The Neumann expansion of Bessel functions (of integer order) of a function g: ℂ→ ℂ corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., g(s)=∑ℓ=0 ∞wℓφℓ(s), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

We propose a new numerical method to solve linear ordinary differential equations of the type δu/δt (t, ϵ) = A(ϵ) u(t,ϵ), where A: C → Cn×n is a matrix polynomial with large and sparse matrix coefficients. The algorithm computes an explicit parameterization of approximations of u(t, ϵ) such that approximations for many different values of ϵ and t can be obtained with a very small additional computational effort. The derivation of the algorithm is based on a reformulation of the parameterization as a linear parameter-free ordinary differential equation and on approximating the product of the matrix exponential and a vector with a Krylov method. The Krylov approximation is generated with Arnoldi's method and the structure of the coefficient matrix turns out to be independent of the truncation parameter so that it can also be interpreted as Arnoldi's method applied to an infinite dimensional matrix. We prove the super linear convergence of the algorithm and provide a posteriori error estimates to be used as termination criteria. The behavior of the algorithm is illustrated with examples stemming from spatial discretizations of partial differential equations.