Two polynomial methods of calculating functions of symmetric matrices

Comparing Leja and Krylov Approximations of Large Scale Matrix Exponentials

“…[7,14]) as the methods of choice. Nevertheless, an alternative class of polynomial methods has been developed since the beginning (cf., e.g., [6,15,13]), which are based on direct interpolation or approximation of the exponential functions on the spectrum (or the field of values) of the relevant matrix. Despite of a preprocessing stage needed to get an estimate of some marginal eigenvalues, the latter are competitive with Krylov-like methods in several instances, namely on large scale, sparse and in general nonsymmetric matrices, arising from the spatial discretization of parabolic PDEs; see, e.g., [4,10,11].…”

A new investigation of the extended Krylov subspace method for matrix function evaluations

“…For large A, a now standard way of approximating (1.1) consists of projecting the original problem onto a subspace of much smaller dimension, which for conveniency reasons is taken to be the Krylov subspace associated with A and v [13], [15], [34], [48]. For particularly challenging problems, however, an unacceptably large approximation space may be required to obtain a sastisfactory approximation.…”

“…The results can also be applied to a linear combination of such integrals with monomials λ ℓ , ℓ ∈ Z as coefficients. This is the case, for instance, for functions such as λ α with α ∈ R\Z, exp(− √ λ), tanh( √ λ)/ √ λ [13]; see also [21], [33] for a similar framework and for further examples. Nonetheless, we stress that the algorithm is also effective when used with other functions that do not belong to this class, as is the case, for instance, for f (λ) = exp(−λ); see, e.g., [45], [1].…”

“…We shall refer to the approximation in (2.1) as the Standard Krylov method, and it reduces to the Spectral Lanczos decomposition method (or SLDM) when A is symmetric (see, e.g., [13], [48]), so that H k is also symmetric and thus tridiagonal.…”

The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations

“…This scheme contains a matrix function, is exact for linear equations with constant inhomogeneity and thus unconditionally stable. In each time step the product of a matrix function with a given vector can be computed by Krylov subspace methods [37,6,21,32,7,16,8,18,36]. The time error of the scheme is of second order uniformly in the frequencies [17] and this allows to choose time steps larger than the smallest wave length.…”