The first part of this thesis is concerned with the harmonic Arnoldi method for evaluating ϕ-functions in exponential integrators. First, we introduce a framework of the harmonic Arnoldi method for ϕ-functions, which is based on the residual and the oblique projection technique. Second, we establish the relation between the harmonic Arnoldi method and Arnoldi method theoretically. Third, we apply the thick-restarting strategy to the harmonic Arnoldi method, and propose a thick-restarted harmonic Arnoldi algorithm for evaluating ϕ-functions. An advantage of this algorithm is that one can compute several ϕ-functions simultaneously in the same subspace. We show the merit of augmenting the search subspace with approximate eigenvectors, and give the relation between the error and the residual. Numerical experiments show the efficiency of our new algorithm for computing ϕ-functions. In the second part, we consider the fast numerical solution for fractional diffusion equations by using an exponential integrator method. A spatial discretization of the fractional diffusion equations by the shifted Grunwald formula leads to a system of ¨ ordinary differential equations, where the resulting coefficient matrix has the Toeplitzlike structure. An exponential quadrature rule is employed to solve such a system of ordinary differential equations. The convergence by the proposed method is theoretically studied. In practical computations, the product of a Toeplitz-like matrix exponential and a vector is calculated by the shift-invert Arnoldi method. Meanwhile, the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method. Numerical results are given to demonstrate the efficiency of the proposed method. In the third part, we study the expansion of search subspaces for non-Hermitian eigenproblems. First, we investigate a residual expansion Arnoldi method for subspace expansions, and our theoretical results justify the use of the refined Ritz vector to expand the search subspace. Second, we prove that the elements of a primitive refined Ritz vector have a decreasing pattern going to zero. We then show that the decreasing pattern exists in any arbitrary primitive approximate eigenvector. An inexact residual expansion Arnoldi method for subspace expansions is derived. Numerical examples show the effectiveness of our theoretical results for subspace expansions.