Let A be the tridiagonal finite-element matrix for the one-dimensional diffusion equation, let $Ax = 0, A \in R^{nxn}$ Implement the weighted-Jacobi and Gauss-Seidel iterations to solve $Ax = 0$. Consider relative reduction in residual norm $\frac{\|0−Ax^l\|}{\|0−Ax^{l−1}\|}, l$ is the number of iteration. What can you say about the “asymptotic” convergence as $l$ gets large? Does this depend on $n$?
My intuition says, that since we are getting closer to the solution with each iteration $\|0−Ax^l\|$ and $\|0−Ax^{l−1}\|$ would be equal for large $l$, so the fraction goes to $1$, and this doesn't seem to depend on $n$. But I'm not sure, can someone give me a formal proof?
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$\begingroup$Without further information on $A$ there is no way to answer your questions. Regarding your intuition, depending on $A$, the methods may not even converge for every initial approximation, so the "getting closer to the solution with each iteration..." may not occur.
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