# Investigation of the factor separation features for basic mathematical functions

Tatiana Sholokhman, Pinhas Alpert

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

## Abstract

FS for some fundamental functions In spite of wide use of the FS method for complex functions, in order to investigate the FS method analytically, we start with only a few examples of mathematical functions chosen from a wide variety of functions. In other words, the FS method is checked out on simple mathematical functions that build up a base for meteorological or, in general, physical fields. A bivariate polynomial A bivariate polynomial is described by the function Let the factors xi0 be multiplied by a varying coefficient ci: It is probably preferable to write the function f(x1, x2) as f(c1x1, c1x2), but here we focus on the variable ci (just simplified writing). It can easily be seen that the values of f and in the different simulations are related through: This function is a form of Taylor series expansion by itself. Hence, this is a trivial case. The four different contributions are naturally expected since the factors' contributions are proportional to the corresponding variables, i.e., ax1, bx2, while the synergy is the term that includes both factors, i.e., kx1x2. An exponential function An exponential function f(x1, x2) = e ax1+bx2+d was chosen. In the same way, the factors xi are multiplied by a varying coefficient ci and consequently we have The four simulations in this case yield And the four contributions are To examine the consistency of the synergism term, the direct method using the Taylor series is applied.

Original language English Factor Separation in the Atmosphere Applications and Future Prospects Cambridge University Press 11-27 17 9780521191739 9780511921414 9780521191739 https://doi.org/10.1017/CBO9780511921414.005 Published - 1 Jan 2011

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