A polynomial approximation to the inverse of a matrix.

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We develop a surprisingly accurate polynomial approximation to the inverse of a large matrix, i.e. p(A) ⇡ A1. This polynomial is implemented using the roots of the GMRES polynomial, or harmonic Ritz values, and can be used to solve multiple right-hand sides as well as trace estimates to the inverse of a matrix. Issues with stability that arise with applying a high-degree polynomial are discussed. In addition, we give bounds on the error of our approximate inverse and illustrate how accurate the approximation can be in practice.

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