[Paper Review] Matrix-Free Approximate Equilibration

Andrew Bradley, Walter Murray|arXiv (Cornell University)|Oct 12, 2011

Matrix Theory and Algorithms16 references3 citations

TL;DR

This paper proposes matrix-free approximate equilibration algorithms for nonsymmetric and symmetric matrices with signed elements, using only matrix-vector products to iteratively balance row and column norms. The key contribution is that just 100–200 matrix-vector products can significantly reduce condition numbers, often achieving near-optimal equilibration without direct matrix access.

ABSTRACT

The condition number of a diagonally scaled matrix, for appropriately chosen scaling matrices, is often less than that of the original. Equilibration scales a matrix so that the scaled matrix's row and column norms are equal. Scaling can be approximate. We develop approximate equilibration algorithms for nonsymmetric and symmetric matrices having signed elements that access a matrix only by matrix-vector products.

Motivation & Objective

• To develop matrix-free algorithms that achieve approximate equilibration for matrices with signed elements, avoiding direct element access.
• To extend equilibration techniques—previously reliant on element access—to matrices where such access is costly or infeasible.
• To improve the conditioning of symmetric and nonsymmetric matrices, especially indefinite ones, using only matrix-vector products.
• To evaluate the effectiveness of these algorithms through numerical experiments on sparse matrices from the University of Florida collection.
• To demonstrate that a small number of matrix-vector products can yield substantial condition number reduction, even for large-scale problems.

Proposed method

• Proposes the snbin and ssbin algorithms for nonsymmetric and symmetric matrices, respectively, using only matrix-vector products.
• Employs iterative updates based on the ratio of row and column norms (or row norms in symmetric case) to adjust scaling vectors.
• Uses a fixed number of iterations (e.g., 128) with no adaptive stopping, relying on convergence behavior observed empirically.
• Applies a switch to snbin-like behavior in symmetric case to address reducibility and improve convergence stability.
• Measures performance via the ratio of largest to smallest row (or column) 2-norms and condition number of the scaled matrix.
• Implements the algorithms in MATLAB and tests them on 741 nonsymmetric and 466 symmetric matrices from the UF Sparse Matrix Collection.

Experimental results

Research questions

• RQ1 Can approximate equilibration be effectively achieved for signed matrices using only matrix-vector products, without direct element access?
• RQ2 How effective are matrix-free equilibration algorithms in reducing the condition number of symmetric and nonsymmetric matrices, especially indefinite ones?
• RQ3 What is the impact of the number of matrix-vector products on the quality of equilibration, and how many are sufficient for practical benefit?
• RQ4 How does the performance of the proposed algorithms compare to exact equilibration or Jacobi scaling in terms of condition number reduction?
• RQ5 To what extent does the algorithm’s output variance affect reliability, and can consistent results be achieved with a small number of matrix-vector products?

Key findings

• The snbin and ssbin algorithms achieve a ratio of largest to smallest row (or column) norm between 1.5 and 6 after 128 iterations, across matrix sizes spanning six orders of magnitude.
• The condition number of the scaled matrix is almost always substantially smaller than that of the original, with many points falling well below the diagonal line in condition number reduction plots.
• For symmetric matrices, the condition number of the scaled matrix is often reduced by a factor of 10 or more, with tight clusters of five runs indicating low variance in output.
• The algorithms converge stably and consistently, with ten individual runs showing closely clustered convergence histories, indicating robustness.
• Switching to snbin-like behavior in the symmetric case improves convergence and avoids stagnation due to reducibility.
• Approximately 100 matrix-vector products are sufficient for symmetric matrices and 200 for nonsymmetric ones to achieve effective equilibration, making the method practical for large-scale problems.