Yuji Nakatsukasa

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 for all the eigenvalues and eigenvectors 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 40 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 decompos...

Optimizing Halley's Iteration for Computing the Matrix Polar Decomposition

We introduce a dynamically weighted Halley (DWH) iteration for computing the polar decomposition of a matrix, and we prove that the new method is globally and asymptotically cubically convergent. For matrices with condition number no greater than

Vector Spaces of Linearizations for Matrix Polynomials: A Bivariate Polynomial Approach

10^{16}

A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

, the DWH method needs at most six iterations for convergence with the tolerance

Computing the common zeros of two bivariate functions via Bézout resultants

10^{-16}

Solving the Trust-Region Subproblem By a Generalized Eigenvalue Problem

. The Halley iteration can be implemented via QR decompositions without explicit matrix inversions. Therefore, it is an inverse free communication friendly algorithm for the emerging multicore and hybrid high performance computing systems.

Backward Stability of Iterations for Computing the Polar Decomposition

We revisit the landmark paper [D. S. Mackey et al. SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a Bezout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of Bezout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis.

Computing Fundamental matrix decompositions accurately via the matrix sign function in two iterations: The power of Zolotarev's functions

The unitary polar factor

On the stability of computing polynomial roots via confederate linearizations

Q=U_p

dqds with Aggressive Early Deflation

in the polar decomposition of