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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.
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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: $$P=\begin{bmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}$$
Dec
7
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.
Dec
6
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.
Dec
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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.
Dec
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revised Modified Cholesky factorization and retrieving the usual LT matrix
Grammer fixes.
Dec
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revised Modified Cholesky factorization and retrieving the usual LT matrix
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Dec
6
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 $P*(LL^T)*P^T=A+E$.
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asked Modified Cholesky factorization and retrieving the usual LT matrix
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accepted Sign change count in modified Sturm sequence for calculating eigenvalue
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answered Sign change count in modified Sturm sequence for calculating eigenvalue
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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.
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11
revised Sign change count in modified Sturm sequence for calculating eigenvalue
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revised Sign change count in modified Sturm sequence for calculating eigenvalue
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asked Sign change count in modified Sturm sequence for calculating eigenvalue