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 \sim50x with respect to LAPACK and ARPACK ,%DIFDELCMD < \sim5x %%% 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  .
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 computes one or all eigenpairs with a complexity per iteration of one matrix-vector or one matrix-matrix multiplication respectively . 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 \sim50x with respect to LAPACK and ARPACK %DIFDELCMD < \sim5x %%% ). For matrices which are not close to being diagonal but have well-separated eigenvalues, IPT can be be used to refine low-precision eigenpairs obtained by other methods. We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices . In a real-world application from quantum chemistry, we find that IPT performs similarly to the Davidson algorithm .
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