PRECONDITIONING VIA GMRES IN POLYNOMIAL SPACE ∗

We propose a class of polynomial preconditioners for solving non-Hermitian linear 3 systems obtained from a least-squares approximation in polynomial space instead of a standard 4 Krylov subspace. The process for building the polynomial relies on an Arnoldi-like procedure in a 5 small dimensional polynomial space and is equivalent to performing GMRES in polynomial space. 6 It is inexpensive and produces results with superior numerical stability. A few improvements to 7 the basic scheme are discussed including the development of a short-term recurrence and the use of 8 compounded preconditioners. Numerical experiments, including a test with challenging 3D Helmholtz 9 equations and a few publicly available sparse matrices, are provided to demonstrate the performance 10 of the proposed preconditioners. 11

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