A finite-precision Lanczos-Golub-Welsch route to probability-table construction in resonance self-shielding

Abstract

This work reformulates Chiba's affine-order prescription as a polynomial-moment problem for a transformed positive measure, and develops an alternative finite-precision construction route based on this reformulation. The proposed construction proceeds through discrete-measure realization, symmetric Lanczos reduction, and Golub--Welsch extraction, replacing the conventional moment--Pade pipeline. The subgroup total levels and probabilities are obtained by a Gauss-type compression step that preserves nonnegative-realness, while the reaction-channel levels are recovered on the compressed nodes by orthogonal-basis matching. In five tested resonance-channel cases, the proposed construction yields lower effective-cross-section errors and avoids the order-induced emergence of complex responses observed in the conventional construction.

Figures (11)

• Figure 1: ROC AUC summary for the diagnostic scores $s_g^{(\mathrm{sep})}$, $s_g^{(\sigma)}$, $s_g^{(\rho)}$, $s_g^{(H)}$, and $s_g^{(V)}$ defined in \ref{['eq:score_sep']}--\ref{['eq:score_V']}, shown as functions of reconstruction order $N$. The dashed horizontal line at $\mathrm{AUC}=0.5$ marks chance-level separation. AUC values are shown only for nondegenerate pre-collapse orders at which both preserving and violating groups are present; the corresponding class counts are reported separately in the Supplementary Material. Bootstrap 95% confidence intervals for the four principal scores are also reported there. The minimum root-separation score, the $\sigma$-space and $\rho$-space root-sensitivity scores, and the Hankel-conditioning score retain substantial discriminatory power over the tested orders, whereas the Vandermonde-conditioning score deteriorates rapidly and approaches chance level, or falls below it, at moderate and high orders.
• Figure 2: Distributions of the four principal diagnostic scores at $N=7$ for nonnegative-real-preserving and nonnegative-real-violating groups: (a) minimum root separation, (b) $\sigma$-space root sensitivity, (c) $\rho$-space root sensitivity, and (d) Hankel conditioning. Each point represents one energy group. This order lies in the mixed regime, so that both classes are represented by substantial samples. Larger scores indicate greater numerical danger.
• Figure 3: Pairwise orthogonality loss of the Lanczos basis versus energy-group index for the three reorthogonalization strategies for the representative case $N=50$. For each energy group, the metric shown is $\max_{i\neq j}|q_i^{\mathsf T}q_j|$, i.e., the maximum off-diagonal inner product among the computed Lanczos vectors. The no-reorthogonalization case exhibits several pronounced spikes, selective reorthogonalization suppresses these excursions markedly, and full reorthogonalization yields the smallest and most uniform values over the full group range.
• Figure 4: Size-resolved runtime comparison between selective and full reorthogonalization. The figure shows the binned ratio $t_{\mathrm{sel}}/t_{\mathrm{full}}$ as a function of the ENDF tabulation-point count $M$ for $N=9$, $30$, and $50$. Faint points represent individual energy groups, while the solid markers and error bars denote the binwise median and interquartile range. For $N=9$, the timing difference is too small to justify replacing full reorthogonalization. For $N=30$ and $50$, a clearer advantage of selective reorthogonalization emerges once $M$ reaches about $5000$.
• Figure 5: Per-group 95th-percentile relative error of the effective cross section for the conventional and proposed constructions, evaluated against the segmentwise analytic reference. Panels (a)--(c) show the results for $N=9$, $N=30$, and $N=50$, respectively. Smaller values indicate better agreement with the segmentwise analytic reference.

Theorems & Definitions (6)

• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Remark 3.4