The extended Krylov subspace method and orthogonal Laurent polynomials
The extended Krylov subspace method and orthogonal Laurent polynomials
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DOI：
10.1016/j.laa.2009.03.006
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发表时间：
2009-07
影响因子：
1.1
通讯作者：
Carl Jagels;L. Reichel
中科院分区：
文献类型：
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作者：
Carl Jagels;L. Reichel
需要计算f(A)v形式的表达式，其中A是一个大的稀疏或结构化对称矩阵，v是一个向量，f是一个非线性函数，在许多应用中出现。扩展的Krylov子空间方法是计算此类表达式近似值的一种有吸引力的方案。该方法将逼近问题投影到一个较小维数的扩展Krylov子空间K，m(A)=span{A- r +1v，…，A-1v,v, v,Av，…，Am-1v}上，然后求解由此得到的小逼近问题。我们回顾了扩展Krylov子空间方法的现有结果，并将它们与Laurent多项式的性质联系起来。投影问题的结构受到特别的关注。我们关注的是m= r和m= 2r的情况。
The need to evaluate expressions of the form f(A)v, where A is a large sparse or structured symmetric matrix, v is a vector, and f is a nonlinear function, arises in many applications. The extended Krylov subspace method can be an attractive scheme for computing approximations of such expressions. This method projects the approximation problem onto an extended Krylov subspace Kℓ,m(A)=span{A-ℓ+1v,…,A-1v,v,Av,…,Am-1v} of fairly small dimension, and then solves the small approximation problem so obtained. We review available results for the extended Krylov subspace method and relate them to properties of Laurent polynomials. The structure of the projected problem receives particular attention. We are concerned with the situations when m=ℓ and m=2ℓ.