Abstract
Theory and applications of multiplicative and Volterra calculi have been evolving rapidly over the recent years. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on multiplicative and Volterra calculi is self-evident. In this paper, the well-known Newton minimization method for one and two variables is developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed minimization methods is exposed by examples, and the results are compared with the original minimization method. One of the striking results of the proposed method is that the rate of convergence and the range of initial values are considerably larger compared to the original method.
Similar content being viewed by others
References
Aniszewska, D.: Multiplicative Runge–Kutta methods. Nonlinear Dyn. 50(1), 265–272 (2007)
Bashirov, A.E., Bashirova, G.: Dynamics of literary texts and diffusion. Online J. Commun. Media Technol. 1(3), 60–82 (2011)
Bashirov, A.E., Kurpınar, E.M., Özyapıcı, A.: Multiplicative calculus and its applications. J. Math. Anal. Appl. 337(1), 36–48 (2008)
Bashirov, A.E., Mısırlı, E., Tandoğdu, Y., Özyapıcı, A.: On modeling with multiplicative differential equations. Appl. Math. J Chin. Univ. 26(4), 425–438 (2011)
Bashirov, A.E., Riza, M.: On complex multiplicative differentiation. TWMS J. Appl. Eng. Math. 1(1), 51–61 (2011)
Bashirov, A.E., Riza, M.: On complex multiplicative integration. arXiv:1307.8293v1 (2013)
Córdova-Lepe, F.: The multiplicative derivative as a measure of elasticity in economics. TMAT Revista Latinoamericana de Ciencias e Ingenieria 2(3) (2006)
Englehardt, J., Swartout, J., Loewenstine, C.: A new theoretical discrete growth distribution with verification for microbial counts in water. Risk Anal. 29(6), 841–856 (2009)
Filip, D.A., Piatecki, C.: A non-Newtonian examination of the theory of exogenous economic growth. CNCSIS-UEFISCSU (project number PNII IDEI 2366/2008) and Laboratoire d’Economie d’Orleans (LEO) (2010)
Florack, L., van Assen, H.: Multiplicative calculus in biomedical image analysis. J. Math. Imaging Vis. 42(1), 64–75 (2012)
Grossman, M.: Bigeometric Calculus. Archimedes Foundation, Rockport, Mass (1983)
Grossman, M., Katz, R.: Non-Newtonian Calculus. Lee Press, Pigeon Cove, Mass (1972)
Jones, J.E.: On the determination of molecular fields from the equation of state of a gas. Proc. R. Soc. Lond. Ser. A 106(738), 463–477 (1924)
Kasprzak, W., Lysik, B., Rybaczuk, M.: Dimensions, Invariants Models and Fractals. Ukrainian Society on Fracture Mechanics, SPOLOM,Wroclaw-Lviv (2004)
Mısırlı, E., Gürefe, Y.: Multiplicative Adams–Bashforth–Moulton methods. Numer. Algo. 57(4), 425–439 (2011)
Mısırlı, E., Özyapıcı, A.: Exponential approximations on multiplicative calculus. Proc. Jangjeon Math. Soc. 12(2), 227–236 (2009)
Riza, M., Özyapıcı, A., Mısırlı, E.: Multiplicative finite difference methods. Q. Appl. Math. 67(4), 745 (2009)
Üzer, A.: Multiplicative type complex calculus as an alternative to the classical calculus. Comput. Math. Appl. 60(10), 2725–2737 (2010)
Volterra, V., Hostinsky, B.: Operations Infinitesimales Lineares. Gauthier-Villars, Paris (1938)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Özyapıcı, A., Riza, M., Bilgehan, B. et al. On multiplicative and Volterra minimization methods. Numer Algor 67, 623–636 (2014). https://doi.org/10.1007/s11075-013-9813-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9813-9