Failure-aware rounded Cholesky factorization

Input a Hermitian positive-definite matrix $A\in {\mathbb{C}}^{n\times n}$, $n\ge 1$, whose real and imaginary components are in the floating-point system, using the convention of Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal; real matrices are included. Initialize every entry of $\widehat{L}$ to zero. Process columns $j=1,\dots ,n$ in increasing order. For the diagonal, initialize $s=0$ and, for $k=1,\dots ,j-1$ in increasing order, compute $r_k=\mathrm{fl}(\mathrm{\Re }({\widehat{l}}_{jk}{)}^2),t_k=\mathrm{fl}(\mathrm{\Im }({\widehat{l}}_{jk}{)}^2),q_k=\mathrm{fl}(r_k+t_k),s=\mathrm{fl}(s+q_k),$ in that order. Then compute $p_j=\mathrm{fl}(a_{jj}-s)$. If an operation fails, or if $p_j$ is not positive and normal, return failure with pivot $j$ and the first failing operation location. Otherwise compute ${\widehat{l}}_{jj}=\mathrm{fl}(\sqrt{p_j})$; if it is not positive and normal, return failure at pivot $j$.

For each $i=j+1,\dots ,n$ in increasing order, initialize the complex accumulator $c_{ij}=0$. For $k=1,\dots ,j-1$ in increasing order, compute the complex product $w_{ik}={\widehat{l}}_{ik}\overline{{\widehat{l}}_{jk}}$ using exactly the six real operations of the complex-product convention in Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add, then update $\mathrm{\Re }(c_{ij})\leftarrow \mathrm{fl}(\mathrm{\Re }(c_{ij})+\mathrm{\Re }(w_{ik})),\mathrm{\Im }(c_{ij})\leftarrow \mathrm{fl}(\mathrm{\Im }(c_{ij})+\mathrm{\Im }(w_{ik}))\mathrm{.}$ After this dot product, compute $r_{ij}=a_{ij}-c_{ij}$ by one rounded real subtraction for each component, then compute each component of ${\widehat{l}}_{ij}=r_{ij}\mathrm{/}{\widehat{l}}_{jj}$ by one rounded real division. The empty dot product is zero. On any failure return failure with pivot $j$ and its first failing operation location (stage, column, row, inner index, component, and operation where applicable), with the diagonal and row loops ordered as stated. If all columns finish, return success(\widehat L). By A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique, exact positive definiteness ensures an exact factor exists, but does not ensure the rounded algorithm succeeds.

The finite column, row and inner-index loops unroll to an initialized arithmetic schedule. Each rounded operation returns a value or failure under the operationwise model, and each pivot test compares already computed values. The separate result Finite arithmetic schedules terminate, with early failure counted separately supplies termination with the specified completion or first-failure return; no arithmetic-success or backward-error conclusion follows from termination alone.