Abstract
It is highly desirable that the numerical differentiation at arbitrary points of a multidimensional space can be done with a reasonable amount of computational labor. A variant of the recently developed method of nonlinear interpolation is shown adequate for numerical differentiation. Computed results indicate that the proposed scheme is indeed feasible. This paper provides a strong case that, to overcome the “vastness of hyperspace”, random samplings are the only and mandatory choice in the numerical analysis of functions of very many variables.
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References
Tsuda, T., Ichida, K.: Nonlinear interpolation of multivariable functions by the Monte Carlo method. J. ACM17, 420–425 (1970).
Ichida, K., Tsuda, T.: Interpolation and numerical differentiation of multivariable functions. Kyoto University Research Institute of Mathematical Sciences Lecture Notes115, 148–160 (April 1971) (in Japanese).
Hammersley, J. M., Handscomb, D. C.: Monte Carlo methods, Chapt. 12 [see especially Eq. (12.2.13) and the subsequent discussion]. London: Methuen & Co. Ltd. 1964.
Tsuda, T.: Interpolation of multivariable functions: An application to dimensionality reduction. (Submitted for journal publication.)
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Tsuda, T. Numerical differentiation of functions of very many variables. Numer. Math. 18, 327–335 (1971). https://doi.org/10.1007/BF01404683
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DOI: https://doi.org/10.1007/BF01404683