GMRES Stagnation: Showing ||rk||2=||r0||2 for k=1,2,3

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In summary, the conversation discusses finding the norm of the residual vector rk in the GMRES iteration process, specifically for iterations k = 1, 2, 3. The matrix A is given, as well as the initial vector x0 and vector b. The individual has attempted to use induction and eigenvalues/eigenvectors, but has not been successful in finding a solution. They are seeking guidance on how to approach the problem.
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Homework Statement


Let A be the 4x4 matrix with elements in positions (1,2), (2,3), (3,4), and (4,1) equal to 1, and all other elements equal to zero. Let b = e1 and x0-0. Show that ||rk||2=||r0||2 for k=1,2,3.

Here, rk=Axk-b where xk is the iterate produced by GMRES in iteration k.

Finding r0 is not a problem for me since x0 is the zero vector. Therefore, ||r0||= b. However, we have not learned how to actually figure out the different iterations, so I'm guessing that we do not have to find x1,x2, and x3 explicitly, but I have no idea how to show that ||rk||2=b for k=1,2,3. Can anyone push me in the right direction? Thank you.
 
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Homework Equations ||rk||2= ||Axk-b||2 The Attempt at a Solution I attempted to use induction, but I'm not sure how to apply it here. I also tried to use the eigenvalues and eigenvectors of A, but that didn't seem to work either.
 

What is GMRES stagnation?

GMRES (Generalized Minimal Residual) stagnation refers to a situation in which the GMRES algorithm fails to converge to a solution. This can occur when the matrix being solved is ill-conditioned or when the initial guess is far from the actual solution.

What does ||rk||2=||r0||2 for k=1,2,3 mean?

This notation represents the residual norm of the GMRES algorithm at the first three iterations. The double bars around the rk and r0 indicate the Euclidean norm, and the subscript k represents the iteration number. The equation ||rk||2=||r0||2 for k=1,2,3 shows that the residual norm remains the same for the first three iterations, indicating stagnation.

Why is it important to show ||rk||2=||r0||2 for k=1,2,3 in GMRES stagnation?

Showing that the residual norm remains the same for the first few iterations is an indication that the GMRES algorithm has reached a stagnation point. This information is important for troubleshooting and determining the cause of the stagnation.

How can GMRES stagnation be resolved?

There are several possible ways to resolve GMRES stagnation: increasing the maximum number of iterations, using a better initial guess, preconditioning the matrix, or using a different algorithm. The best approach will depend on the specific problem being solved.

Are there any drawbacks to using GMRES for solving linear systems?

While GMRES is a popular and efficient algorithm for solving linear systems, it does have some limitations. It can be computationally expensive for large matrices, and it may not converge if the matrix is too ill-conditioned or if the initial guess is too far from the actual solution. In these cases, other algorithms may be more suitable.

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