Growth factors of orthogonal matrices and local behavior of Gaussian elimination with partial and complete pivoting
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Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided last year by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors stay at most polynomial with very high probability when using small Gaussian perturbations. GE with complete pivoting (GECP) has also seen a lot of recent interest, with recent improvements to lower bounds on worst-case GECP growth provided by Edelman and Urschel earlier this year. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. We will also study systems when GECP leads to larger growth than GEPP, which will lead to new empirical lower bounds on how much worse GECP can behave compared to GEPP in terms of growth. We also present an empirical study on a family of exponential GEPP growth matrices whose polynomial behavior in small neighborhoods limits to the initial GECP growth factor.
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