METHOD OF ONE-DIMENSIONAL DISCRETE INTERPOLATION, USING COORDINATES OF THREE POINTS OF NUMERIC SEQUENCES, IN THE CASE OF EXPONENTIAL FUNCTIONS
Ключові слова:discrete modeling, geometric images, finite difference method, geometric apparatus of superposition, exponential functions
The purpose of discrete geometric modeling is a discrete representation and definition of geometric images (any engineering objects, processes or phenomena).
Solving of most engineering problems requires constructing and analyzing geometric models, processes, phenomena in a discrete form. The main demands for such models are adequacy, clarity, simplicity and accuracy. Created models with a given accuracy should reflect all the characteristic features of objects and at the same time be as accessible as possible during research. A discrete geometric interpretation of numerical methods, in particular the finite difference method, is closely related to specific applied problems, gives more visualization to the numerical methods and makes them an effective tool for designing geometric objects.
An implementation of the discrete geometric modeling process involves, in particular, a development of effective algorithms of transition from a discretely presented image to its continuous analogue and vice versa. It is necessary, because the most significant theoretical and applied results of creating modeling techniques were obtained for continuous forms of input data, but the most of input data, target conditions and requirements of applied tasks, their presentation, processing and analysis of data are discrete. Using the geometric apparatus of superposition allows to perform such transitions in the simplest way.
Using the geometric apparatus of superpositions in combination with the classical method of finite differences can significantly increase efficiency and expand capabilities of the process of geometric images discrete modeling. In particular, it allows investigating a possibility of using not only parabolic, but also any other functional dependencies as interpolants.
By the example of the exponential function, it is shown that the obtained formulas for calculating the superposition coefficients values of given three nodal points for selected computational schemes allow to solve problems of continuous discrete interpolation and extrapolation by numerical sequences of any one-dimensional functional dependences (to determine ordinates of desired points of discrete curves by three given ordinates of nodal points) without laborious operations of compiling and solving huge systems of linear equations.
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