Is accuracy of block LU factorization and column by column version (I am not sure about the name but I mean the original version) equal? I think the column by column version should be more accurate than block version.
TL;DR : the “accuracy” of the methods Is the same, their “stability” depends on details of the implementation.
Long version : Both methods are “direct” methods, which means that, in “exact arithmetics”, they should both return the exact solution after a finite number of operations.
In floating point arithmetics, the solution Is not exact because of truncation errors affecting both entries of the matrix and of the RHS vector, that are amplified (at least) by a factor equal to the “condition number” of the matrix, further amplification depends on “instabilities” in the numerical algorithm that can be mitigated by specifico techniques, so the actual magnitude of the error depends on implementation details.
Thanks. I am seeing a table (5.1) in this paper which by increasing the block size res(x) has a lot of changes. So I understand by increasing the block size the stability will be affected. Is it correct?
And by using pivoting we can improve the stability.
I have another question. Here we are using the block version to improve the performance and parallel computation (BLAS3). What will happen if we want to have complete pivoting in the block version? Because we have to update all columns after the current column to select the next pivot.