In Section 1.2, we gave the following description for the Cholesky factorization of matrix A:
where and are scalars and and are vectors of length n-1 . The indicates the symmetric part of A . Now,
This in turn yields the equations
We conclude that the following steps will implement the Cholesky factorization, overwriting the lower triangular portion of A with L :
- compute the Cholesky factorization of updated recursively, yielding .
Given the PLAPACK level-1 and level-2 BLAS operations described in previous chapters, a complete implementation is given in Figure 8.1. In this implementation, the ``current'' matrix A is referenced by acur. The routine Take_sqrt overwrites the view of sub-matrix by its square-root. The comments to the right of the algorithm give essentially the description of the algorithm as presented in the text. Notice that indeed the code is line-for-line a translation of this algorithm.