Stability of the Cholesky factorization

The Cholesky factorization algorithm is very stable. The entries of $L$ cannot grow. This is true even if we do not pivot.

This can be seen from the equations:

$$A=L{L}^T,\sum _j{l}_{ij}^2=a_{ii}$$

The entries of $L$ are bounded by the square root of the diagonal entries of $A.$ From our backward error analysis, this implies that the factorization is backward stable.

Stability of the LU factorization, Backward error analysis for LU, Row pivoting, Cholesky factorization, Existence of the Cholesky factorization