Why Use LU Factorization Despite Increased Matrix Density?

[Natalie89]

Hello Everyone,

I have a question about LU factorization.

I understand that LU factorization provides an upper and lower traingular matrices of matrix A. In matlab, a large matrix was generated, and we plotted the sparsity of A and then the sparsity of L+U and it was less sparse.

My question is:

Why would we want to do this? Is it used to just save time when finding the inverse of a matrix?

 

[Ray Vickson]

Hello Everyone,

I have a question about LU factorization.

I understand that LU factorization provides an upper and lower traingular matrices of matrix A. In matlab, a large matrix was generated, and we plotted the sparsity of A and then the sparsity of L+U and it was less sparse.

My question is:

Why would we want to do this? Is it used to just save time when finding the inverse of a matrix?

Usually inverting a matrix is just about the worst way of solving a set of linear equations (unless the problem has some special structure). LU decomposition is nothing other than the familiar high-school method of solving the equations by Gaussian elimination, with the difference that we store the steps we use (so we can re-use them later if we want); these steps are put into the matrix L. Really good programs do not use "pure" LU decomposition, but may apply permutations as well to get results that are numerically more stable by avoiding pivots on small numerical values. It not uncommon for the final results to be "denser" than the original matrix, as you have observed.

RGV