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Additional resources for Advances in Computers, Vol. 21
UNIVARIATE PROBLEMS 29 The radius of information is given by and as in general, it is a lower bound on the error of any algorithm where is the class of all algorithms using N. 1 The following equation holds where C is the class of all sequential information. To prove this theorem we need two lemmas. , k, and We let Li : T —>• 7£ be linearly independent linear functionals. 1 For every 6, 0 < 8 < 1, and every family of intervals Ii C [0,1], with diameter diam(/;) = 6, i = ! , . . , k are linearly independent on Ck- 30 CHAPTER 2.
U;,-(a;)|. , for r < 4 it is bounded by k • Cs, where Cy, denotes the number of arithmetic operations to find the minimizer of a cubic polynomial over a given interval. 4 General Error Criterion in C00 and Wr°° One may wish to solve a nonlinear equation by utilizing a different error criterion than the absolute or residual criteria analyzed in previous sections. 2. We assume for simplicity that a > 0. 1 we used two function with the same information whose zeros were arbitrarily close to the endpoints of [a, b].
We make this assumption to normalize our cost function. In addition we assume that comparisons and evaluations of certain elementary function also have unit cost. We define the worst case cost of a method M = ( , N), as: where the cost(N(f)) is the cost of computing the information vector y = N ( f ) , and cost( (N(/))) is the cost of combining the information y to compute M(f) =
Advances in Computers, Vol. 21