Schur-Cohn Algorithm Research Articles

Multidimensional Systems: BIBO Stability Test Based on Functional Schur Coefficients

This paper presents a multidimensional extension of the Schur-Cohn algorithm for testing BIBO stability in N variables. This new method only needs a unique condition to be checked, as an alternative to the set of conditions of the well-known Huang-Jury-Anderson stability test. The proposed algorithm is based on a generalization of the Schur coefficients recently obtained by the authors. In the two-variable case a simplification to the Sturm algorithm is presented, which can also be used in the case of a polynomial with complex coefficients. This approach is illustrated on several numerical examples.

A Fast Version of the Schur–Cohn Algorithm

In this paper we describe a fast algorithm for counting the number of roots of complex polynomials in the open unit disk, classically called the Schur–Cohn problem. For degree d polynomials our bound is of order dlog2d in terms of field operations and comparisons, but our approach nicely allow us to integrate the bit size as a further complexity parameter as well. For bit size σ of input polynomials of Z[X] our running time is of order d2σ·log(dσ)·loglog(dσ)·log(d) in the multi-tape Turing machine model, thus improving all previously proposed approaches.

A Schur's modified sequence and the number of zeros of a real polynomial in the unit disk: Testing its numerical quality

With a perturbation method, implemented on an arithmetic machine, and an stochastic arithmetic model, it is possible to determine the number of zeros of real polynomial in the unit disk, using the modified Schur-Cohn algorithm in combination with a test to measure goodness of fit. Numerical and algebraic experiments are also given.

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 O(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.

On polynomials nonnegative on the unit circle and related questions

This paper is concerned with the polynomial zero location problem with respect to the unit circle (| z| = 1) for complex parareciprocal polynomials, i.e. polynomials of the form p( z) = ∑ n i=0p iz i = ϵ ∑ n i=0p̄ n-iz i where ϵ is a unit modulus complex number. It is shown, in particular, that the problem can be solved by repeatedly applying the standard Schur-Cohn algorithm to a degree decreasing sequence of polynomials, which are recursively defined from the given polynomial p( z). The resulting algorithm yields, as a side result, a well-defined factorization of p( z). Furthermore, two new algorithms based on this factorization property are proposed, which involve linear arithmetic operations only: a trigonometric polynomial nonnegativity test and a characterization of positive (real) functions. These results extend to the unit disk (| z| < 1) similar results originally obtained in the right half plane within the context of the generalized Routh-Hurwitz algorithm.

On the root distribution of general polynomials with respect to the unit circle

A modification and an extension of Schur-Cohn's algorithm and Jury's table are presented. The new versions of the classical algorithms allow us to associate a sequence of coefficients k j to every polynomial P. We establish that the number of zeros of P outside the unit circle equals the number of k j satisfying ¦k j¦ > 1 . A simple expression for the number of zeros of P on the unit circle is also established. An extension of the modified algorithm which introduces an arbitrary parameter allows us to study the critical situations: 1 − ε ⩽ ¦k j¦ ⩽ 1 + ε , where ε is an arbitrary small positive number.

On the convergence domains of the p-cyclic SOR

For the solution of the linear system Ax = b, where A is block p- cyclic, the block SOR iterative method is to be considered. Suppose that the block Jacobi iteration matrix B, associated with A, has eigenvalues whose pth powers are all real of the same sign. The problem of the determination of the precise convergence domains of the SOR method in case A is also consistently ordered was solved by Hadjidimos, Li and Varga by using the Schur-Cohn algorithm. The same convergence domains were later recovered by other approaches too; specifically, Wild and Niethammer and also Noutsos, independently, used hypocycloidal curves. In this manuscript it is assumed that A is not consistently ordered but A T is. By using the Schur-Cohn algorithm we successfully determine, not only: (i) the precise SOR convergence domains, but also (ii) intervals for ϱ( B), the spectral radius of B, that directly imply that the optimal value of the SOR relaxation factor ω is equal to 1. In this work new results are obtained, some well-known ones are recovered or confirmed and a number of theoretical examples are investigated further. It is worth noting that among the new results, we derived something not quite expected; specifically, in many cases there exist pairs ( ϱ( B), ω) for which the SOR method associated with the matrix A we consider converges while the corresponding SOR for the p-cyclic consistently ordered matrix A T does not!

AN APPLICATION OF THE SCHUR-COHN ALGORITHM TO TIME SERIES ANALYSIS

. Standard least squares analysis of autoregressive moving-average (ARMA) processes with errors-in-variables entails the construction of a new set of parameters which are functions of the original ARMA parameters, and requires that derivatives of these new parameters of order three or less with respect to the ARMA parameters exist and be bounded. The boundedness of these derivatives in turn depends critically on the nonsingularity of a matrix B which is a function of the ARMA parameters via the new parameters in the model. A particular version of the classical Schur–Cohn algorithm enables us to establish this nonsingularity.

Schur-Cohn Sub-transforms of a Polynomial

Considering the exponential growth of the size of the coefficients of the Schur-Cohn transforms of a polynomial, we define new polynomials proportional to the latter and whose coefficients remain small. These Schur-Cohn sub-transforms replace the Schur-Cohn transforms in the computation of the number of roots of a polynomial in a disk. This considerably improves the speed of the Schur-Cohn algorithm.

On a generalization of the Szego-Levinson recurrence and its application in lossless inverse scattering

Predictor polynomials corresponding to nested Toeplitz matrices are known to be connected by the Szego-Levinson recurrence relation. A generalization of that result, where the relevant reduction process for Toeplitz matrices (of decreasing order) is defined by an elementary one-parameter linear transformation, is addressed. The descending and ascending versions of the corresponding generalized Szego-Levinson recurrence relations are discussed in detail. In particular, these relations are shown to be essentially the same as the extraction formulas for canonical Schur and Brune sections in the Dewilde-Dym (1984) recursive solution of the lossless inverse scattering problem. Some extensions of the Levinson algorithm for linear prediction and of the Schur-Cohn algorithm for polynomial stability test are presented. >

Methods of Search for Solving Polynomial Equations

The problem of determining a zero of a given polynomial with guaranteed error bounds, using an amount of work that can be estimated a priori, is attacked here by means of a class of algorithms based on the idea of systematic search. Lehmer''s machine for solving polynomial equations is a special case. The use of the Schur-Cohn algorithm in Lehmer''s method is replaced by a more general proximity test which reacts positively if applied at a point close to a zero of a polynomial. Various such tests are described, and the work involved in their use is estimated. The optimality and non-optimality of certain methods, both on a deterministic and on a probabilistic basis, are established.