The Static Analysis of Linear Loops

In the first part of the paper, we consider the problem of generation of polynomial invariants of iterative loops with operator of initialization of loop and non-singular linear operator in the loop body. In the article we also show the algorithm for calculating the basic invariants for linear operator of the Jordan cell, and an algorithm for calculating the basic invariants of diagonalizable linear operator with an irreducible minimal characteristic polynomial. The second part presents a new method for proving the invariance of the system of linear inequalities and of termination of certain linear iterative loops of imperative programs whose data are elements of the constructive linearly ordered field. The theoretical material of the paper is illustrated by examples.