Geometrical inverse preconditioning for symmetric positive definite matrices. (English) Zbl 1360.65093
Summary: We focus on inverse preconditioners based on minimizing \(F(X)=1-\cos(XA,I)\), where \(XA\) is the preconditioned matrix and \(A\) is symmetric and positive definite. We present and analyze gradient-type methods to minimize \(F(X)\) on a suitable compact set. For this, we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of \(F(X)\) on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.
MSC:
| 65F08 | Preconditioners for iterative methods |
Keywords:
preconditioning; cones of matrices; gradient method; minimal residual method; inverse preconditioners; symmetric and positive definite matrices; numerical resultsReferences:
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