Estimating the Solutions of Slowly Varying Recursions
A. van der Sluis · SIAM Journal on Mathematical Analysis · 1976
If the eigenvalues $\lambda _1 , \cdots ,\lambda _n $, of an $n \times n$ matrix A are distinct, then the recursion $x_{i + 1} = Ax_i $ has n linearly independent solutions, one for each $\lambda _j $, such that the solution corresponding to $\lambda _j $ grows by a factor $\lambda _j $ per step. In this paper, similar results are obtained for the recursion $x_{i + 1} = A_ix_i $ if the eigensystem of $A_i $ changes slowly as a function of i. The conditions about the eigenvalues are relaxed and practicable error bounds are given.