zeros of polynomials

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1: 24.12 Zeros

… ►

§24.12(i) Bernoulli Polynomials: Real Zeros

… ►Let

R(n)

be the total number of real zeros of

B_{n}\left(x\right)

. … ►

§24.12(ii) Euler Polynomials: Real Zeros

… ►

§24.12(iii) Complex Zeros

… ►

§24.12(iv) Multiple Zeros

…

2: 18.16 Zeros

… ►

§18.16(ii) Jacobi

… ►

Inequalities

… ►

§18.16(iii) Ultraspherical, Legendre and Chebyshev

… ►

§18.16(iv) Laguerre

… ►

Asymptotic Behavior

…

3: 29.20 Methods of Computation

… ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … ►

§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). …

4: 1.11 Zeros of Polynomials

§1.11 Zeros of Polynomials

… ►

Horner’s Scheme

… ►

Extended Horner Scheme

… ►

§1.11(ii) Elementary Properties

… ►

Descartes’ Rule of Signs

…

5: 31.15 Stieltjes Polynomials

… ►

§31.15(ii) Zeros

►If

z_{1},z_{2},\dots,z_{n}

are the zeros of an

n

th degree Stieltjes polynomial

S(z)

, then every zero

z_{k}

is either one of the parameters

a_{j}

or a solution of the system of equations … ►If

t_{k}

is a zero of the Van Vleck polynomial

V(z)

, corresponding to an

n

th degree Stieltjes polynomial

S(z)

, and

z_{1}^{\prime},z_{2}^{\prime},\dots,z_{n-1}^{\prime}

are the zeros of

S^{\prime}(z)

(the derivative of

S(z)

), then

t_{k}

is either a zero of

S^{\prime}(z)

or a solution of the equation … ► … ►See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials. …

6: 3.8 Nonlinear Equations

… ►

§3.8(iv) Zeros of Polynomials

… ►

Bairstow’s Method

… ►For further information on the computation of zeros of polynomials see McNamee (2007). … ►

§3.8(vi) Conditioning of Zeros

… ►For moderate or large values of

n

it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem. …

7: 18.2 General Orthogonal Polynomials

… ►

§18.2(vi) Zeros

…

8: 28.9 Zeros

… ►For

q\to\infty

the zeros of

\operatorname{ce}_{2n}\left(z,q\right)

and

\operatorname{se}_{2n+1}\left(z,q\right)

approach asymptotically the zeros of

\mathit{He}_{2n}\left(q^{1/4}(\pi-2z)\right)

, and the zeros of

\operatorname{ce}_{2n+1}\left(z,q\right)

and

\operatorname{se}_{2n+2}\left(z,q\right)

approach asymptotically the zeros of

\mathit{He}_{2n+1}\left(q^{1/4}(\pi-2z)\right)

. …

9: 29.12 Definitions

… ►In consequence they are doubly-periodic meromorphic functions of

z

. ►The superscript

m

on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of

z

-zeros of each Lamé polynomial in the interval

(0,K)

, while

n-m

is the number of

z

-zeros in the open line segment from

K

K+\mathrm{i}{K^{\prime}}

. … ►The polynomial

P(\xi)

is of degree

n

and has

m

zeros (all simple) in

(0,1)

and

n-m

zeros (all simple) in

(1,k^{-2})

. … ►Let

\xi_{1},\xi_{2},\dots,\xi_{n}

denote the zeros of the polynomial

P

in (29.12.9) arranged according to … ►

29.12.13

{\frac{\rho+\frac{1}{4}}{\xi_{p}}+\frac{\sigma+\frac{1}{4}}{\xi_{p}-1}+\frac{% \tau+\frac{1}{4}}{\xi_{p}-k^{-2}}+\sum_{\begin{subarray}{c}q=1\\ q\neq p\end{subarray}}^{n}\frac{1}{\xi_{p}-\xi_{q}}=0},

p=1,2,\dots,n

ⓘ

Symbols:: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : real parameter, ( $0$ or $\tfrac{1}{2}$ ): parameter $\rho$ , ( $0$ or $\tfrac{1}{2}$ ): parameter $\sigma$ , ( $0$ or $\tfrac{1}{2}$ ): parameter $\tau$ and $\xi_{n}$ : zeros
Referenced by:: §29.20(iii)
Permalink:: http://dlmf.nist.gov/29.12.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §29.12(iii), §29.12 and Ch.29

…

10: 18.24 Hahn Class: Asymptotic Approximations

… ►Asymptotic approximations are also provided for the zeros of

K_{n}\left(x;p,N\right)

in various cases depending on the values of

p

and

\mu

. … ►For asymptotic approximations for the zeros of

M_{n}\left(nx;\beta,c\right)

in terms of zeros of

\operatorname{Ai}\left(x\right)

(§9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … ►Corresponding approximations are included for the zeros of

P^{(\lambda)}_{n}\left(nx;\phi\right)

. …