# 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 $B_{n}\left(x\right)$. …
##### 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). …
##### 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
31.15.2 $\sum_{j=1}^{N}\frac{\gamma_{j}/2}{z_{k}-a_{j}}+\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{n}\frac{1}{z_{k}-z_{j}}=0,$ $k=1,2,\dots,n$.
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
###### 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. …
##### 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 $K$ to $K+\mathrm{i}{K^{\prime}}$. … 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 ${\frac{\rho+\frac{1}{4}}{\xi_{p}}+\frac{\sigma+\frac{1}{4}}{\xi_{p}-1}+\frac{% \tau+\frac{1}{4}}{\xi_{p}-k^{-2}}+\sum_{\begin{subarray}{c}q=1\\ q\neq p\end{subarray}}^{n}\frac{1}{\xi_{p}-\xi_{q}}=0},$ $p=1,2,\dots,n$.
##### 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 $P^{(\lambda)}_{n}\left(nx;\phi\right)$. …