# For symmetric matrices, is the Cholesky decomposition better than the SVD? [closed]

I am inverting a sparse, symmetric, ill-conditioned matrix. I have used both SVD and the LDL decomposition. I find that my results are better with the latter. Why?

I understand that LDL decomposition works only on symmetric matrices. But that doesn't necessarily say why it works better than a "more general" algorithm.

• What do you mean that your results are "better"? – Sycorax says Reinstate Monica Feb 11 '18 at 14:15
• @Sycorax I am using these decompositions to find the least squares solution to a system of linear equations. I can generate systems with known ground truth values and see which of the two methods is more reliable in recovering them. – rhombidodecahedron Feb 11 '18 at 14:26
• I think it is important to note here that you are having sparse matrices. In general, Cholesky should be better in terms of time-complexity. Cholesky has time-complexity of order $\frac{1}{3}O(n^3)$ instead $\frac{8}{3}O(n^3)$ which is the case with the SVD. These go a bit out of the window now that you are talking about sparse matrices because the sparsity pattern changes the rules of the game. I strongly suspect you are using CHOLMOD for the sparse Cholesky and that is a great work-horse, but the sparse SVD, maybe ARPACK, maybe straight-up MKL? (cont.) – usεr11852 Feb 11 '18 at 19:40
• What are dimensions of the matrices involved and what percentage of the entries are non-zero? Are you explicitly setting them as sparse matrices (the obvious thing)? Do you have any reproducible examples? – usεr11852 Feb 11 '18 at 19:43
• I think that the question could be moved to the Computational Science SE scicomp.stackexchange.com – usεr11852 Feb 11 '18 at 19:44