Verified computation for the geometric mean of two matrices

被引：0
作者：
Shinya Miyajima
机构：
[1] Iwate University,Faculty of Science and Engineering
来源：
Japan Journal of Industrial and Applied Mathematics | 2021年 / 38卷
关键词：
matrix geometric mean; continuous-time algebraic Riccati equation; verified numerical computation; 15A24; 39B42; 65G20;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.
引用
收藏
页码：211 / 232
页数：21
相关论文
共 19 条
[1]  
Arndt H(2007)On the interval systems Reliab. Comput. 13 245-259
[2]  
Frommer A(2009) and the powers of interval matrices in complex interval arithmetics SIAM J. Matrix Anal. Appl. 31 1279-1302
[3]  
Hashemi B(2014)Verified computation of square roots of a matrix Linear Algebra Appl. 456 199-213
[4]  
Frommer A(2017)Computing enclosures for the inverse square root and the sign function of a matrix J. Comput. Appl. Math. 313 515-535
[5]  
Hashemi B(1987)Methods for verified stabilizing solutions to continuous-time algebraic Riccati equations Linear Algebra Appl. 88 405-430
[6]  
Sablik T(2016)Computing real square roots of a real matrix Numer. Linear Algebra Appl. 23 208-229
[7]  
Haqiri T(2006)The geometric mean of two matrices from a computational viewpoint SIAM J. Matrix Anal. Appl. 29 54-66
[8]  
Poloni F(2014)The matrix golden mean and its applications to Riccati matrix equations SIAM J. Matrix Anal. Appl. 35 1205-1225
[9]  
Higham NJ(2015)Fast enclosure for all eigenvalues and invariant subspaces in generalized eigenvalue problems Linear Algebra Appl. 467 116-135
[10]  
Iannazzo B(2015)Fast enclosure for a matrix inverse square root Jpn. J. Ind. Appl. Math. 32 529-544
12