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Gaussian Elimination with Randomized Complete Pivoting
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Gaussian elimination with partial pivoting (GEPP) has long been among the most widely used methods for computing the LU factorization of a given matrix. However, this method is also known to fail for matrices that induce large element growth during the factorization process. In this paper, we propose a new scheme, Gaussian elimination with randomized complete pivoting (GERCP) for the efficient and reliable LU factorization of a given matrix. GERCP satisfies GECP (Gaussian elimination with complete pivoting) style element growth bounds with high probability, yet costs only marginally higher than GEPP. Our numerical experimental results strongly suggest that GERCP is as reliable as GECP and as efficient as GEPP for computing the LU factorization.
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Convergence rates for pivoted QR and LU
Under approximate greedy pivoting, pivoted QR/LU residuals are bounded by the geometric mean of leading singular values, yielding algebraic and geometric convergence rates for matrices and bivariate functions.
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