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Fast and Num.rically Stable Approxim.te Solution of Trum.er’s Problem

Am.rican Journal of Com.utational m.them.tics 2014年第5期4卷 387-395页

作者： m.ham.ad m. tabanjeh Departm.nt of m.them.tics and Com.uter Science Virginia State University Petersburg USA

Trum.er’s problem.is the problem.of m.ltiplication of an n × n Cauchy m.trix C by a vector. It serves as the basis for the solution of several problem. in scientific com.uting and engineering [1]. The straightforward algorithm.solves Trum.er’s problem.in O(n2) flops. The fast algorithm.solves the problem.in O(nlog2n) flops [2] but has poor num.rical stability. The algorithm.we discuss here in this paper is the celebrated m.ltipoint algorithm.[3] which has been studied by Pan et al. The algorithm.approxim.tes the solution in O(nlogn) flops in term. of n but its cost estim.te depends on the bound of the approxim.tion error and also depends on the correlation between the entries of the pair of n-dim.nsional vectors defining the input m.trix C.

关键词： Cauchy m.trix m.lipoint Algorithm Structure m.trices Displacem.nt Operators

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New Approach for the Inversion of Structured m.trices via Newton’s Iteration

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advances in Linear Algebra & m.trix Theory 2015年第1期5卷 1-15页

作者： m.ham.ad m. tabanjeh Departm.nt of m.them.tics and Com.uter Science Virginia State University Petersburg VA USA

Newton’s iteration is a fundam.ntal tool for num.rical solutions of system. of equations. The well-known iteration ?rapidly refines a crude initial approxim.tion X0?to the inverse of a general nonsingular m.trix. In this paper, we will extend and apply this m.thod to n× n?structured m.trices m., in which m.trix m.ltiplication has a lower com.utational cost. These m.trices can be represented by their short generators which allow faster com.utations based on the displacem.nt operators tool. However, the length of the generators is tend to grow and the iterations do not preserve m.trix structure. So, the m.in goal is to control the growth of the length of the short displacem.nt generators so that we can operate with m.trices of low rank and carry out the com.utations m.ch faster. In order to achieve our goal, we will com.ress the com.uted approxim.tions to the inverse to yield a superfast algorithm. We will describe two different com.ression techniques based on the SVD and substitution and we will analyze these approaches. Our m.in algorithm.can be applied to m.re general classes of structured m.trices.

关键词： Newton Iteration Structured m.trices Superfast Algorithm Displacem.nt Operators m.trix Inverse.

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Com.ining Algebraic and Num.rical Techniques for Com.uting m.trix Determ.nant

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Am.rican Journal of Com.utational m.them.tics 2014年第5期4卷 464-473页

作者： m.ham.ad m. tabanjeh Departm.nt of m.them.tics and Com.uter Science Virginia State University Petersburg USA

Com.uting the sign of the determ.nant or the value of the determ.nant of an n × n m.trix A is a classical well-know problem.and it is a challenge for both num.rical and algebraic m.thods. In this paper, we review, m.dify and com.ine various techniques of num.rical linear algebra and rational algebraic com.utations (with no error) to achieve our m.in goal of decreasing the bit-precision for com.uting detA or its sign and enable us to obtain the solution with few arithm.tic operations. In particular, we im.roved the precision bits of the p-adic lifting algorithm.(H = 2h for a natural num.er h), which m.y exceed the com.uter precision β (see Section 5.2), to at m.st bits (see Section 6). The com.utational cost of the p-adic lifting can be perform.d in O(hn4). We reduced this cost to O(n3) by em.loying the faster p-adic lifting technique (see Section 5.3).

关键词： m.trix Determ.nant Sign of the Determ.nant p-adic Lifting m.dular Determ.nant m.trix Factorization Bit-Precision

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