Here is some code for stable updating/downdating of a Cholesky factor after simple changes to the system matrix. Our modifications leave the routines numerically identical, while making them faster (by using BLAS routines).
A method is presented for updating the Cholesky factorization of a band symmetric matrix modified by a rank-one matrix which has the same band width.
When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.
LINPACK is a software library for performing numerical linear algebra on digital computers.
LINPACK makes use of the BLAS (Basic Linear Algebra Subprograms) libraries for performing basic vector and matrix operations.
LAPACK (Linear Algebra PACKage) is a software library for numerical linear algebra.
Imagine doing linear regression, but being given the datapoints one-by-one.
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If A, C are fixed, and B is variable but nice (low-rank), then you want what is called "Cholesky update".
If A, B, C are fixed, then probably you should not be picky about how the blocking is done, and you want to use a standard "block Cholesky".
(A, and C are also pos def) There is a formula for carrying out block Cholesky decomposition. So we have already calculated $A^$, and $C^$ (It is therefore straightforward to calculate the inverses $A^$, and $C^$ using forward substitution). The problem is indeed technical in its origin , but I'd hoped (perhaps naively) that the problem would also be of interest to other mathematicians.
Rewriting the Q in terms of these quantities we now have. The problem is related to the training a machine learning algorithm.