On Restart and Error Estimation for Krylov Approximation of $w=f(A)v - Tal-Ezer, Hillel (SIAM journal on scientific computing, 2007)
Krylov algorithms are widely used for the solution of large scale linear systems. A Krylov approach is implemented also for the approximation of the vector $w$ which results from operating with a function of a matrix on a vector $v$. For example, when solving a set of linear ODEs we have $w = f (A)v$, where $f(z) = {\rm exp}(tz)$. The main drawback of the Krylov algorithm lies in the need to store all the vectors spanning the approximation space. When solving linear systems, it is possible to overcome this drawback by restarting. In this paper we show how to apply restarts in the general case of approximating $w=f(A)v$. This new algorithm allows a much more efficient approximation of the exponential operator than the standard Krylov algorithm, and it is especially useful in the case of functions which cannot be factored into a product of functions. Yet another interesting subject is the error estimate. The scheme given here provides error estimates "for free," a trait which enables us to stop the algorithm whenever the desired accuracy is achieved.