An algorithm for locating all zeros of a real polynomial

Locher, Franz; Skrzipek, Michael-Ralf

doi:10.1007/bf02238233

Cited by 6 publications

(5 citation statements)

References 6 publications

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“…in some algorithms for determining or locating zeros ofq n , see, for example, [6], or in stability tests [5], [7].…”

Section: Some Numerical Remarksmentioning

confidence: 99%

Polynomial evaluation and associated polynomials

Skrzipek

1998

Numerische Mathematik

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We show a connection between the Clenshaw algorithm for evaluating a polynomial q n , expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These results enable us to get the derivatives of q n (z) analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomialq n (z) given in monomial (!) representation can be evaluated for z ∈ C using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded in terms of Chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients. Classification (1991): 65D20, 42C10 Mathematics Subject

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“…in some algorithms for determining or locating zeros ofq n , see, for example, [6], or in stability tests [5], [7].…”

Section: Some Numerical Remarksmentioning

confidence: 99%

Polynomial evaluation and associated polynomials

Skrzipek

1998

Numerische Mathematik

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“…The idea of this construction is based on the work of Schelin (1983) who first used Chebyshev polynomials to construct Sturm-like sequence to count zeros of real polynomials. A similar construction using Chebyshev polynomials appears in the works of Locher and Skrzipek (1995) and Gleyse (1997). The examination of the number of sign changes and the sign repetitions in the built-off Sturm sequences in this work using 4q-Boubaker polynomials finally leads to define the complete protocol to achieve the goal of computing the number of complex zeros of real polynomials.…”

Section: Introductionmentioning

confidence: 81%

“…For a proof, refer to Collins and Rudiger (1983). While the usual Sturm theorem and related works on Sturm-like sequence using Chebyshev polynomials in literature (Schelin, 1983;Locher and Skrzipek, 1995;Gleyse, 1997) target only the number of real zeros of real coefficient polynomials in open unit disk or other annulus, we demonstrate through examples in the next section that the proposed protocol -an extension to the theorem -can be used to count number of complex zeros of real polynomials in open unit disk.…”

Section: The Boubaker Polynomialsmentioning

confidence: 98%

An efficient numerical method for computation of the number of complex zeros of real polynomials inside the open unit disk

Shaikh

Boubaker²

2016

Journal of the Association of Arab Universities for Basic and A

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In this paper, a simple and efficient numerical method is proposed for computing the number of complex zeros of a real polynomial lying inside the unit disk. The proposed protocol uses the Boubaker polynomial expansion scheme (BPES) to build sequence of polynomials based on the concept of Sturm sequences. The method is used in a direct way without using any restrictions in reference to other existing methods. The protocol is applied to some example polynomials of different orders and utility of the algorithm is noticed.

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“…As has zeros on |z| = 1 and by the symmetry of its zeros with respect to the real axis, there are 2 zeros for t ∈ (0, ) . Thus, r ∕2 ∶= gcd(r , r −1 ) ∈ Π ∕2 can be calculated by the Euclidean algorithm for Chebyshev expansions which we have introduced in [25]. This algorithm yields r ∕2 as an U -expansion.…”

Section: Zeros On the Unit Circlementioning

confidence: 99%

“…by Newton's method with deflation. The deflation can be done by the Euclidean algorithm for Chebyshev expansions [25], too, or by the Clenshaw algorithm in the version which we have described in [41, p. 605 f.], where q (1) n−1 from there is the deflated polynomial and y from there denotes the zero. Alternatively, the algorithm in [6] calculates the roots as eigenvalues of a matrix, whose coefficients depend on the recurrence coefficients of the U and on the coefficients of r ∕2 in terms of the U .…”

Section: Zeros On the Unit Circlementioning

confidence: 99%

Parameter estimation and signal reconstruction

Skrzipek

2021

Calcolo

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In frequency analysis an often appearing problem is the reconstruction of a signal from given samples. Since the samples are usually noised, pure interpolating approaches are not recommended and appropriate approximation methods are more suitable as they can be interpreted as a kind of denoising. Two approaches are widely used. One uses the reflection coefficients of a finite sequence of Szegő polynomials and the other one the zeros of the so called Prony polynomial. We show that both approaches are closely related. As a kind of inverse problem, it’s not surprising that they have in common that both methods depend very sensitive on sampling errors. We use known properties of the signal to estimate the positions of the zeros of the corresponding Szegő- or Prony-like polynomials and construct adaptive algorithms to calculate these ones. Hereby, we get the corresponding parameters in the exponential parts of the signal, too. Then, the coefficients of the signal (as a linear combination of such exponential functions) can be obtained from a system of linear equations by minimizing the residuals with respect to a suitable norm as a kind of denoising.

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An algorithm for locating all zeros of a real polynomial

Cited by 6 publications

References 6 publications

Polynomial evaluation and associated polynomials

Polynomial evaluation and associated polynomials

An efficient numerical method for computation of the number of complex zeros of real polynomials inside the open unit disk

Parameter estimation and signal reconstruction

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