Randomized preprocessing versus pivoting

Gaussian Elimination with Randomized Complete Pivoting

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 ...

Randomized Preprocessing of Homogeneous Linear Systems

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations. In the case of Toeplitz inputs, we decrease the solution time from quadratic to nearly linear, and our tests show dramatic decrease of the CPU time as well. We prove numerical stability of our randomized algorithms and extend our approach to solving nonsingular line...

Stability of the Diagonal Pivoting Method with Partial Pivoting

LAPACK and LINPACK both solve symmetric indefinite linear systems using the diagonal pivoting method with the partial pivoting strategy of Bunch and Kaufman [Math. Comp., 31 (1977), pp. 163–179]. No proof of the stability of this method has appeared in the literature. It is tempting to argue that the diagonal pivoting method is stable for a given pivoting strategy if the growth factor is small....

Randomized preprocessing of configuration space for path planning: articulated robots

Randomized Preprocessing of Homogeneous Linear Systems of Equations

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations, which leads to significant acceleration of the known algorithms in the cases of both general and structured input matrices. E.g., in the case of Toeplitz inputs, we decrease the estimated solution time from quadratic to nearly linear, and our tests show dramatic dec...