Ordered sparse Cholesky with explicit pivot and range statuses

Input a Hermitian positive-definite $A\in {\mathbb{C}}^{n\times n}$ in the convention of Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal, its Hermitian structural graph from Undirected structural graph of a Hermitian matrix, and an ordering $\pi =(v_0,\dots ,v_{n-1})$. Use the operation model of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add. Let $P$ be the permutation matrix whose row $j$ is row $v_j$ of $I_n$, so the ordered matrix is $B=PA{P}^{\ast }$ and $B_{ij}=A_{v_i{v}_j}$. Relabel the filled graph on the original vertices by positions: $\{i,j\}\in E_{\pi }^{+}$ exactly when $\{v_i,v_j\}$ is an edge of the filled graph. In every displayed sum, accumulate in increasing $k$ from left to right.

Preallocate only the diagonal and lower-triangular positions permitted by $E_{\pi }^{+}$. The support guarantee for complex Hermitian Schur complements is proved in Hermitian Schur-complement support stays inside the symmetric filled graph. Initialize those positions from the lower triangle of $B$, using zero where a position is not present in the original pattern. For $j=0,\dots ,n-1$, perform these instructions:

1. Compute $d_j=B_{jj}-{\sum }_{k<j}{l}_{jk}\overline{l_{jk}}$ using only already computed entries, with an empty sum equal to zero. If an operation reports range failure, return range-failure$(j)$. If $d_j$ is not a finite positive real number, return invalid-pivot$(j,d_j)$.
2. Set $l_{jj}=\sqrt{d_j}$. If the square root is not finite and positive, return range-failure$(j)$.
3. For each $i>j$ such that $\{i,j\}\in E_{\pi }^{+}$, compute $l_{ij}=\frac{B_{ij}-{\sum }_{k<j}{l}_{ik}\overline{l_{jk}}}{l_{jj}}\mathrm{.}$ If an operation reports range failure, return range-failure$(j)$. Set structurally absent entries to zero.

If all columns finish, return success$(P,L)$. In exact arithmetic, the intended factor convention is $PA{P}^{\ast }=L{L}^{\ast }$ with $L$ lower triangular and positive real diagonal. The finite-arithmetic procedure makes no claim that every positive-definite input returns success: a rounded pivot can fail the stated test.