Approximating the large sparse matrix exponential using incomplete orthogonalization and Krylov subspaces of variable dimension

Krylov subspace approximations to the matrix exponential are popularly used with full orthogonalization instead of incomplete orthogonalization, even though the latter strategy is known to reduce the cost by truncating the recurrences of the modified Gram-Schmidt process. This study combines such a strategy with an adaptive step-by-step integration scheme that allows both the stepsize and the dimension of the Krylov subspace to vary. A convergence analysis is done. Numerical results on test problems drawn from systems biology and computer systems show a significant speedup over the standard implementation with full orthogonalization and fixed dimension.