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1: 1.11 Zeros of Polynomials

§1.11 Zeros of Polynomials

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§1.11(v) Stable Polynomials

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Hurwitz Criterion

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2: Bibliography S

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Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

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Szegő polynomials from hypergeometric functions. Proc. Amer. Math. Soc. 138 (12), pp. 4259–4270.

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ISSN 0002-9939, Link, MathReview (Andrei Martínez Finkelshtein), ZentralBlatt

Referenced by:

§18.33(iv).

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A stable quotient-difference algorithm. Math. Comp. 34 (150), pp. 515–519.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§3.10(ii).

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Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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Abstract

Referenced by:

§18.16(v).

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

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3: 18.40 Methods of Computation

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§18.40(i) Computation of Polynomials

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … … ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let $N$ be a positive integer and define …See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. …

4: 18.39 Applications in the Physical Sciences

… ►The associated Coulomb–Laguerre polynomials are defined as … ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the $x_i$ and $w_i$, from the J-matrix as in §3.5(vi), straightforward. … ►

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

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The Coulomb–Pollaczek Polynomials

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§18.39(v) Other Applications

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5: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

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An inequality of Turán type for Jacobi polynomials. Proc. Amer. Math. Soc. 32, pp. 435–439.

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ISSN 0002-9939, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.14(ii).

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

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Orthogonal Polynomials: Applications and Computation. In Acta Numerica, 1996, A. Iserles (Ed.), Acta Numerica, Vol. 5, pp. 45–119.

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MathReview (Sven Ehrich), ZentralBlatt

Referenced by:

§3.5(v).

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The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.

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ISSN 0002-9939, Link, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§18.28(xi).

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6: 22.19 Physical Applications

… ► $\theta$ being the angular displacement from the point of stable equilibrium, $\theta =0$. … ►for the initial conditions $\theta (0)=0$, the point of stable equilibrium for $E=0$, and $d\theta (t)/dt=\sqrt{2E}$. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. … ►Hyperelliptic functions $u(z)$ are solutions of the equation $z={\int }_0^u{(f(x))}^{-1/2}dx$, where $f(x)$ is a polynomial of degree higher than 4. …