Let m be a polynomial satisfying some mild conditions. Given a set R ⊂ C, a continuous function f on R and its best approximation p*n- 1 from IIn with respect to the maximum norm, we show that p*n p*n is a best approximation to f o m on the inverse polynomial image 5 of R, i.e. m(S) = R, where the extremal signature is given explicitly. A similar result is presented for constrained Chebyshev polynomial approximation. Finally, we apply the obtained results to the computation of the convergence rate of Krylov subspace methods when applied to a preconditioned linear system. We investigate pairs of preconditioners where the eigenvalues are contained in sets 5 and R, respectively, which are related by m(S) = R.
|Journal||Electronic Transactions on Numerical Analysis|
|Number of pages||11|
|Publication status||Published - 01.12.2001|