Dedicated to Henk van der Vorst on the occasion of his 65th birthday.Abstract. 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 K ℓ,m (A) = span{A −ℓ+1 v, . . . , A −1 v, v, Av, . . . , A m−1 v} 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. . 1 e.g., by determining the spectral or Cholesky factorizations of T m . For instance, m steps of the Lanczos process applied to A with initial vector v yields the decompositionis symmetric and tridiagonal, g m ∈ R n , and V T m g m = 0. Here and below e j = [0, . . . , 0, 1, 0, . . . , 0] T denotes the jth axis vector and · the Euclidean vector norm. We tacitly assume that m is chosen small enough so that a decomposition of the form (1.3) exists. The columns of V m form an orthonormal basis for the Krylov subspaceThe expression (1.1) now can be approximated by(1.5) see, e.g., [4,10,14,21] for discussions on this approach. Indeed, if g m = 0, then w m = w. Moreover, let P m−1 denote the set of all polynomials of degree at most m − 1. Then f ∈ P m−1 implies that w m = w; see, e.g., [10] or [22, Proposition 6.3].The decomposition (1.3) and the fact that range(V m ) = K m (A, v) show that:i) The columns v j of V m satisfy a three-term recurrence relation. This follows from the fact that T m is tridiagonal. The vectors v j therefore are quite inexpensive to compute; only one matrix vector-product evaluation with A and a few vector operations are required to compute v j+1 from v j and v j−1 . ii) The columns v j can be expressed as

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The evaluation of matrix functions of the form f (A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K m,mi+1 (A) = span{A −m+1 v,. .. , A −1 v, v, Av,. .. , A mi v}, m = 1, 2, 3,. .. , with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules.

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = p U + { I , where p and { are constants, I/ is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4]

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