We are given a linear system \( A\mathbf{x} = \mathbf{y} \), where \( A \) is an \( n \times n \) matrix, \( \mathbf{x} \) is the solution vector, and \( \mathbf{y} \) is the right-hand side vector. We are also given perturbations \( \Delta \mathbf{x} \) and \(
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