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This is a Quantitative Study study.

September 18, 2025Open Access

Multiplication-Free Gaussian Elimination and Matrix Inversion via Bitplane Semantics: Exact Trailing Updates with Boolean GEMM, GF(2) Gauss–Jordan, Bareiss, and Modular CRT

Key Points

This method executes gaussian elimination without any scalar multiplications, increasing efficiency.
The approach integrates boolean gemm with three pivoting methods, providing versatility in matrix operations.
All updates are bit-sliced and designed to handle numerical stability across different number systems.
Executable code is provided alongside numerical validation for the algorithms, ensuring practical application.

Abstract

We extend our multiplication-free bit-sliced paradigm from matrix multiplication and determinants to Gaussian elimination and matrix inversion. All trailing updates are bilinear and can be executed by a Boolean (bit-sliced) GEMM with bitwise AND, population count, shifts and additions, yielding zero scalar multiplications at the matrix level. We integrate the bit-sliced core with three exact pivot/inversion regimes: (i) fully Boolean Gauss-Jordan over F2; (ii) fraction-free Bareiss over Z; (iii) modular LU over primes with CRT reconstruction. We provide executable code and small numerical checks; all GEMM-shaped updates are multiplication-free.

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Cite This Study

Michael Rey (Tue,) studied this question.

synapsesocial.com/papers/68d463db31b076d99fa62b25 https://doi.org/https://doi.org/10.20944/preprints202509.1122.v1

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