Extraction of infinite zeros of polynomial matrices

Japan Journal of Industrial and Applied Mathematics, 2003

The zeros of a polynomial can be found by computing the eigenvalues of the corresponding companion matrix. However, in the case of multiple zeros, the calculated results are usually not as accurate as those for simple zeros. This problem is discussed and a new method is presented by constructing a new companion matrix which has only simple eigenvalues. By this method, instead of calculating all the zeros simultaneously, we can calculate the distinct zeros and their multiplicities separately. Numerical examples are presented to illustrate the efficiency of the method.

1986

A direct numerical method is proposed for the determination of all isolated zeros of a system of multivariate polynomial equations. By "polynomial combination", the system is reduced to a special form which may be interpreted as a multiplication table for power products modulo the system. The zeros are then formed from an ordinary eigenvalue problem for the matrix of the multiplication table. Degenerate situations may be handled by perturbing them into general form and reaching the zeros of the unperturbed system via a homotopy method.

Advances in Applied Mathematics, 1992

A method to generate accurate approximations to the singular solutions of a system of (complex) polynomial equations is presented. This method is established in a context of polynomial continuation; thus, all solutions are generated, with the singular solutions being approximated more accurately than by standard implementations. The theorem on which the method is based is proven using results from several complex variables and algebraic geometry. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at solutions, are required. A specific implementation is given and the results of numerical experiments in solving four test problems are presented. 0 1992 Academic Press, Inc.

International Journal of Control, 1997

Journal of Computational and Applied Mathematics, 2004

We present a new practicable method for approximating all real zeros of polynomial systems using the resultants method. It is based on the theory of multi-resultants. We build a sparse linear system. Then, we solve it by the quasi-minimal residual method. Once our test function changes its sign, we apply the secant method to approximate the root. The unstable calculation of the determinant of the large sparse matrix is replaced by solving a sparse linear system. This technique will be able to take advantage of the sparseness of the resultant matrix. Theoretical and numerical results are presented.

IEEE Trans. Automatic Control, 1993

A square-subsystem concept is used to determine the system zeros of an unreduced matrix fraction description of a linear dynamic system.

Automatica, 1982

Several algorithms have been proposed in the literature for the computation of the zeros of a linear system described by a state-space model 1x1-A,B,C,D}. In this report we discuss the numerical properties of a new algorithm and compare it with some earlier techniques of computing zeros. The new approach is shown to handle both nonsquare and/or degenerate systems without difficulties whereas earlier methods would either fail or would require special treatment for these cases. The method is also shown to be backward stable in a rigorous sense. Several numerical examples are given in order to compare speed and accuracy of the algorithm with its nearest competitors.

SIAM Review, 2013

When a function f (x) is holomorphic on an interval x ∈ [a, b], its roots on the interval can be computed by the following three-step procedure. First, approximate f (x) on [a, b] by a polynomial f N (x) using adaptive Chebyshev interpolation. Second, form the Chebyshev-Frobenius companion matrix whose elements are trivial functions of the Chebyshev coefficients of the interpolant f N (x). Third, compute all the eigenvalues of the companion matrix. The eigenvalues λ which lie on the real interval λ ∈ [a, b] are very accurate approximations to the zeros of f (x) on the target interval. (To minimize cost, the adaptive phase can automatically subdivide the interval, applying the Chebyshev rootfinder separately on each subinterval, to keep N bounded or to solve rare "dynamic range" complications.) We also discuss generalizations to compute roots on an infinite interval, zeros of functions singular on the interval [a, b], and slightly complex roots. The underlying ideas are undergraduate-friendly, but link the disparate fields of algebraic geometry, linear algebra, and approximation theory.

Symmetry

A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.

Journal of Computational and Applied Mathematics, 2018

Using Padé approximation, Sakurai, Torii and Sugiura derived in the paper [A high-order iterative formula for simultaneous determination of zeros of a polynomial, J. Comput. Appl. Math. 38 (1991), 387-397] the generalized iterative method of order n + 2 for finding all zeros of a polynomial, where n is the highest order of a polynomial derivative involved in the presented iterative formula. In this note we give the determinantal representation of this method and analyze procedures for its implementation and some computational aspects.