Reducing Internal State in Eigenvalue-Only Divide-and-Conquer Tridiagonal Eigensolvers
For numerical linear algebra practitioners needing fast eigenvalue computation on modern hardware, this work provides a more memory-efficient and faster D&C variant.
The paper proposes a boundary-row divide-and-conquer algorithm for eigenvalue-only computation that reduces memory from quadratic to linear space and eliminates unnecessary work, achieving up to 2x speedup over standard D&C and QR methods on CPUs and GPUs.
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.