[Paper Review] Recursive Method for the Solution of Systems of Linear Equations

Gennadi Malaschonok|arXiv (Cornell University)|Mar 29, 2017

Algebraic and Geometric Analysis2 references3 citations

TL;DR

This paper proposes a recursive algorithm for solving systems of linear equations over commutative integral domains with computational complexity matching that of matrix multiplication, specifically $O(n^{ ext{exponent of matrix multiplication}})$. The method uses recursive partitioning and determinant-based reductions to transform the coefficient matrix into diagonal form, enabling efficient solution via Cramer’s rule with minimal arithmetic operations when corner minors are non-zero and non-divisors.

ABSTRACT

New solution method for the systems of linear equations in commutative integral domains is proposed. Its complexity is the same that the complexity of the matrix multiplication.

Motivation & Objective

• To develop a solution method for systems of linear equations over commutative integral domains that matches the complexity of matrix multiplication.
• To generalize existing methods like forward/backward substitution and one-pass algorithms by unifying them under a recursive framework.
• To reduce the computational complexity of solving linear systems from $O(n^3)$ to the same asymptotic order as matrix multiplication, leveraging recursive decomposition and determinant arithmetic.
• To ensure applicability in any commutative ring by requiring only that corner minors are non-zero and not zero divisors.

Proposed method

• The algorithm recursively partitions the coefficient matrix into upper and lower submatrices, applying recursive reduction to each part.
• It uses recursive determinant computation via minors $\delta^{k} = \det \mathbf{A}_{kk}^{k}$ and cofactor matrices $\delta^{k}_{ij}$ to maintain structural invariance during transformation.
• A key identity $\delta^{k} \cdot \hat{A}_{2}^{2} = \delta^{s} \cdot A_{2}^{2} - A_{1}^{2} \cdot G_{2}^{1}$ enables transformation of the lower block while preserving determinant structure.
• The method reduces the matrix to diagonal form $\tilde{A} \rightarrow (\delta^{l}I_{l-k}, \hat{G})$, where $\delta^{l}$ is the leading minor and $\hat{G}$ contains generalized cofactors.
• Recursive steps are applied until the entire matrix is reduced, with the final solution derived from Cramer’s rule using the computed minors and cofactor matrices.
• The algorithm supports both forward/backward substitution (when partitions are single-row) and dichotomous partitioning (for optimal complexity matching matrix multiplication).

Experimental results

Research questions

• RQ1 Can a recursive algorithm achieve the same asymptotic complexity as matrix multiplication for solving linear systems over integral domains?
• RQ2 How can the solution of linear systems be unified under a single recursive framework that generalizes one-pass and forward/backward methods?
• RQ3 What recursive decomposition strategy ensures that the complexity of solving linear systems matches that of matrix multiplication?
• RQ4 Under what conditions on the ring and matrix structure does the algorithm remain valid and efficient?

Key findings

• The algorithm achieves a computational complexity matching that of matrix multiplication, specifically $O(n^{\log_2 7})$ using Strassen’s algorithm, significantly improving over the classical $O(n^3)$ complexity.
• For $n = 2^p$ and $m = n+1$, the number of multiplication/division operations is $MD^{S}_{n,n+1} = \frac{7}{15}n^{\log_2 7} + n^2(\log_2 n - \frac{2}{3}) + n(2\log_2 n + \frac{1}{5})$, demonstrating near-optimal scaling.
• The method generalizes both the one-pass method and forward/backward substitution: when partitions are single-row, it reduces to these known techniques.
• The solution is derived via Cramer’s rule using computed minors $\delta^n$ and cofactor matrices $G$, with the final system represented as $\delta^n I_n$ and $\hat{G}$.
• The algorithm is valid in any commutative integral domain as long as all corner minors $\delta^k$ are non-zero and not zero divisors.
• A numerical example over $\mathbb{Z}$ confirms the correctness: for a $4 \times 4$ system, the algorithm reduces the matrix to diagonal form with $\delta^4 = 27$, yielding the solution vector $x = (1, 2, -2, -1)^T$.