| Abstract [eng] |
This Master‘s work covers a mathematical analysis system which can visualize multivariate data layers, approximate multi-dimensional functions by polynomials, estimate approximation accuracy and present few the most effective aproximation models. Multivariate approximation theory is an increasingly active research area today. It encompasses a wide range of tools for multivariate approximation such as multi-dimensional splines and finite elements, shift-invariant spaces and radial-basis functions. Approximation theory in the multivariate setting has many applications including numerical analysis, wavelet analysis, signal processing, geographic information systems, computer aided geometric design and computer graphics. The field is fascinating since much of the mathematics of the classical univariate theory does not straightforwardly generalize to the multivariate setting, so new tools are required. Graphs of one variable functions are frequantly displayed as curves, bivariate functions - as contour plots. In generally it is very hard to display or realize function in the multivariate setting. However, some efforts have been made to render functions of precisely three variables. Two obvious approaches suggest themselves: 1. Display a number of cross sections where one of the variables is held constant, or, 2. display contour surfaces where the value of function equals some constant. We will use the first method modification in this Master‘s work. All function variables except one will be taken as constants. Changing the constants values we will be given certain data layers. Other approximation method is based on multivariate plynomial interpolation. Instead of interpolant we will get smooth function while decreasing each variable maximum degree. Approximation accuracy will be estimated using some standard errors characteristics and three quality coefficients. Also it is possible to compare suitability of two polynomial models using F-test. The mathematical analysis system covered in this paper realizes previously mentioned approximation methods, also allows fast data layers visualization and these layers approximation. This paper includes technical documentation of the system and theory of mathematical part. Also, there is an example demonstrating work and capabilities of the system. |
