Krylov subspace methods of approximate solving differential equations from the point of view of functional calculus - Kurbatov, Vladimir Georgievich; Kurbatova, Irina Vladimirovna (Eurasian Mathematical Journal, 2012)
The paper deals with projection methods of approximate solving the problem \( Fx’(t) = Gx(t) + bu(t), y(t) = x(t), d \), which consist in passage to the reduced-order problem \( Fx’(t) = Gx(t) + bu(t), y(t) = x(t), d \), where \( F = Λ F V, G = Λ G V, b = Λ b, d = V^* d \). It is shown that if \( V \) and \( Λ \) 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 \( λ λ F - G \), of the exponential function by its rational interpolation satisfying some interpolation conditions. Special attention is paid to the case when \( F \) is not invertible.