Reducing Internal State in Eigenvalue-Only Divide-and-Conquer Tridiagonal Eigensolvers

2026-05-26

AI Summary

This work addresses the high memory overhead and redundant computations inherent in traditional divide-and-conquer methods for eigenvalue problems, which still require storing or replaying the full transformation matrix even when only eigenvalues are needed. The authors propose a boundary-row divide-and-conquer algorithm that propagates only the necessary boundary rows and columns during recursion, thereby avoiding the construction of the full eigenvector matrix. This approach achieves, for the first time in an eigenvalue-only divide-and-conquer solver, a reduction in memory complexity from O(n²) to O(n) and eliminates redundant matrix-vector operations associated with lazy backtransformation. Combined with recursive propagation optimizations and efficient CPU/GPU implementations, the method significantly reduces memory consumption and improves computational efficiency while preserving numerical accuracy, outperforming both QR and conventional divide-and-conquer routines in standard libraries.

Abstract

Divide and Conquer (D&C) is a widely used algorithmic strategy for symmetric eigenvalue decomposition. Its natural parallelism makes D&C attractive on modern multicore CPUs and GPUs, but existing eigenvalue-only routines often default to QR-based methods because conventional D&C still materializes or replays large transformation matrices during the conquer phase. This paper proposes a boundary-row D&C algorithm for eigenvalue-only computation. The key observation is that the conquer phase only needs selected boundary rows/columns rather than the full accumulated eigenvector matrix. By propagating these boundary rows directly through the recursion, the proposed algorithm reduces the memory requirement from quadratic to linear space while also eliminating unnecessary matrix-vector work in the conventional lazy-replay formulation. We provide the algorithm, its time and space complexity analysis, correctness and stability arguments, optimized CPU and GPU implementations, and an evaluation against QR and D&C routines in standard numerical libraries.

Problem

• Divide-and-Conquer
• eigenvalue-only
• tridiagonal eigensolver
• memory reduction
• boundary rows

Innovation

• Divide-and-Conquer
• eigenvalue-only
• boundary-row propagation
• linear space complexity
• tridiagonal eigensolver