Bounds for an interval polynomial

Rokne, Jon G.

doi:10.1007/bf02253209

Cited by 25 publications

(8 citation statements)

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“…The only difference from our previous results in [8] is that we now allow A and B to be arbitrary points and consider the line segment 0 rather than <A thus indicating linear We note that the same proof shows that dist ([._Jk, U)-k convergence to the hull of the polynomial.…”

Section: S=0mentioning

confidence: 72%

The range of values of a complex polynomial over a complex interval

Rokne

1979

Computing

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--ZusammenfassungThe Range of Values of a Complex Polynomial Over a Complex Interval. We discuss algorithms for the computation of the range of values of a complex interval polynomial over a complex interval. The mathematical results needed are based upon a result by Rivlin [7] valid for the range of values of a complex polynomial over the line segment [0,1]. In the present work we extend his results to an arbitrary line segment in the complex plane. Based upon these results we then generate algorithms suitable for both line segments and rectangular regions in the complex plane executed in rectangular complex interval arithmetic. The algorithms are then tested on a variety of complex interval polynomials and compared both to the true rafige of values and to the values obtained by the Horner scheme. Der Wertevorrat eines komplexen Polynoms in einen komplexen Interval AnalysisInterval analysis is now a well established subject [31, [5] and here we will only introduce some notation. The set of closed intervals on the real line R will be denoted by I (R). The set of polynomials with coefficients in 1 (R) will be denoted by I P (R).The complex plane is denoted by C. If zl, z 2 ~ C, re (zl) <_ re (Zz) and im (zl) _< im (z2) then we call [zl, z21 a rectangular complex interval. The set of rectangular complex intervals is denoted by R C (C) (see [81, [-10]). If z s C and r s R then [z, r] is called a circular complex interval. The set of circular complex intervals is denoted by C I (C) (see [31, [4]).The elementary operations on elements from 1 (R) and 1P (R) are described in [51.The operations on elements from RI (C) are described in [101 and the operations 0010-485X/79/0022/0153/$ 03.40

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Section: S=0mentioning

confidence: 72%

The range of values of a complex polynomial over a complex interval

Rokne

1979

Computing

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“…For a polynomial p, (x) s P (R) of degree n it was shown in [5] that it could be written in the Bernstein form of degree k over I = I-a, b] as…”

Section: "mentioning

confidence: 99%

“…In a recent paper [5] we explored the use of Bernstein algorithms by [1] and [4] for the evaluation of the bounds for an interval polynomial over an interval In, b]. We will here show that the coefficients required in the Bernstein expansion over an arbitrary interval can be computed using a scheme similar to the one in [1] for [0, 1] without having to magnify (shrink) the polynomial first.…”

Section: Introductionmentioning

confidence: 97%

A note on the Bernstein algorithm for bounds for interval polynomials

Rokne

1979

Computing

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“…Let a polynomial p,(x) of degree n with real coefficients {a~} for the power basis {xl}, i=0(1)n be given by Rivlin's coefficient sequence forms the basis for an algorithm for finding the maxima and minima of a polynomial on an interval and has been implemented and tested by Rokne [3]. The technique of increasing the order of the Bernstein polynomial has also found application in Computer-Aided Geometric Design (CAGD).…”

Section: Introductionmentioning

confidence: 99%