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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.
Introduction Theory HOWTO Error Analysis Examples Questions Applications in Engineering Matlab Maple In this topic, we see that under certain circumstances, we may factor a matrix M in the form M = LL.
Such a decomposition is called a Cholesky decomposition.
It requires half the memory, and half the number operations of an PLU decomposition, but it may only be applied in restricted circumstances, namely when the matrix M is real, symmetric, and positive definite.
For example, in a 2D problem with an n-by-n grid, the 1D separator front partitioning the domain into two pieces has $s = O(n)$ degrees of freedom.
Thus amount of memory required is $O(n^2)$, the same order of magnitude as storing the right hand side.
\end Code: Matlab code to do this is, for example, If you only want to consider the nonzero eigenvalues, you can just sum over the nonzero entries of the diagonal of $R$.
Test case: I tested this against Matlab's build in det() function on the 2D finite difference Laplace operator with Dirichlet B.
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.Suppose that the n × n matrix M given by a system of linear equations is both real and symmetric.The symmetry suggests that we can store the matrix in half the memory required by a full non-symmetric matrix of the same size.C.'s, and got exact agreement for matrices up to around 400-by-400, after which point using matlab's det() yielded Inf, whereas the cholesky method had no problem going up to million-by-million scale on my laptop.Here is the code, for reference: Memory considerations: Since this method is based on the Cholesky factorization, it inherits the speed and stability from that.