Samik R
Reputation
Top tag
Next privilege 250 Rep.
 Oct7 awarded Popular Question Aug11 awarded Teacher Jul2 awarded Notable Question Nov20 awarded Popular Question Dec7 comment Modified Cholesky factorization and retrieving the usual LT matrix Continued: The $M=P*L$ thus becomes: $$M=\begin{bmatrix}1.2247 & 2.1213 & 0 & 1 \\ 1.633 & 1.4142 & 2.3094 & 0 \\ 3.266 & -1.4142 & 1.5877 & 3.1325 \\ 2.4495 & 0 & 0 & 0 \end{bmatrix}$$ This matrix is different than your result in command [4]. May be I am not getting the correct rotation matrix. Dec7 awarded Commentator Dec7 comment Modified Cholesky factorization and retrieving the usual LT matrix I am still trying to understand what the rot() command is doing in MatMate (or rather, how it is doing what it is doing). I assumed that, by rotation, the resultant matrix would be the dot product of a rotation matrix (given by 'list' in the command), and the original matrix. So, for e.g., if I take [4]-th command in the first part of your comment (M = rot(L,"drei",2´3´4´1)), for the given list, I think the rotation matrix would be: $$P=\begin{bmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}$$ Dec7 comment Modified Cholesky factorization and retrieving the usual LT matrix Thanks for the further explanations. Let me digest and get back - I think I now have enough information. Summary: I need to apply rotation in order to get back the vanilla Cholesky factor. The rotations that are needed are, I am guessing, to be somehow found from the permutation matrix, ostensibly from the order of pivoting that has been performed in the algorithm. Note that: the algorithm in the paper produces both the L (modified Cholesky factor matrix) and P (the permutation matrix). The P matrix is not unknown, as you mention in the last line. Dec6 comment Modified Cholesky factorization and retrieving the usual LT matrix Thanks for your detailed comments. However, I am feeling slightly lost. The first part of your post seem to suggest that there is a way to recover the vanilla Cholesky factor given the modified Cholesky factor and the permutation matrix. But I did not get the procedure (or "steps") to do that. Regarding the second part of your post: you are correct, that is how the algorithm proceeds (by pivoting on the maximum diagonal), but I am not sure what your example demonstrates. Sorry, can you please help me out some more. Dec6 comment Modified Cholesky factorization and retrieving the usual LT matrix @J. M.: I just added an example of a 4X4 matrix. Please let me know if it makes sense now. Thanks again. Dec6 revised Modified Cholesky factorization and retrieving the usual LT matrix Grammer fixes. Dec6 revised Modified Cholesky factorization and retrieving the usual LT matrix added 3 characters in body Dec6 comment Modified Cholesky factorization and retrieving the usual LT matrix @J.M.: Thanks for replying. My implementation of the algorithm did not give the same lower triangular matrix as the vanilla Cholesky for pd matrices. I have also checked the implementation with a few other and the answers match. I think the reason is that, PD or not, the algorithm involves pivoting and hence the answer does not remain same. The only relationship that holds is $P*(LL^T)*P^T=A+E$. Dec6 asked Modified Cholesky factorization and retrieving the usual LT matrix Apr25 accepted Sign change count in modified Sturm sequence for calculating eigenvalue Apr15 answered Sign change count in modified Sturm sequence for calculating eigenvalue Apr11 comment Sign change count in modified Sturm sequence for calculating eigenvalue @J.M.: Thanks a lot for the comments and pointer. I did not count 0->1 in the original sequence, since that is how it is stated in Golub-VanLoan (Theorem 8.5.1, Ch. 8, Matrix Computations, 3rd Ed). I did not get what you meant in the line "The substitution with a tiny ...". Although I do not have any zero in the subdiagonal elements, the zero was appearing in the calculation of $q_3(\mu)$, for $k=3$. So, I am unable to calculate $q_4(\mu)$, without using $\delta$. I will check the Algol code in the paper. Apr11 revised Sign change count in modified Sturm sequence for calculating eigenvalue added 31 characters in body Apr11 revised Sign change count in modified Sturm sequence for calculating eigenvalue added 13 characters in body Apr10 awarded Student