DOI: https://doi.org/10.20535/1810-0546.2018.6.151759

### Program of Simplification of High-Level Polynomial at the Example of Simplification of Wilson’s Formula

#### Abstract

**Background.** Since many formulas have the form of high-level polynomials and their use leads to a large number of computations, which slows down the speed of obtaining results, the technology of simplification of high-level polynomials is considered.

**Objective.** The aim of the paper is to obtain a technology for the simplification of high-level polynomials based on the application of the theory of experiment planning to the Wilson’s formula.

**Methods.** To simplify high-level polynomials, combination and consistent application of experiment planning and least squares methods are proposed. For the field of input values, matrix of rotatable central composite plan Box of second order for three factors is constructed. To the constructed matrix the least squares method was applied, by which the coefficients of the simplified formula can be found. The resulting simplified formula will have the form of a polynomial of the 2nd degree.

**Results.** The Wilson’s formula, which has the form of a polynomial of degree 4, is simplified to the form of a polynomial of degree 2. Having broken down the entire definition domain for Wilson's formula on the parts and constructed a simplified formula for a particular part, we obtained a result that, using the simplified formula, one can calculate the speed of sound almost 25 times faster than using the Wilson's formula, with only a slight deviation in the results.

**Conclusions.**When simplifying polynomials of high degree, the reduction of the ranges of input parameters is decisive for obtaining a satisfactory deviation between the calculated values. The proposed approach to simplifying the formulas worked quite well on the example of Wilson's formula. It can also be used to simplify other formulas that have the form of high-level polynomials. One of the options for further use of the results of this work is the creation of a technology that would enable the parallel calculation of the sound speed based on the simplified formulas obtained for each of the parts to which the definition area for the Wilson's formula is divided.

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N.B. Vargaftik, *Handbook on the Thermophysical Properties of Gases and Hydroxides*. Moscow, SU: Nauka, 1972, 721 p.

V.I. Babiy, *Problems and Prospects of Measuring the Speed of Sound in the Ocean*. Sevastopol, Ukraine: Scientific and Production Center “EKOSI-Hidrofizika”, 2009, 142 p.

C.C. Leroy, “A new equation for the accurate calculation of sound speed in all oceans”, *J. Acoust. Soc. Am.*, vol. 124, no. 5, pp. 2774–2782, 2008. doi: 10.1121/1.2988296

W.D. Wilson, “Equation for the speed of sound in sea water”, *J.** **Acoust.** **Soc.** **Am.*, vol. 32, no. 10, p. 1357, 1960. doi: 10.1121/1.1907913

Yu.P. Adler, *Introduction to Experiment Planning*. Moscow, SU: Metalurgia, 1968, 155 p.

V.I. Asaturyan, *The Theory of Experiment Planning*. Moscow, SU: Radio i Sviaz’, 1983.

V.V. Nalimov, *Theory of the Experiment*. Moscow, SU: Nauka, 1971.

Yu.V. Linnik. *The Method of least Squares and the Fundamentals of the Mathematical-Statistical Theory of Processing Observations*, 2nd ed. Leningrad, SU: Fizmatgiz, 1962.

L.I. Turchak, *Fundamentals of Numerical Methods*. Moscow, Russia: Fizmatlit, 2005, 304 p.

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