Iteration are the stationary values of the same

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Unformatted text preview: es The eigenvalues of A are the stationary values for x ∈ IRm of the Rayleigh quotient r (x) = xx Ax x The eigenvalue estimates (Ritz values) associated with step n of the Lanczos iteration are the stationary values of the same function r (x) if x is restricted to the Krylov subspace Kn Perfect parallel of what we have shown that the solution x∗ of Ax = b is the minimal point in IRm of the scalar function φ(x), and the CG iterate xn is the minimal point of the same function φ(x) if x is restricted to Kn 10 / 25 Conjugate gradients and polynomial approximation Connection between Krylov subspace iteration and polynomials of matrices The Arnoldi and Lanczos iterations solve the Arnoldi/Lanczos approximation problem Find p n ∈ P n such that p n (A)b = minimum The GMRES iteration solves the GMRES approximation problem Find pn ∈ Pn such that pn (A)b = minimum For CG, the appropriate approximation problem involves the A-norm of the error Find pn ∈ Pn such that pn (A)e0 A = minimum where e0 denotes the initial error e0 = x∗ − x0 = x∗ ,...
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