monic

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1: 29.21 Tables

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$ , $n=1(1)30$ . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

2: 18.4 Graphics

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See accompanying text — Figure 18.4.7: Monic Hermite polynomials $h_{n}(x)=2^{-n}H_{n}\left(x\right)$ , $n=1,2,3,4,5$ . Magnify

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3: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Jacobi Formula

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Gauss–Laguerre Formula

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Gauss–Hermite Formula

… ►All the monic orthogonal polynomials

\{p_{n}\}

used with Gauss quadrature satisfy a three-term recurrence relation (§18.2(iv)): …

4: 1.11 Zeros of Polynomials

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§1.11(ii) Elementary Properties

… ►Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. …

5: 18.2 General Orthogonal Polynomials

… ►then two special normalizations are: (i) orthonormal OP’s:

h_{n}=1

k_{n}>0

; (ii) monic OP’s:

k_{n}=1

. … ►If the OP’s are monic, then

a_{n}=1

(

n\geq 0

). …

6: 18.38 Mathematical Applications

… ►The scaled Chebyshev polynomial

2^{1-n}T_{n}\left(x\right)

n\geq 1

, enjoys the “minimax” property on the interval

[-1,1]

, that is,

|2^{1-n}T_{n}\left(x\right)|

has the least maximum value among all monic polynomials of degree

n

. …

7: 32.8 Rational Solutions

… ►where the

Q_{n}(z)

are monic polynomials (coefficient of highest power of

z

1

) satisfying …