UKRAINIAN
Astronomical School’s Report, 2020, Volume 16, Issue 2, Pages 38–42
Estimation of accuracy of the values defined from functional dependencies
Fys M.M., Brydun A.M., Sohor A.R.
Abstract
The generalization of the formula of a priori accuracy estimation for the case of implicitly given functions is performed. This is based on the classical definition of the root mean square error, which is given as the sum of the squares of the products of partial derivatives of the arguments and the errors of their definition. Differentiation of functions is carried out with the involvement of the theory of implicit functions of many variables, which does not require explicit assignment of the function by an analytical expression. The corresponding derivatives are determined by the differentiation of the equations in which the investigated function appears, according to the corresponding variables, including the function itself. Only the values of the function and arguments for which the accuracy is evaluated are required for the calculations. These values are found in different ways, including approximate methods for solving nonlinear equations (for example, Newton's method, half-division method). This approach is generalized to the case of several functions, which are already determined by a set of nonlinear equations. Their differentiation gives a linear system, which solutions are elements in the formulas for estimating the accuracy of each function. The solution of this system is determined by Cramer's method. Since the matrix of coefficients is the same for all linear systems, it is advisable to use the inverse matrix method to solve it. This significantly reduces the calculations. The values of the functions for which errors are determined are obtained from the set of equations that connect them. Finding them is much more problematic than for one variable. Thus, a strict priori estimate of accuracy is obtained without any restrictions on the studied functions, for example, in the form of their approximate representation by Taylor series or approximate estimates when solving equations. Proposed method is tested on test examples, which include the assessment of accuracy for both one and two variables, and is considered in the first case. The results of the calculations confirm the feasibility of using this technique. Therefore, with the traditional approach, the above algorithm can be used in more complex cases, i.e. for the case of implicit definition of the function.
Keywords: implicit function; partial derivatives; error theory
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