Special functions are widely used in many scientific fields such as physics,chemistry,computer science,and engineering.Due to the complicated forms of most special functions,it is difficult to evaluate and calculate their exact values and therefore,in practice,many numerical methods are used to approximate the special functions.In most of the computer systems where the IEEE 754 floating-point standard is widely used,due to limitations in storage space and computer word length,roundoff error occurs in numerical calculations and the error will grow bigger and bigger as the calculation process accumulates.On the other hand,most of the expansions for special function approximations are infinite series or continued fractions,and truncation errors occur when a finite sum is taken.Both round-off error and truncation error can cause inaccuracies in evaluation of special functions.In the thesis,we study an error-free computational environment and analyze the design of exception handling mechanism in this environment;for evaluations of special functions,we present a complete analysis on two families of special functions,i.e.,the exponential integrals and Bessel functions.The work of this thesis includes1.Based on the universal number(Unum)formats recently introduced for an error-free computation environment,we analyze the representation of special values,include overflow,underflow,true(infinity),Na N(Not a Number)etc.,that are often encountered in the floating-point computation,and design a new mechanism on computation on these special values.This bottom-level design enables to build a mathematically reliable computing environment for further mathematical operations.2.We present a complete analysis on the validated evaluation on exponential integrals using different approximation methods.Through numerical experiments in the computer algebra system Maple,we compare the relative error of different evaluation methods over different interval for functions of different orders,and we explore the optimal approximation of the exponential integrals.3.We present a complete analysis on the Bessel function of the first kind of both integer orders and semi-odd orders.We compare the relative errors of different evaluation methods over different intervals and find the optimal approximation for the Bessel functions in discussion.In addition,we explore the effect of various evaluation methods on the Bessel functions of different orders. |