2008, Volume 10, Issue 2
Strategic Study of CAE >> 2008, Volume 10, Issue 2
High-precision Numerical Computation of High-degree Gauss quadrature Nodes
1. Anhui Institute of Optics and Fine Mechanics, CAS, Hefei 230031, China ;
2. Key Lab of Environment Optics and Technology of CAS, Hefei 230031, China ;
3. Hefei Institute of Physical Sciences, CAS, Hefei 230031, China
Next Previous
Abstract
Gauss quadrature is used widely in many fields such as the engineering numerical computation, X-ray diffraction profile analysis, spectroscopy,and so on. The nodes and weight factors of Gauss-quadrature are essential data to the numerical integration. A method to compute the zeroes of the high-degree Legendre, Laguerre and Hermite polynomials, which are the nodes of Gauss-Legendre, Gauss-Laguerre and Gauss-Hermite Quadrature, respectively, is studied, and a very efficient algorithm scan-iteration method(SIM) is given. According to the properties of Legendre, Laguerre and Hermite polynomials, their definitions are modified a little, and the stable recursive relations to compute their value are obtained. To extract these polynomials, their root intervals are searched with a certain step within a certain range. After the intervals of all roots are obtained, the roots with the desired precision can be gotten by the general iteration methods such as secant or bisection method. Numerical experiments indicate that the method is very efficient and the high-precise roots of Legendre, Laguerre and Hermite polynomials can be extracted.
Keywords
Gauss quadrature ; Legendre polynomial ; Laguerre polynomial ; Hermite polynomial ; extract roots
Figures
图 1
References
[ 1 ] 雷银照.时谐电磁场解析方法[M].北京:科学出版社,2000.253,266
[ 2 ] Kincaid D , Cheney W . Numerical Analysis : Mathematics of Scientific Computing [ M] . Beijing : China Machine Press , 2003 . 493
[ 3 ] 易大义,陈道琦.数值分析引论[M].杭州:浙江大学出版社,1998.176~178
[ 4 ] Abramowitz M , Stegun I A . Handbook of Mathematical Functions With Formulas Graphs , and Mathematical Tables [ M] . The Fifth Printing , Washington :U .S .Government Printing Office , 1964 . 919 ~ 924
[ 5 ] 数学手册编写组.数学手册[M].北京:高等教育出版社,1979.297~300
[ 6 ] 图马JJ,沃尔什RA.工程数学手册[M].欧阳芳锐,张玉平译,北京:科学出版社,2002.282,424.
[ 7 ] Recktenwald G . Numerical Methods With Matlab : Implementation and Application [M] . Translated by Wu Weiguo , Wan Qun , Zhang Hui , et al . Beijing : China Machine Press , 2004 . 439
[ 8 ] Laurie D P . Computation of Gauss-type quadrature formulas [ J] . Computationla and Applied Mathematics , 2001 , 127 : 201 ~ 217 link1
[ 9 ] Milovanovic＇ G V . Calculation of Gaussian-type quadratures with multiple nodes [A] . Spalevic＇ M M , Cvetkovic＇ A S . Mathematical and Computer Modelling [C] . 2004 , 39 : 325 ~ 347 link1
[10] Lether F G , Wenston P R . Minimax approximations to the zeros of Pn ( x ) and Gauss-Legendre quadrature [ J ] . Computational and Applied Mathematics , 1995 , 59 : 245 ~ 252 link1
[11] Swarztrauber P N . On computing the points and weights for Gauss- Legendre quadrature [J] . Siam J Sci Comput ,2002 , 24(3) : 945 ~ 954 link1
[12] Babolian E , MasjedJamei M , Eslahchi M R . One numerical improvement of Gauss-Legendre quadrature rules [ J ] . Applied Mathematics and Computation , 2005 , 160 : 779 ~ 789 link1
京公网安备 11010502051620号
