# 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
$${\mathrm{\mathit{H}\mathit{p}}}_{n,m}(a,{q}_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell}(a,{q}_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

►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

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###### 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 ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials ${C}_{n}^{(\lambda )}\left(x\right)$, 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 ${T}_{n}\left(x\right)$, $n=0,1,\mathrm{\dots},N$, are orthogonal on the discrete point set comprising the zeros ${x}_{N+1,n},n=1,2,\mathrm{\dots},N+1$, 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)). …##### 5: 3.8 Nonlinear Equations

###### §3.8 Nonlinear Equations

… ►###### Bisection Method

… ►###### Secant Method

… ►###### Eigenvalue Methods

… ►###### Bairstow’s Method

…##### 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

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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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An electrostatics model for zeros of general orthogonal polynomials.
Pacific J. Math. 193 (2), pp. 355–369.
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More on electrostatic models for zeros of orthogonal polynomials.
Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.
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Bound on the extreme zeros of orthogonal polynomials.
Proc. Amer. Math. Soc. 115 (1), pp. 131–140.
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On the asymptotic analysis of the Painlevé equations via the isomonodromy method.
Nonlinearity 7 (5), pp. 1291–1325.
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##### 9: 18.38 Mathematical Applications

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