  This paper develops a new class of exponential-type integrators where all the
matrix exponentiations are performed in a single Krylov space of low dimension.
The new family, called Lightly Implicit Krylov-Exponential (LIKE), is well
suited for solving large scale systems of ODEs or semi-discrete PDEs. The time
discretization and the Krylov space approximation are treated as a single
computational process, and the Krylov space properties are an integral part of
the new LIKE order condition theory developed herein. Consequently, LIKE
methods require a small number of basis vectors determined solely by the
temporal order of accuracy. The subspace size is independent of the ODE under
consideration, and there is no need to monitor the errors in linear system
solutions at each stage. Numerical results illustrate the favorable properties
of new family of methods.
