# 代数 Algebra MATHS4072_1

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$$\rho\left(J_{p}\right)<\frac{p-1-s}{p-2}$$
then the regions of convergence of the SOR method $\left(\rho\left(\mathscr{L}{\omega}\right)<1\right)$ are For $s=1, \quad \omega \in\left(0, \frac{p}{p-1}\right)$ and for $s=-1, \omega \in\left(\frac{p-2}{p-1}, \frac{2}{1+\rho\left(J{p}\right)}\right)$.

## MATHS4072_1 COURSE NOTES ：

\begin{aligned} &\dot{x}^{k}(t)=G\left(t, x^{k}(t), x^{k-1}(t)\right), t \in[0, T], \ &x^{k}(0)=x_{0} \end{aligned}
for $k=1,2, \ldots .$ Here, the function $x^{k-1}$ is known and $x^{k}$ is to be determined.
Note that the familiar Picard iteration
\begin{aligned} &\dot{x}^{k}(t)=F\left(t, x^{k-1}(t)\right), t \in[0, T], \ &x^{k}(0)=x_{0} \end{aligned}