计算正余弦矩阵的两种算法  

Two algorithms for computing the matrix sine and cosine

作　　者：张芳妹 晏云 ZHANG Fang-mei;YAN Yun(School of Mathematics and Statistics,Minnan Normal University,Zhangzhou,Fujian 363000,China;Fujian Key Laboratory of Granular Computing and Applications,Zhangzhou,Fujian 363000,China)

机构地区：[1]闽南师范大学数学与统计学院,福建漳州363000 [2]福建省粒计算及其应用重点实验室,福建漳州363000

出　　处：《宁德师范学院学报(自然科学版)》2024年第4期444-448,共5页Journal of Ningde Normal University(Natural Science)

基　　金：福建省自然科学基金面上项目(2022J01897);福建省中青年教师教育科研项目(JAT190374);闽南师范大学校长基金(KJ18019)。

摘　　要：提出两种新算法来计算正余弦矩阵函数。首先,使用Euler公式将正弦矩阵和余弦矩阵用复指数矩阵表示出来。其次,将计算实指数矩阵的精细积分方法推广应用于复指数矩阵的计算,并给出了基于Taylor级数展开和基于Padé级数展开的两种算法。最后,利用Euler公式将复指数矩阵的计算结果应用于正弦矩阵和余弦矩阵的求解。为了测试提出的新算法,给出两个数值算例,结果表明这两种新算法的计算精度很高,与基于倍角公式的精细积分方法相差不大,但这两种新算法的计算效率更高。因此,可以考虑将提出的算法应用于需要求解正余弦矩阵的相关领域。Two new algorithms are presented for calculating the two types of matrix functions,sine and cosine matrices.Firstly,the sine and cosine matrices are represented as complex exponential matrices using Euler’s formula.Then,the precise integration method for calculating real exponential matrices is extended to complex exponential matrices,and two algorithms based on Taylor series expansion and Padéseries expansion are pro⁃posed.Finally,the calculation results of complex exponential matrices are applied to the solution of sine and co⁃sine matrices again with the help of Euler’s formula.In order to test the new algorithms,two numerical ex⁃amples are given,which show that the new algorithms have high accuracy,are comparable to the precise integra⁃tion method based on double-angle formulas,but have higher efficiency.Therefore,it is possible to consider ap⁃plying the algorithms proposed in this paper to related fields where sine and cosine matrices need to be solved.

关 键 词：精细积分法 EULER公式 Padé级数 TAYLOR级数 正余弦矩阵 指数矩阵 

分 类 号：O151.21[理学—数学]