Cholesky decomposition and the cost of elimination

In class today we saw the Cholesky decomposition for symmetric matrices, which at the same time can serve as an alternative derivation of the method to compute the LU decomposition.

The formulas for the Cholesky decomposition are 

        (        2     2          2    )1/2
  l   = ( a   - l   - l   - .. - l     )
   ii   (  ii    i1    i2         ii-1 )

and for i > j:

        (                                            )  /
  l   = ( a   - l   l   - l   l   - .. - l     l     ) / l
   ij   (  ij    i1  j1    i2  j2         ij-1  jj-1 )/   jj
This gives the entries l(i,j) of the lower triangular matrix L, such that A = L*L'.

The chol command of MATLAB returns U = L' instead of L.

We investigated the cost of solving a triangular system, the cost of the LU decomposition, and the cost of determinant computation via expanding rows or columns. An illustration of the discrepancy between order n^3 (cost of LU decomposition) and n! (cost of expansion of determinant) can be found at the web pages for this course