Ltd., Wellesley (2006)
Wolfe M.A.: On first zero crossing points. Appl. Math. Comput. 150, 467–479 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sergeyev, Y.D. Higher order numerical differentiation on the Infinity Computer. Optim Lett 5, 575–585 (2011). https://doi.org/10.1007/s11590-010-0221-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-010-0221-y