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

1. Introduction

Elimination and inversion reduce to panel factorizations and trailing matrix–matrix updates [1,2]. Our bit-sliced (bitplane) GEMM reconstructs integer products from Boolean plane products using AND+POPCNT and power-of-two shifts, thus removing all scalar multiplications from matrix products. This paper applies the same mechanism to Schur-complement updates in elimination/inversion and combines it with exact pivot regimes: Gauss–Jordan over ${\mathbb{F}}_2$, Bareiss over $\mathbb{Z}$, and modular LU + CRT. We emphasize exactness and reproducibility with simple Python reference code.

Contributions.

(1) Bit-sliced Boolean trailing updates inside blocked LU (zero scalar multiplications for all GEMMs); (2) a pure-Boolean Gauss–Jordan over ${\mathbb{F}}_2$; (3) fraction-free Bareiss with bit-sliced numerators; (4) modular LU + CRT path with bit-sliced updates modulo primes; (5) executable code and verified small-case numerics.

2. Related Work

Foundational accuracy and stability are treated by Higham and by Golub–Van Loan [1,2]. Bareiss introduced fraction-free elimination [3]. CRT and residue-number reconstruction for exact linear algebra are classical [4,5]. Bit-sliced/bit-serial multiplication appears in hardware/software works such as BISMO [6] and in popcount-oriented references [7,8].

3. Method

3.1. Bit-Sliced Boolean GEMM (Recap)

Let $A={\sum }_b{A}^{\left(b\right)}{2}^b$, $B={\sum }_b{B}^{\left(b\right)}{2}^b$, with $A^{\left(b\right)},B^{\left(b\right)}\in \{0,1\}$. With row-packing for $A^{\left(b\right)}$ and column-packing for $B^{\left(b\right)}$ along the shared inner dimension,

$$AB=\sum _{b_1,b_2}{2}^{b_1+b_2}\mathrm{popcount}\left(A_{\mathrm{rows}}^{\left(b_1\right)}\wedge B_{\mathrm{cols}}^{\left(b_2\right)}\right),$$

(1)

using only Boolean operations and shifts. Exactness holds in integer/modular settings provided accumulator widths (or CRT) are sufficient.

3.2. Where GEMM Appears in Elimination

In a blocked LU, after forming a panel with pivots, the trailing update is

$$A_{22}\leftarrow A_{22}-L_{21}{U}_{12},$$

(2)

a GEMM addressed by the bit-sliced core. Pivoting (row swaps) is compatible; we keep all panel solves and scalar inverses outside the matrix-product core.

3.3. Exact Regimes

GF(2) Gauss–Jordan. Addition is XOR and $1^{-1}=1$, so row scaling vanishes; the algorithm is purely Boolean.

Bareiss (fraction-free). Divisions are exact by construction; numerators are bilinear (bit-sliced); exactness follows from Bareiss divisibility [3].

Modular LU + CRT. Perform LU modulo several primes; panel inverses are modular scalars; updates are bit-sliced modulo p; reconstruct over $\mathbb{Z}$ via CRT once the product of primes exceeds a bound [4,5].

4. Results: Multiplication Counts

We count matrix–matrix multiplications (GEMMs) only. Trailing updates in a blocked LU are 1 GEMM per block traditionally and 0 in the bit-sliced model (Boolean GEMM). Gauss–Jordan over ${\mathbb{F}}_2$ uses row ops/XOR only (0 GEMMs).

Task	Traditional GEMMs	Bit-sliced GEMMs
Trailing update $A_{22}\leftarrow A_{22}-L_{21}{U}_{12}$	1 per block	0
Blocked LU (integer/mod p)	many	0 (all trailing)
Gauss–Jordan over ${\mathbb{F}}_2$	0	0

5. Numerical Illustrations

Executable tests: (i) bit-sliced GEMM equals NumPy integer GEMM; (ii) GF(2) inverse on an invertible example; (iii) blocked LU driver that calls Boolean GEMM for each trailing update and reports GEMM counts; (iv) modular CRT inverse sanity check; and (v) Bareiss $2\times 2$ step with exact divisibility.

6. Discussion and Limitations

The bit-sliced model eliminates scalar multiplications from all matrix products within elimination/inversion. Exactness over $\mathbb{Z}$ follows by Bareiss or modular LU + CRT. Practical speed depends on POPCNT throughput and bandwidth. Large problems need careful blocking, packing, and prime/bound selection for CRT.

Appendix A.1. Bit-Sliced Boolean GEMM, GF(2) Inverse, Blocked LU Driver, CRT Inverse, Bareiss, Tests