The Richardson Extrapolation Process with a Harmonic Sequence of Collocation Points

• Authors: Avram Sidi
• Venue: SIAM Journal on Numerical Analysis
• Year: 2000

Let $A(y)\sim A+\sum^\infty_{k=1}\alpha_ky^{\sigma_k}$ as $y\rightarrow 0+$, where y is a discrete or continuous variable. Assume that $\sigma_k$ are known numbers that may be complex in general and that A(y,) is known for $y\in(0,b]$ for some b0. The aim is to find or approximate A, the limit or antilimit of A(y) as $y\rightarrow0+$. One very effective way of approximating A is by the Richardson extrapolation process that is defined via the linear systems $A(y_l)=A^{(j)}_n + \sum^n_{k=1} \overline{\alpha}_ky^{\sigma_k}_l, \ j\leq l\leq j+n$. Here $A^{(j)}_n$ are approximations to A and $\overline{\alpha}_k$ are additional unknowns. The yl are picked such that y0 y1 y2 . . . 0 and $\lim_{l\rightarrow \infty} y_l = 0$. In this paper we give a detailed analysis of the convergence and stability of the column sequences ${\{A^{(j)}_n\}}^\infty_{j=0}$ with n fixed, when $y_l=c/(l+\eta)^q$ for some positive c, $\eta$, and q. Specifically, we prove that convergence takes place as $j\to\infty$ and give the precise rate at which it does. We also prove that the process is unstable and quantify its instability asymptotically. This instability may be dealt with numerically by using high-precision floating-point arithmetic.