# monic polynomial

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## 7 matching pages

##### 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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##### 3: 3.5 Quadrature

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###### Gauss–Legendre Formula

… ►###### Gauss–Jacobi Formula

… ►###### Gauss–Laguerre Formula

… ►###### 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: 18.38 Mathematical Applications

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►The scaled Chebyshev polynomial
${2}^{1-n}{T}_{n}\left(x\right)$, $n\ge 1$, 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 $n$.
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##### 6: 18.2 General Orthogonal Polynomials

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►then two special normalizations are: (i)

*orthonormal OP’s*: ${h}_{n}=1$, ${k}_{n}>0$; (ii)*monic OP’s*: ${k}_{n}=1$. …##### 7: 32.8 Rational Solutions

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►where the ${Q}_{n}(z)$ are monic polynomials (coefficient of highest power of $z$ is $1$) satisfying
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