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学术讲座【Highly accurate doubling algorithms for M-matrix algebraic Riccati equations】

时间:2017-02-23浏览:274设置

时间:2017年2月25日(星期六)09:00 - 10:30

地点:旗山校区理工北楼601报告厅

主办:数学与计算机科学学院、福建省分析数学及应用重点实验室、数学研究中心

主讲:复旦大学 薛军工教授 

专家简介:薛军工,复旦大学数学学院教授、博导,研究领域为数值代数、排队论、随机微分方程数值解,1999年获德国洪堡基金,2004年入选教育部新世纪优秀人才计划,2006年入选上海市浦江计划。主持国家自然科学基金面上项目3项。

报告摘要:The doubling algorithms are very efficient iterative methods forcomputing the unique minimal nonnegative solution to an M-matrix algebraic Riccati equation (MARE).They are globally and quadratically convergent, except forMARE in the critical case at which it converges linearly with the linear rate 1/2. However,the initialization phase and the doubling iteration kernel of any doubling algorithm involve invertingnonsingular M-matrices. In particular for MARE in the critical case, the M-matrices inthe doubling iteration kernel,although nonsingular, move towards singular M-matrices at convergence.A nonsingular M-matrix can be inverted by the GTH-like algorithm to almost full entrywiserelative accuracy, provided a triplet representation of the matrix is known.Recently, Nguyen and Poloni (Numer. Math., 130(4):763--792, 2015) discovereda way to construct triplet representations in a cancellation-free manner for all involved M-matrices inthe doubling iteration kernel, fora special class of MAREs arising from Markov-modulated fluid queues.In this paper, we extend Nguyen's and Poloni's work to all MAREs byalso devising a way to construct the triplet representations cancellation-free.Our construction, however, is not a straightforward extension of theirs. It is made possible byan introduction of novel recursively computable auxiliary nonnegative vectors.As the second contribution, we propose an entrywise relative residual for an approximatesolution. The residual has an appealing feature of being able to reveal the entrywiserelative accuracies of all entries, large and small, of the approximation. This is in marked contrast to the usual legacy normalized residualwhich reflects relative accuracies of large entries well but not so much those of very tiny entries.Numerical examples are presented to demonstrate and confirmour claims.

 

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