Jacobi Polynomials are defined for fixed parameters $\alpha,\beta>-1$ by their orthogonality relation $$\int_{-1}^1 (1-x)^\alpha(1+x)^\beta J_n^{(\alpha,\beta)}(x)J_m^{(\alpha,\beta)}(x) dx = c_n^{(\alpha,\beta)}\delta_{n,m}$$ where $c$ is some irrelevant normalization. It can be shown that all $n$ roots of the polynomial $J_n^{(\alpha,\beta)}$ of degree $n$ are distinct, real, and lie inside the interval $[-1,1]$. Is there any good approximation of the location of the roots $x_k$?


For the special case $\alpha=\beta=0$ (Legendre Polynomials) there are very precise formulas like $$x_k = \left(1-\frac{n-1}{8n^3} + O(n^{-4})\right)\cos\left(\frac{4k-1}{4n+2}\pi\right)$$ for $k=1,\dots,n$. And for $\alpha,\beta\in[-\frac{1}{2},\frac{1}{2}]$ there are at least some (non-overlapping) bounds on $x_k$. But what about general $\alpha,\beta>-1$? The purpose of such approximations is to use them as initial value for a Newton-Raphson method in order to find the precise roots of $J_n^{(\alpha,\beta)}$ numerically. I know the roots (which are important for certain quadrature methods) are usually computed as the eigenvalues of certain tridiagonal matrices, but I'm interested in this alternative method.


A starting approximation I frequently use for the $k$-th root of the Jacobi polynomial $P_n^{(\alpha,\beta)}(x)$ for further polishing with Newton-Raphson or Halley is


This can be derived from the first term of the asymptotic approximation for $P_n^{(\alpha,\beta)}(\cos \theta)$.

There are other references with more precise estimates, but they (often) depend on the zeroes of Bessel functions; see e.g. Gatteschi's paper, Frenzen/Wong's paper, Baratella/Gatteschi's paper, or Szegő's (now classical) book.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.