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
►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities .
These solutions are the Heun polynomials.
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3: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …4: 18.3 Definitions
§18.3 Definitions
… ►For explicit power series coefficients up to for these polynomials and for coefficients up to for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials , , are orthogonal on the discrete point set comprising the zeros , of : … ►Bessel polynomials
►Bessel polynomials are often included among the classical OP’s. …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 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: Bibliography I
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The factorization method.
Rev. Modern Phys. 23 (1), pp. 21–68.
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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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8: Bibliography Q
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Asymptotic expansion of the Krawtchouk polynomials and their zeros.
Comput. Methods Funct. Theory 4 (1), pp. 189–226.
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“Best possible” upper and lower bounds for the zeros of the Bessel function
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Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
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Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials.
Modern Physics Letters A 26, pp. 1843–1852.