Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 142-157. Chebyshev Spectral Methods and the Lane-Emden Problem John P. Boyd 1*1 Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109-2143, USA. Received 1 July 2010; Accepted (in revised version) 16 November 2010 Abstract The three-dimensional spherical polytropic Lane-Emden problem is $y_{rr}+(2/r) y_{r} + y^{m}=0, y(0)=1, y_{r}(0)=0$ where $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0, y(1)=0$. We find that a Newton-Kantorovich iteration always converges from an $m$-independent starting point $y^{(0)}(x)=\cos([\pi/2] x), \xi^{(0)}=3$. We apply a Chebyshev pseudospectral method to discretize $x$. The Lane-Emden equation has branch point singularities at the endpoint $x=1$ whenever $m$ is not an integer; we show that the Chebyshev coefficients are $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. However, a Chebyshev truncation of $N=100$ always gives at least ten decimal places of accuracy - much more accuracy when $m$ is an integer. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table. AMS subject classifications: 65L10, 65D05, 85A15 Key words: Lane-Emden, Chebyshev polynomial, pseudospectral. *Corresponding author. Email: jpboyd@umich.edu (J. P. Boyd)