Lets say we have a block matrix $ M =\left( \begin A & B\ B^ & C \end \right)$ where M is positive definite. The matrix $M = LU$ can be decomposed in an algebraic manner into $L = \begin A^ & 0 \ B^ A^ & Q^ \end$ where $\begin Q = C - B^ A^ B \end$ $*$ indicates transpose in this case Now lets say we have already carried out the cholesky decomposition for A, and C.
(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.
If pivot = TRUE, then the Choleski decomposition of a positive semi-definite x can be computed.
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where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L.
Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition.The Cholesky factorization of a stiffness matrix can be updated after modifying a local stiffness matrix which can be written as a sum of a few rank-one matrices.This factorization is called the Cholesky decomposition A, and the .... 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.Problems which could involve applications of such a method arise frequently in plasticity and structural optimization where repeated solutions of a band algebraic system with a changing matrix are needed.Or in other words can I use $C^$ to help me in the calculation of $Q^$. I'm trying to work out if I can parallelise the training.