Algebraic Geometry I: Complex Projective Varieties by David Mumford

By David Mumford

From the stories: "Although a number of textbooks on glossy algebraic geometry were released meanwhile, Mumford's "Volume I" is, including its predecessor the purple booklet of sorts and schemes, now as prior to some of the most very good and profound primers of contemporary algebraic geometry. either books are only precise classics!" Zentralblatt

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They are direct attempts to apply certain results of linear algebra to operator equations. The first method we shall consider ll9, 120, 126]. It generalizes is due to Petryshyn the Gauss-Seidel and Jacobi methods of matrix theory [ll] to a certain class of Kpd operators. calls this method the generalized or the go-method for short. here is due to Kellogg the Peaceman-Rachford [81]. overrelaxation [llT, Petryshym iterative method The other method we wish to consider It is a direct attempt at generalizing method for matrix equations [35] to operator equations.

Proof. We first observe that for n ~ 0 : rn+ I = y - AXn+ I = y - A(xn+tnr n) = rn-tnArn while ro= y . Thus II~n+lll 2= Ilrn-tnArnll 2= Ilrnll 2 - 2 II~nll 2 This means that the sequence Hence the sequence converges {llrnll]is monotonically to some real number decreasing. p , 0_ p _< IlYll • This implies that 2 lim ~ = 0 . 3 (~)~ llrnf • 0 Since A has a bounded inverse, (23) 51 Because llxll~ ~ llAxll f o r all Y ing simple estimate for the e r r o r : llXn-X*ll ~ ~ x ~ H , we have the follow- llAxn-Yll • In [141] Sobolevskii gives the following definition.

But this Ikl < 1 , as required. The error estimates (ii) and (12) follow from (7) and (8), since (9) is equivalent to NlXn+ 1 = N2x n + ~y where NI=D-~S and NR=(I-w)D+~Q with N=NI'N2 =~A "l The presentation given here is due to Petryshyn [ll7]. 4 In A, D, 61 S, and Q satisfying conditions (a), (b), and (c), ~(T ) c [z c C : Izl < l] if and only if A is Kpd. In view of Lemma (5), this means that the go-method converges to the unique solution x* of (3) if and only if A tions are given by Petryshyn in [126].

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