Exact Solution of a Linear Difference Equation in a Finite Number of Steps

Abstract

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

A Simple Iterative Method for Exact Solution of Systems of Linear Algebraic Equations

“…A method to find the exact solution of SLAE in a finite number of iterations is reported for the first time. Some points of the method are stated in [8][9][10], in particular, the iterative equations for a secondorder SLAE represented in analytical form are given in [10]. Here we present a theorem and prove it.…”

On the Deepest Fallacy in the History of Mathematics: The Denial of the Postulate about the Approximation Nature of a Simple-Iteration Method and Iterative Derivation of Cramer’s Formulas

“…The basics of the method for transforming a matrix spectrum were expounded in [21] [22]. The method for solving a linear difference equation in a finite number of steps was presented in several journals, for example, in [23]. In this way, the possibility to find the exact solution of an SLAE by using iterative procedure with a stationary matrix has been confirmed.…”