To perform simple calculations to compute certain quantities relating to Brownian motion, and to understand how these quantities can be important in pricing financial derivatives

这是一份Bath巴斯大学MA50170作业代写的成功案

by $\S$ 4. Apply successive point-overrelaxation (point SOR) to, with the particular (over)relaxation factor
$$
\text { (12) } \omega_{b}=2 /\left(1+\sqrt{1-\mu^{2}}\right)=1+\left[\mu /\left(1+\sqrt{1-\mu^{2}}\right)\right]^{2} \text {. }
$$
Kahan has proved that for this $\omega_{b}$,
$$
\omega_{b}-1 \leqq \rho\left(L_{\omega_{b}}\right) \leqq \sqrt{\omega_{b}-1}
$$
hence this $\omega_{b}$ is a good relaxation factor, since $\rho\left(L_{\omega}\right) \geqq \omega_{b}-1$ for any relaxation factor. For $\rho(B)=1-\varepsilon$, where $\varepsilon$ is small, the asymptotic convergence rate $\gamma=-\log \rho\left(L_{\omega_{\Delta}}\right)$ therefore satisfies
$$
\sqrt{2 \varepsilon}=-\frac{1}{2} \log \left(\omega_{b}-1\right) \leqq \gamma \leqq \log \left(\omega_{b}-1\right)=2 \sqrt{2 \varepsilon}
$$

MA50170 COURSE NOTES ：

has the desired property. This was shown by L. F. Richardson, Lanczos , and Stiefel. (Here $C_{m}(t)=\cos \left(m \cos ^{-1} t\right)$ on $(-1,1)$.) Moreover, from classic recursion formulas for the Chebyshev polynomials, it follows that where the $r$ th relaxation factor is
$$
\omega_{r}=1+C_{r-2}(1 / \rho) / C_{r}(1 / \rho), \quad \rho=\rho(B)
$$
Furthermore, as was first observed by Golub and Varga
$$
\lim {r \rightarrow \infty} \omega{r}=\omega_{b}=2 /\left[1+\left(1-\rho^{2}(B)\right)^{1 / 2}\right]
$$