Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

Spectral divide and conquer algorithms solve the eigenvalue problem by recursively computing an invariant subspace for a subset of the spectrum and using it to decouple the problem into two smaller subproblems. A number of such algorithms have been developed over the last forty years, often motivated by parallel computing and, most recently, with the aim of achieving minimal communication costs. However, none of the existing algorithms has been proved to be backward stable, and they all have a significantly higher arithmetic cost than the standard algorithms currently used. We present new spectral divide and conquer algorithms for the symmetric eigenvalue problem and the singular value decomposition that are backward stable, achieve lower bounds on communication costs recently derived by Ballard, Demmel, Holtz, and Schwartz, and have operation counts within a small constant factor of those for the standard algorithms. The new algorithms are built on the polar decomposition and exploit the recently developed QR-based dynamically weighted Halley algorithm of Nakatsukasa, Bai, and Gygi, which computes the polar decomposition using a cubically convergent iteration based on the building blocks of QR factorization and matrix multiplication. The algorithms have great potential for efficient, numerically stable computations on computing architectures where the cost of communication dominates the cost of arithmetic.