Certified real-component Hermitian one-by-one and two-by-two LDL* kernels

Input stored Hermitian $D\in {\mathbb{C}}^{k\times k}$, $B\in {\mathbb{C}}^{t\times k}$ and $C\in {\mathbb{C}}^{t\times t}$, with $k\in \{1,2\}$, $t\ge 0$, threshold $0<\theta \le 1$, unit roundoff $0<u<1$, and a finite certificate budget. Use Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add. Divide complex values by a nonzero real denominator by separately dividing their real and imaginary components, using the model's real operations; no complex division is used. Any invalid operation, nonfinite output or nonzero exact intermediate outside the normal range returns range-failure. Hermitian counterparts, sign changes and conjugation are exact copies.

For a complex value $z=x+iy$, $\mathrm{\mid }z\mathrm{\mid }$ denotes its exact mathematical modulus $\sqrt{x^2+y^2}$; this convention does not add a modulus operation to the floating-point model. Moduli in guard specifications and error bounds have this meaning, while the algorithm obtains only the rigorous bounds produced by the rounded scaled-norm procedure below.

The pivot controller uses exact real-RAM arithmetic on the stored real components and the supplied exact $u,\theta$, separately from the rounded numerical kernel. Its allowed operations are real addition, subtraction, multiplication, division by a certified nonzero real denominator, fixed nonnegative integer powers, and comparisons. Each field operation or comparison consumes one unit of the nonnegative integer controller budget; a fixed power is expanded into its stated finite chain of multiplications. Before any controller operation, if no unit remains, return pivot-inconclusive. Exact sign/absolute-value and maximum decisions use comparisons and conditional sign changes. For binary/rational stored values and rational $u,\theta$, exact rational arithmetic implements this controller with finite integers; no bit-complexity bound is claimed. There is no optional unspecified certifier or sign oracle. For complex modulus guards, a successful two-component scaled norm first scales by the larger absolute real component, then rounds both divisions, squares, sum, positive square root and rescaling. Its returned $h$ gives bounds $h\mathrm{/}(1+u{)}^4\le \mathrm{\mid }z\mathrm{\mid }\le h\mathrm{/}(1-u{)}^4$; zero is exact. Maxima use the corresponding lower/upper bounds and an empty maximum is zero. Evaluate modulus guards once per distinct stored complex value in row-major order, using only the displayed rounded scaled norm; its finite floating operations carry their own range status. No exact square root or additional modulus operation is used by the controller. Evaluate each displayed rational bound left to right, entrywise in row-major order; evaluate column maxima in increasing row order, then the at-most-two-term matrix-vector bounds in increasing column order, and finally the acceptance comparisons in row order. Compute each of $(1-u{)}^4,(1+u{)}^4,(1+u{)}^5$ by left-associated repeated multiplication. These prescribed finite expression trees and finite input loops fix the guard procedure and its budget count. If acceptance is not established, evaluate the stated lower-bound failure comparisons; if neither conclusion follows, return pivot-inconclusive without refinement.

For $k=1$, write $D=[d]$, $d$ real. If $d=0$ return pivot-failure. Certify ${\max }_i\mathrm{\mid }{B}_{i1}\mathrm{\mid }\le \mathrm{\mid }d\mathrm{\mid }\mathrm{/}\theta$ using the guard bounds. A rigorous lower maximum exceeding $\mathrm{\mid }d\mathrm{\mid }\mathrm{/}\theta$ returns pivot-failure; an unresolved comparison returns pivot-inconclusive. Set ${\widehat{M}}_{i1}$ by rounded real-component divisions of $B_{i1}$ by $d$.

For $k=2$, write $D=\left(\begin{array}{cc}a& z\\ \bar{z}& c\end{array}\right)$ with real $a,c$, and set $s=\max (\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }c\mathrm{\mid },\mathrm{\mid }\mathrm{Re}z\mathrm{\mid },\mathrm{\mid }\mathrm{Im}z\mathrm{\mid })$. If $s=0$ return pivot-failure. Compute the four real entries of $\widehat{E}=\mathrm{fl}(D\mathrm{/}s)$ and the conjugate counterpart by copying. In order compute $p=\mathrm{fl}(\widehat{a}\widehat{c}),q_1=\mathrm{fl}((\mathrm{Re}\widehat{z}{)}^2),q_2=\mathrm{fl}((\mathrm{Im}\widehat{z}{)}^2),q=\mathrm{fl}(q_1+q_2),\widehat{d}=\mathrm{fl}(p-q)\mathrm{.}$ Put ${ϵ}_d=3((1+u{)}^5-1)$ and $\widehat{J}=\mathrm{adj}(\widehat{E})=\left(\begin{array}{cc}\widehat{c}& -\widehat{z}\\ -\bar{\widehat{z}}& \widehat{a}\end{array}\right)$ by copies. Unless $\mathrm{\mid }\widehat{d}\mathrm{\mid }>{ϵ}_d$ is certified, return pivot-inconclusive. For each entry obtain rigorous modulus bounds $j_{ij}^{-}\le \mathrm{\mid }{\widehat{J}}_{ij}\mathrm{\mid }\le j_{ij}^{+}$ and set $U_{ij}=\frac{j_{ij}^{+}}{s(1-u)(\mathrm{\mid }\widehat{d}\mathrm{\mid }-{ϵ}_d)},V_{ij}=\frac{j_{ij}^{-}}{s(1+u)(\mathrm{\mid }\widehat{d}\mathrm{\mid }+{ϵ}_d)}\mathrm{.}$ Let $c_j^{-},c_j^{+}$ bound ${\max }_i\mathrm{\mid }{B}_{ij}\mathrm{\mid }$. Accept only if $U{c}^{+}\le {\theta }^{-1}(1,1{)}^T$ componentwise. If any coordinate of $V{c}^{-}$ exceeds ${\theta }^{-1}$ return pivot-failure; otherwise an unresolved test returns pivot-inconclusive. On acceptance compute ${\widehat{B}}_s=\mathrm{fl}(B\mathrm{/}s)$, $K=\mathrm{fl}({\widehat{B}}_s\widehat{J})$ by sequential two-term complex dots, then $\widehat{M}=\mathrm{fl}(K\mathrm{/}\widehat{d})$ by real-component divisions. No explicit reciprocal $1\mathrm{/}s$ is formed.

For either block size, compute each allocated lower-triangle entry of $\widehat{Z}=\mathrm{fl}(\widehat{M}{B}^{\ast })$ by sequential $k$-term dots. Compute $\mathrm{fl}(C-\widehat{Z})$ there, retain only its real component on the diagonal, and obtain upper entries by exact conjugation. Return success with stored $D,\widehat{M},\widehat{S}$ and the pivot certificates. An empty trailing block returns empty $\widehat{M},\widehat{S}$. In the sparse application, allocate the original block-neighbor clique and retain positions even when their numerical value is zero. Additional sparsity drops must be recorded separately as additive perturbations.