A quadratically convergent QR-like method without shifts for the Hermitian eigenvalue problem

We propose a new QR-like algorithm, symmetric squared QR (SSQR) method, that can be readily parallelized using commonly available parallel computational primitives such as matrix–matrix multiplication and QR decomposition. The algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts. We provide a rigorous convergence analysis of SSQR and derive structures for several of the important quantities generated by the algorithm. We also discuss various practical implementation issues such as stopping criteria and deflation techniques. We demonstrate the convergence behavior of SSQR using several numerical examples.