1. It can be shown that the spectral radius of the iteration matrix GS for the Gauss-Seidel iteration is GS2 Show that Explain why for large values of the Gauss-Seidel method converges faster. 2....

1.

It can be shown that the spectral radius of the iteration matrix

_GS
for the Gauss-Seidel iteration is

_GS

²


Show that

Explain why for large values of

the Gauss-Seidel method converges faster.

2.

There is a means of handling the Gauss-Seidel iteration for the one-dimensional Poisson equation that involves coloring half the unknown values red and half black. This method allows parallelism, whereas the iteration
does not, and the modification leads to best case convergence results that depend on it
Color

₂
light gray (represents red),

₃
black, and so forth, as shown in

If a grid point is red, what are the colors of its neighbors
If a grid point is black, what is the color of its neighbors?

To apply Gauss-Seidel, begin with

₂
and apply the difference formula to all red points. Now start at

₃
and apply the difference formula to all the black points. Sketch an algorithm that uses this coloring to implement the Gauss-Seidel iteration.

Explain why this ordering allows the iteration to be parallelized.