By de Boor C. (ed.)

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Show that in the CG method the ith residual is A-orthogonal to any j th residual if | j − i| > 1. 8. Explain why, after the restart of the minimal residual method, one typically observes the increase of the residual. 9. Is it possible that the residual in the QMR method is increasing? main 2014/6/13 page 36 36 Chapter 1. 1 Convergence The (generalized) minimal residual method is the Krylov subspace method and gives the exact solution of Ax = b in a finite number of steps. For this reason, it is the direct method for solving a system of algebraic equations.

45) Thanks to the Lipschitz condition, it holds that f (xk ) − Jk ≤ c max xk − ξ j ∞ 1≤ j ≤n ∞ ≤ c xk − z ∞. 45) yield the inequality (a). If z − xk ∞ ≤ , then z − xk+1 ∞ ≤ (γ ) ≤ . Since the estimate (a) is valid for every k, we immediately get estimate (b). 39) for finding zero of f (x), one tries to exploit the information from the m steps preceding the current step (k → k + 1). For the fixed point method, m = 0. Setting m > 0 may help to find a better iterate x k+1 . This can be done by a variety of (vector) extrapolation methods, which produce a better convergent sequence of vectors given x k , .

For the analysis, it is often convenient to reformulate optimization problems in a way such that a minimum is sought over a vector space rather than over an affine space. For one-dimensional optimization, the equivalent form is y = argmin { f˜(x) : x ∈ span {r i }}, r i = b − Ax i , x i +1 = x i + y, where f˜(x) is the error or residual functional for the system Ax = r i . This simple observation will be exploited also for minimization over spaces of higher dimensions. 1 Minimization and projection Many iterative algorithms for finding a minimum of a functional f solve a local minimization problem on every iteration.