Bairstow method (for zeros of polynomials)

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1: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

… ►

Normalization

… ►

Orthogonal Invariance

… ►

Summation

… ►

Mean-Value

…

2: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.2

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)=\mathit{H\!% \ell}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

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Defines:: $\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},\NVar{q_{n,m}};\NVar{-n},\NVar{% \beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun polynomials
Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.5 and Ch.31

►is a polynomial of degree

n

, and hence a solution of (31.2.1) that is analytic at all three finite singularities

0,1,a

. These solutions are the Heun polynomials. …

3: 24.1 Special Notation

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

B_{n}\left(x\right)

, is not clear. … ►

Euler Numbers and Polynomials

… ►The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

4: 18.3 Definitions

§18.3 Definitions

… ►For exact values of the coefficients of the Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

, the ultraspherical polynomials

C^{(\lambda)}_{n}\left(x\right)

, the Chebyshev polynomials

T_{n}\left(x\right)

and

U_{n}\left(x\right)

, the Legendre polynomials

P_{n}\left(x\right)

, the Laguerre polynomials

L_{n}\left(x\right)

, and the Hermite polynomials

H_{n}\left(x\right)

, see Abramowitz and Stegun (1964, pp. 793–801). … ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials

T_{n}\left(x\right)

n=0,1,\dots,N

, are orthogonal on the discrete point set comprising the zeros

x_{N+1,n},n=1,2,\dots,N+1

, of

T_{N+1}\left(x\right)

: … ►For another version of the discrete orthogonality property of the polynomials

T_{n}\left(x\right)

see (3.11.9). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

6: 29.20 Methods of Computation

§29.20 Methods of Computation

… ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices

\mathbf{M}

given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). … ►

§29.20(iii) Zeros

►Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …

7: 18.40 Methods of Computation

§18.40 Methods of Computation

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. ►However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …

8: Bibliography I

… ►

M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia)
Referenced by:: §10.21(v).
Encodings:: BibTeX

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M. E. H. Ismail (2000a) An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

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External Links:: ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt
Referenced by:: §18.39(ii).
Encodings:: BibTeX

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M. E. H. Ismail (2000b) More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.

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External Links:: ISSN 0163-0563, FullText, MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §18.39(ii).
Encodings:: BibTeX

… ►

M. E. H. Ismail and X. Li (1992) Bound on the extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 115 (1), pp. 131–140.

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External Links:: ISSN 0002-9939, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: (18.16.12), (18.16.13).
Encodings:: BibTeX

►

A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:: ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

…

9: 18.38 Mathematical Applications

… ►

Approximation Theory

… ►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the

n

th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding

2n-1

. … ►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). This process has been generalized to spectral methods for solving partial differential equations. … ►

Integrable Systems

…

10: Bibliography Q

… ►

W. Qiu and R. Wong (2004) Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

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External Links:: ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)
Referenced by:: §18.24.
Encodings:: BibTeX

►

C. K. Qu and R. Wong (1999) “Best possible” upper and lower bounds for the zeros of the Bessel function

J_{\nu}(x)

. Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:: ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX