Numerical Methods for Evaluating Fermi-Dirac Functions

• Main Authors: Jin-Wei Lein, 連晉緯
• Other Authors: Fu-Sen Lin
• Format: Others
• Language: zh-TW
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/50896628451092766247

碩士 === 國立臺灣海洋大學 === 資訊工程學系 === 98 === This paper mainly investigates numerical methods for evaluating the Fermi-Dirac function. The Fermi-Dirac function, which arises from semi- conductor physics and astrophysics, is defined by F_{j}(x)=\frac{1}{\Gamma(1+j)}\int_{0}^{\infty}\frac{t^j}{e^{t-x}+1}\,dt , −∞ < x < ∞ , where j is integer or half of integer bearing a particular meaning in physics. Since the class of functions involves an integral over infinite range, there is no corresponding function (closed form) to this improper integral except for some integers of j. In general, it is difficult to find the exact value, we therefore have to rely on numerical methods to approximate the values of the class of functions. Over several decades, many numerical methods were proposed, how- ever, the accuracy and efficiency of the existing schemes are quite re- stricted. According to our experiments, our new approach can perform highly accurately and efficiently, provided a suitable choice of parame- ters, M (positive constant), h (step size), and N (the number of function evaluations). We also compared our method with the popular schemes, such as Goano’s algorithm, Lether’s method, and the lastest approach of Mohankumar. It is evident that our scheme works quite well and competitively.