The polar decomposition A = UpH finds many uses in applications, and it is a fundamental tool for computing the symmetric eigenvalue decomposition and the singular value decomposition via a spectral divide-and-conquer process. Conventional algorithms for these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with a...

An analytic parahermitian matrix admits in almost all cases an eigenvalue decomposition (EVD) with eigenvalues and eigenvectors. We have previously defined a discrete Fourier transform (DFT) domain algorithm which has been proven to extract the eigenvalues. The selection of as functions guarantees turn existence unique one-dimensional eigenspaces eigenvectors can exist. Determining such is not ...

We present a fast condition estimation algorithm for the eigenvalues of a class of structured matrices. These matrices are low rank modifications to Hermitian, skew-Hermitian, and unitary matrices. Fast structured operations for these matrices are presented, including Schur decomposition, eigenvalue block swapping, matrix equation solving, compact structure reconstruction, etc. Compact semisepa...

In the present study, we demonstrate how to perform, using quantum annealing, the singular value decomposition and the principal component analysis. Quantum annealing gives a way to find a ground state of a system, while the singular value decomposition requires the maximum eigenstate. The key idea is to transform the sign of the final Hamiltonian, and the maximum eigenstate is obtained by quan...