Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus

Abstract

The paper deals with projection methods of approximate solving the problem

\( Fx'(t) = Gx(t) + bu(t), \quad y(t) = \langle x(t), d \rangle \),

which consist in passage to the reduced-order problem

\( \hat{F}\hat{x}'(t) = \hat{G}\hat{x}(t) + \hat{b}u(t), \quad \hat{y}(t) = \langle \hat{x}(t), \hat{d} \rangle \),

where

\( \hat{F} = \Lambda F V, \hat{G} = \Lambda G V, \hat{b} = \Lambda b, \hat{d} = V^{*} d \).

It is shown that if \( V \) and \( \Lambda \) are constructed on the basis of Krylov’s subspaces, a projection method is equivalent to the replacement in the formula expressing the impulse response via the exponential function of the pencil \( \lambda \mapsto \lambda F - G \), of the exponential function by its rational interpolation satisfying some interpolation conditions. Special attention is paid to the case when \( F \) is not invertible.