The extended Krylov subspace method and orthogonal Laurent polynomials

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.