A fast iterative algorithm for near-diagonal eigenvalue problems
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We introduce a novel iterative eigenvalue algorithm for near-diagonal matrices termed iterative perturbative theory (IPT). Built upon a "perturbative" partitioning of the matrix into diagonal and off-diagonal parts, IPT has the complexity of one matrix-vector (when one eigenpair is requested) or one matrix-matrix multiplication (when all eigenpairs are requested) per iteration. Thanks to the high parallelism of these basic linear algebra operations, we obtain excellent performance on multi-core processors and GPUs, with large speed-ups over standard methods (up to ∼50x with respect to LAPACK and ARPACK, ∼5x with respect to Davidson). When off-diagonal elements are comparable to eigengaps, IPT diverges like the quadratic map outside the Mandelbrot set; for such general matrices IPT can nevertheless be used to refine low-precision eigenvectors obtained by other methods. We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices.
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