Accurate computation of generalized eigenvalues of regular SR-BP pairs

：黄荣（湖南科技大学）
：2021-12-11 10:00
：腾讯会议ID：653 937 001（无密码）

报告人：黄荣（湖南科技大学）

时  间：12月11日上午10:00

地  点：腾讯会议ID：653 937 001（无密码）

内容摘要:

In this paper, we consider the generalized eigenvalue problem (GEP) for bidiagonal-product (BP) pairs with sign regularity (SR), which include structured pairs associated with ill-conditioned matrices such as Cauchy and Vandermonde matrices that arise in many applications. A sufficient and necessary condition is provided for an SR-BP pair to be regular. For regular SR-BP pairs having both matrices singular, we establish sharp relative perturbation bounds to show that all the generalized eigenvalues including infinite and zero ones can be accurately determined by their BP representations. By operating on the BP representations, a new method is developed to accurately compute generalized eigenvalues of such a regular SR-BP pair. Our method first transforms the pair into an equivalent pair with a certain sign regularity. Then a technique is proposed to implicitly deflate all the infinite eigenvalues. After deflating infinite eigenvalues, finite eigenvalues are computed by reducing the deflated GEP into a standard eigenvalue problem. An attractive property of our method is that all the generalized eigenvalues are computed to high relative accuracy, specially all the infinite and zero eigenvalues are computed exactly. Error analysis and numerical experiments are performed to confirm the claimed high relative accuracy.

个人简介：

黄荣，教授、博士、博士生导师，现为湖南省芙蓉学者、湖南省杰出青年基金获得者、湖南省普通高校学科带头人、湖南省新世纪121人才工程人选、湖南省普通高校青年骨干教师、湖南省数学会副理事长等，主要从事数值代数方面的研究工作，目前在研主持国家自然科学基金面上项目等，已完成主持国家自然科学基金面上项目、国家自然科学基金青年项目、教育部博士点基金、中国博士后基金、湖南省杰出青年基金项目、湖南省教育厅重点项目、湖南省科技计划项目、湖南省教育厅优秀青年项目、湖南省自然科学基金项目等，研究成果全部以第一作者或独著方式发表在Math. Comp.、SIAM. J. Matrix Anal. Appl.、J. Sci. Comput.、Adv. Comput. Math.、Appl. Numer. Math.、BIT、J. Comput. Appl. Math.、Numer. Linear Algebra Appl.、Numer. Algor.等上。

 

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