The Schur–Cohn Algorithm Revisited,

The aim of the Schur–Cohn algorithm is to compute the number of roots of a complex polynomial in the open unit disk, each root counted with its multiplicity. Unfortunately, in its original form, it does not work with all polynomials. Using technics similar to those of the sub-resultants, we construct a new sequence of polynomials, the Schur–Cohn sub-transforms. For this, we propose an algorithm in only (d2) arithmetical operations (dbeing the degree of the polynomial studied), which is well adapted to computer algebra and supports specialization. We then show how to use bezoutians and hermitian forms to compute the number of roots in the unit disk with the help of the Schur–Cohn sub-transforms we have built.