Accurate approximation of time-varying matrices of linear differential systems

Abstract

We consider the problem of the identification of the time-varying matrix A ( t ) of a linear m -dimensional differential system y′ = A ( t )y. We develop an approximation A n , k = ∑ n j − 1 c j { Y ( t k + τ j ) Y −1 ( t k ) − I } to A ( t k ) for grid points t k = a + kh , k = 0,…, N using specified τ j = θ j h , 0 < θ j < 1, j = 1, …, n , and show that for each t k , the L 1 norm of the error matrix is O ( h n ). We demonstrate an efficient scheme for the evaluation of A n , k and treat sample problems.