Research On Fast Validated Evaluation Of Certain Classes Of Special Functions

Special functions are pervasive in all fields of science.The most well-know application areas are in physics,engineering,chemistry,computer science and statistics.Special func-tions are particular mathematical functions with special properties and because of their importance in applications,a large number of researches have been devoted to the study and computation of special functions.Due to the complicated forms of special functions,how to get reliable evaluation of these functions has become a challenging task.In prac-tical terms,the numerical method based on floating-point arithmetic has been playing the dominant role in evaluating the special functions.So far,there is no shortage of numerical routines for evaluating many of the special functions in widely used mathematical software packages,systems and libraries.However,the algorithms are limited and ineffective,and none of these contains reliable,or validated routines,due to rounding-off errors accumu-lated caused by the limitation of work length and memory space in floating-point system under IEEE standard.In the thesis,we investigate the following problems on the automatization of error analysis,error control and efficiency involved in validated evaluation of special functions.Based on the principles of floating-point arithmetics and error analysis,we imple-ment in the computer algebra language Maple an automatic error analysis tool for compound expressions involving basic arithmetic operations.To show the utility of this tool,we apply it on the error analysis for the series and continued fraction ex-pressions of selected special functions.We perform a complete evaluation analysis on inverse trigonometric functions,er-ror functions and Polygamma functions.We investigate the influence of different expansion representations on approximation accuracy and figure out the optimal ap-proximation over different domains.Moreover,we explore some properties of these functions and propose improved evaluation algorithms for intervals where the general approximation methods become a bit problematic.And the resulting relative errors of the improvement could be less than 10-100 with less expansion terms.We investigate and apply the technique of chains of recurrences(CR)on the evaluation of Trigamma functions to achieve a fast algorithm.The results reveal that under the condition of generating the results of same accuracy,the new CR expressions accelerate the evaluation process more than 10 times.