Timeline for Why $\alpha I +A$ can improve the condition nubmer of a SPD matrix $A$?
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Nov 22, 2019 at 16:46 | comment | added | Tihskirap Ayayhdapu | You are right about the condition number depending on $h$. Check the answer again. I added a "proof". | |
| Nov 22, 2019 at 16:43 | history | edited | Tihskirap Ayayhdapu | CC BY-SA 4.0 |
added 827 characters in body
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| Nov 22, 2019 at 11:49 | comment | added | Happy | But, the condition number of matrix $A$ is $O(h^{-2})$, why the condition number of $\alpha I+A$ is independent on mesh size? I cannot understand it, can you give me some more details or some proof? Thanks. | |
| Nov 22, 2019 at 11:39 | vote | accept | Happy | ||
| Nov 22, 2019 at 10:02 | comment | added | Tihskirap Ayayhdapu | MATLAB's gallery() with the 'poisson' arguement returns a matrix with the stencil values (-1,4,-1) and is independent of the number of discretization points or the grid spacing h. Even if you scale this matrix by multiplying by $1/(h^2)$, both the maximum and minimum eigenvalues get scaled equally and hence, there is no change in the condition number. | |
| Nov 22, 2019 at 0:17 | comment | added | Happy | Thanks for your reply. By the way, Is the condition number of $\alpha I+A$ independent on step size $h$? Why? Because in my numerical examples, when the system size increases, the condition number does not increase much. | |
| Nov 21, 2019 at 13:25 | history | answered | Tihskirap Ayayhdapu | CC BY-SA 4.0 |