Cholesky factorization

The LU factorization's specialization to a symmetric positive-definite matrix: A = LL^T, using roughly half the arithmetic and never needing a pivot swap.

For a symmetric positive-definite matrix $A$ , the Cholesky factorization is $A=LL^T$ for a lower-triangular $L$ with strictly positive diagonal entries, computed by $l_{kk}=\sqrt{a_{kk}-\sum_{j<k}l_{kj}^2}$ and $l_{ik}=\frac{1}{l_{kk}}(a_{ik}-\sum_{j<k}l_{ij}l_{kj})$ for $i>k$ . Positive definiteness guarantees every quantity under a square root stays positive, so partial pivoting is never required.

Examples

For $A=\begin{pmatrix}4&2&2\\2&5&3\\2&3&6\end{pmatrix}$ : $l_{11}=\sqrt4=2$ , $l_{21}=2/2=1$ , $l_{22}=\sqrt{5-1}=2$ , and so on -- every square root stays real because $A$ is positive definite.