Schurcriterium

The Schur criterion is a collection of tests for determining whether all zeros of a polynomial lie inside the unit circle in the complex plane. It is used to assess Schur stability in discrete-time systems, serving as the analogue of the continuous-time Routh–Hurwitz criterion.

For a polynomial p(z) = a0 z^n + a1 z^{n-1} + ... + an with a0 ≠ 0, p is Schur-stable if every zero satisfies |z| < 1. The most common form is the Schur–Cohn criterion, which provides a finite sequence of algebraic checks. One formulation uses the reciprocal polynomial p*(z) = z^n overline{p(1/\bar z)} and a Schur transform that maps p to a new polynomial whose zeros are the reflections of the original zeros across the unit circle; applying this transform decreases the degree by one. Repeating this process yields a sequence p0, p1, ..., pn, and p is Schur-stable precisely when all steps are well defined and the associated determinants (the Schur–Cohn determinants) are positive, or equivalently when the corresponding reflection coefficients lie inside the open unit disk.

In practice, the criterion provides a numerically robust alternative to root location, widely used in digital filter design, digital control, and signal processing to guarantee stability. It also relates to the Schur class of analytic functions mapping the unit disk to itself; Schur's work connects coefficient conditions with boundedness in the unit disk.

Historically, the criterion is named after Issai Schur and was developed in the early 20th century as part of his work on analytic and algebraic function theory.