Structural Preserving Numberical Methods for Eigenvalue Problems

  • Li, Ren Cang (PI)

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EigcnproblclllS 1-.LIJpearubiquitously all across applied science and engineering, and their solutions are routinel,}' sought. and arc critical in one way or another to various scientific COlllputa. t.ional ta.sks. A lot. of progress both ill algorithmic ad\"anccs and software development has been made. but even more remains to be donC'. especially for large and sparse rnatrix computational problems which remain challenging and are going to be that way for a long time. Often a large matrix cOInputatiollal problem is soh'cd by some subs pace projections - lllOst COIlllllOllly Krylov subspace type projections. The basic idea is to project the original probleln~ (matrices) of high dimensions onto certain sub~paces to arrive at smaller and manageable O1H-;S.and the smaller reduced problems CRll then bE-' solved by one of the dense matrix algorithms such as those in LAPACK. For linear generalized eigenvalue problem A - )"B from various applications) it oftcn enjoy~ certain structural properties associa.ted with its underlying practical background. But existing Krylov subspace methocb typically project implicitly n-l A (or a similar one. e.g.~ after incorpora. ting a shift) into a much smaller 11latrix T whose eigcIlvalues are c01nputcd as approximations to sorne of A - AH: such projections willlikcly leave no trace of the block struct.ures in A and B to T. i.cH physical meaningful substructures in A and B are destroyed. There are ca.scs where structural preserving methods is far superior to those that arc blind to the inherent structures. The OBJECTIVE of this proposal is to exploit in depth struetnral properties of matrices front the standpoint of their application backgrounds and to develop accurate and efficient structural preserving nU1l1ericai methods for eigenvalue and related problems of practical signif. icance. Our motivations include generalized eigenvalue problems A - AD from applications such as the modified no(ial analysis of RLC circuits and linearized quadratic eigenvalue problem~ from strllctnral d.',llCunics. \Yc lwlievc that approxirnating a problem by one of its own kind would do bet.ter. nllr investigation. if carried out succe~sfully, will advance significantly the under1yin~ engineering
Effective start/end date7/1/057/1/06


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