Rational Matrix Functions and Rank-1 Updates
Rational Matrix Functions and Rank-1 Updates
复制标题
有理矩阵函数和 Rank-1 更新
DOI：
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发表时间：
2000
影响因子：
1.5
通讯作者：
C. Loan
中科院分区：
文献类型：
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作者：
D. Bernstein;C. Loan
假设f = p/q是两个多项式的商，并且p具有rp度，q具有rq度。假设F（a）和f（a+uvt）是在{Mathbb r}^{n imes n} $中$ a中定义的，{mathbb r}^n $ in {Mathbb r}^n $，而Mathbb r}中的$ v给出 ^n $，并设置r = max {rp，rq）。我们展示了如何在O（rn2）中计算F（a+UVT），假设F（a）可以与“分母矩阵” Q（a）的适当分解一起使用。中心结果可以解释为众所周知的Sherman-Morrison公式的概括。对于应用程序，我们考虑在涉及矩阵指数的反问题中产生的雅各布计算。有了某些假设，可以通过有效利用本文开发的Rank-1更新公式来减少设置Jacobian矩阵所需的工作。
Suppose f=p/q is a quotient of two polynomials and that p has degree rp and q has degree rq. Assume that f(A) and f(A+uvT) are defined where $A in {mathbb R}^{n imes n}$, $u in {mathbb R}^n$, and $v in {mathbb R}^n$ are given and set r = max{rp,rq). We show how to compute f(A+uvT) in O(rn2) flops assuming that f(A) is available together with an appropriate factorization of the "denominator matrix" q(A). The central result can be interpreted as a generalization of the well-known Sherman--Morrison formula. For an application we consider a Jacobian computation that arises in an inverse problem involving the matrix exponential. With certain assumptions the work required to set up the Jacobian matrix can be reduced by an order of magnitude by making effective use of the rank-1 update formulae developed in this paper.