Computational Spinor Linearization of Nonlinear Algebraic Systems

Abstract

This paper presents a scalable and robust computational architecture for the algebra-valued linearization and subsequent solution of square nonlinear algebraic systems. Building upon the historical foundation of the spinor method (Kuznetsov & Pshenichnikov, 1985), the proposed approach replaces scalar nonlinearities with associative algebra-valued coefficients, rendering the system linear in the unknowns while conditionally preserving the exact scalar root set.Key technical contributions include: Singular Pencil Architecture: The explicit and potentially unstable matrix inversion at rank-deficient points is replaced by an augmented matrix pencil formulation, preserving finite and infinite eigeninformation. Low-Rank Tensor Compression: Storage and coefficient growth in high-dimensional settings are controlled via low-rank Tensor-Train (TT) formats and Matrix Product Operators (MPO). Certified Computing Framework: False roots introduced by squaring rules are systematically filtered using branch signatures, while accepted isolated roots are validated through modular lifting, rational univariate representations, and Krawczyk interval inclusion tests. Reproducibility Protocol: A strict validation suite is established, measuring solver reliability through multiplicity-weighted missing and false root fractions. The complete theoretical framework is demonstrated and validated on a classic non-linear hydraulic two-loop engineering network problem.