Parallel Structured Gaussian Elimination (PSGE)

• PSGE is a block-based, panel-partitioned variant of Gaussian elimination designed to efficiently solve large sparse linear systems.
• It employs structured preconditioning to ensure numerical stability without pivoting, reducing fill-in and harnessing GPU architectures.
• Benchmark results show 4–6× speedup over classical methods, making PSGE ideal for high-throughput symbolic and algebraic computations.

Parallel Structured Gaussian Elimination (PSGE) is a block-based, panel-partitioned variant of Gaussian elimination designed for highly parallel computation, particularly in the context of solving large sparse linear systems arising from multivariate polynomial reductions and symbolic computation tasks. PSGE eliminates the need for data-dependent pivoting, leverages structured preprocessing for safety and numerical stability, and exploits memory architecture features to minimize fill-in and optimize throughput. The technique is an overview of algebraic block elimination, randomized or structured preconditioning, and GPU-aware kernel engineering, providing both theoretical guarantees and practical speedups (Gokavarapu, 11 Jan 2026, Pan et al., 2015).

1. Matrix Structure, Block Partitioning, and Parallel Panel Elimination

PSGE operates on a sparse Macaulay matrix $A\in\mathbb{F}_p^{r\times N}$ constructed from a sorted monomial dictionary $\mathcal{T}$ and a set of shifted polynomial reducers $\mathcal{R}$, as commonly encountered in Gröbner basis algorithms. Columns are divided contiguously into panels $P_1,\ldots,P_s$, each of width $b_\ell$, so that panel $P_\ell$ comprises columns indexed from $c_{\min} = \sum_{k<\ell} b_k + 1$ to $c_{\max} = \sum_{k\leq \ell} b_k$. The elimination procedure within PSGE is structured as follows:

1. Pivot selection: Rows with a leading nonzero in $P_\ell$ are flagged in parallel using CSR‐style data structures, and sorted by leading column index.
2. Dense block elimination: The submatrix formed by pivot rows and panel columns is extracted into a local tile in shared memory/registers, enabling dense Gaussian elimination via small-block GEMM.
3. Sparse trailing update: Each non-pivot row is updated in parallel, with fill-in constrained strictly to columns beyond the current panel.

This block-centric scheduling aligns naturally with GPU SIMT architectures and minimizes inter-thread communication (Gokavarapu, 11 Jan 2026).

2. Structured Preconditioning: Safety and Numerical Stability Without Pivoting

Traditional GEPP (Gaussian elimination with partial pivoting) incurs high communication and unpredictable data transfers in parallel regimes. PSGE, by preconditioning the input matrix $A$ through multiplication with a random or structured nonsingular, well-conditioned multiplier $H$ (e.g., full Gaussian, circulant, or bidiagonal-derived), realizes two key properties:

• Safety: With probability one, all principal minors of $AH$ (or $HA$) are nonzero, preventing division-by-zero, provided $H$ is sampled from an adequate ensemble [(Pan et al., 2015), Theorem 4.1, 4.3].
• Numerical Safety: Strong well-conditioning of all leading principal blocks is achieved with overwhelming probability, bounding entry growth and controlling round-off propagation [(Pan et al., 2015), Theorem 4.2, 4.4].

Structured preconditioners (circulant, Toeplitz-type, Givens-chain+DFT, Householder+DFT, sparse $f$-circulant), as well as augmentation constructs (Sherman–Morrison–Woodbury formula) further broaden the design space for PSGE [(Pan et al., 2015), Section 2.1–2.2, 3.5].

3. Data Layout, Memory Coalescence, and Kernel Engineering

To achieve peak throughput and minimize memory latency, PSGE employs several layout and access strategies:

• Structure-of-Arrays (SoA) Polynomial Storage: Terms and coefficients are stored in separate arrays (mon_key[], coeff[]), with offsets and lengths for each row segment; this enables warp-level coalesced access for both monomial keys and coefficients in 128-byte bursts.
• Sorted Monomial Dictionary: dict_keys[] maintains a flat, sorted array of key values. Parallel merge-path joins are used for row assembly and column indexing, optimizing writes.
• SELL-C-σ Bucketing: Rows with similar length and signature are grouped into buckets of size $C$ and padded to height $\sigma$, regularizing thread divergence in kernels and assembly.
• Panel Tiling: Panels are sized so the corresponding dense tile fits in shared memory or registers, ensuring optimal block GEMM utilization [(Gokavarapu, 11 Jan 2026), Section 3].

4. Fill-in Bound, Computational Complexity, and Parallel Scaling

Unlike classical sparse GE, which can incur unbounded fill-in up to $O(rN)$, PSGE confines fill-in to block panels. For each panel $\ell$, the fill created during elimination satisfies

$$\mathrm{fill}_\ell \le b_\ell (r - |\mathrm{pivot\_rows}_\ell| ) - M_{\ell,\mathrm{init}}$$

where $M_{\ell,\mathrm{init}}$ is the pre-elimination nonzero count in panel $\ell$ (Gokavarapu, 11 Jan 2026). Empirical fill ratios $\kappa = (\mathrm{nnz}(L)+\mathrm{nnz}(U))/M_0$ are consistently observed in the range $1.1$–$1.4$ (as opposed to $>10$ for classical GE). Complexity for PSGE (panel width $b$) is $O(M_0b + rb^2)$ work and $O(s\log b + \mathrm{depth}(\mathrm{GEMM}(b)))$ span—far superior to classical methods for $b \ll r$, and readily exploitable on multi-core and GPU architectures [(Gokavarapu, 11 Jan 2026), Table 1].

Method	GFLOPS	nnz(A) Fill Ratio	Speedup × Classical GE
Classical GE	150	7.8	1.0
Block Wiedemann	520	1.0	2.7
PSGE (b=64)	1020	1.2	5.3

5. Register-Resident Finite Field Arithmetic and Kernel Fusion

All arithmetic in PSGE uses register-local finite field update kernels, ensuring high occupancy and minimal instruction latency:

• Barrett reduction: Employs a precomputed $\mu$ and a single 64-bit high-multiply followed by predicated subtraction, yielding $a \gets a + bc \bmod p$.
• Montgomery multiplication: Uses $R = 2^w$, $p'$ as the modular inverse, mask, shift, and subtraction for the modular update in place [(Gokavarapu, 11 Jan 2026), Section 6].

Explicit pseudo-PTX kernels demonstrate fusion of FMA updates and reductions into a single pipeline, with large unroll factors amortizing overhead.

6. Symbolic Preprocessing Integration and Pipeline Structure

PSGE is directly integrated into a two-phase pipeline with symbolic preprocessing (FBSP):

• FBSP Output: $$(\mathrm{dict\_keys}, \mathrm{row\_ptr}, \mathrm{col\_ind}, \mathrm{val}, \mathrm{row\_meta})$$
• PSGE Consumption: Dictates panel boundaries, matrix storage (CSR/SELL), and row signature information.
• Pipeline Dependencies: Panel boundaries calculated from dict_keys; admissible rows labeled with signature filtering upfront.

This strictly separates combinatorial structure compilation from numeric computation, enabling deterministic scheduling under the SIMT model and guaranteeing correctness and reproducibility (Gokavarapu, 11 Jan 2026).

7. Experimental Performance and Practical Outcomes

Benchmarks on NVIDIA A100 (64-bit Montgomery kernels) confirm the effectiveness of PSGE:

• On Cyclic-6 ($p=2^{31}-1$): PSGE (b=64) achieves $1.7\times 10^8$ row-reductions/sec (vs $3.5\times 10^7$ for classical GE), with nnz$(L)+$nnz$(U)\approx 1.2$×nnz$(A)$.
• Block Wiedemann achieves lower but still significant speedups (2.7×), with negligible fill.
• On Katsura-10 and other test sets, panel-based PSGE consistently delivers 4–6× speedup over classical GE.

A plausible implication is that the decoupling of symbolic and numeric phases via PSGE opens new avenues for high-throughput algebraic reduction pipelines on modern GPU hardware, without sacrificing fill control or numerical stability (Gokavarapu, 11 Jan 2026, Pan et al., 2015).