An O(n)-randomness perturbation combining a dense deterministic pattern matrix with a non-uniform sparse dependent perturbation reduces condition numbers to O(n) for any input matrix.
R., Stiefel, E.,et al.Methods of conjugate gradients for solving linear systems
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Well-Conditioned Oblivious Perturbations in Linear Space
An O(n)-randomness perturbation combining a dense deterministic pattern matrix with a non-uniform sparse dependent perturbation reduces condition numbers to O(n) for any input matrix.