I have a problem where I am given three simple equations composed of $x_1$, $x_2$, and $x_3$.
I am then asked to use two iterations of Jacobi's Method and also two iterations of Gauss-Seidel Method to solve for $x$.
Once doing this I have found approximations for $x$ using both methods.
The problem then states to compute the $L^2$ Norm of the error for your solutions.
This is where I am stuck.
The L2 Norm is $$ \|x\|_2 = \sqrt{\sum_{k=1}^n |x_k|^2 } $$ Is $x_k$ simply the $x$ values I solved for?
Or since these are very simple equations, am I supposed to find the individual error for each $x$, thus making $x_k$ the absolute value of the real answer minus the found answers? I don't see how this would make sense when considering larger, more complicated problems.
You have, presumably, a $3×3$ system $Ax=b$ where $A$ is diagonal dominant. $x$ is the vector $(x_1,x_2,x_3)^T$.
You iterate the methods to obtain vectors $x^{[1]},x^{[2]},x^{[3]}$ and then some $x^{[n]}$ with a larger number $n$ as the numerical limit. Each $x^{[k]}$ is here a vector $x^{[k]}=(x_1^{[k]},x_2^{[k]},x_3^{[k]})^T$.
You can now estimate how good $x^{[3]}$ is as a solution approximation by computing the norm of the residual $\|Ax^{[3]}-b\|_2$ or by comparing it to the far better approximation $x^{[n]}$ by computing the distance $\|x^{[3]}-x^{[n]}\|_2$.