4 research outputs found
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For Which Error Criteria Can We Solve Nonlinear Equations?
For which error criteria can we solve a nonlinear scalar equation f(x) = 0 where f is a real function on the interval [a,b]
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Study of Linear Information for Classes of Polynomial Equations
Linear adaptive information for approximating a zero of f is studied where f belongs to the class of polynomials of unbounded degree. A theorem on constrained approximation of smooth functions by polynomials is established
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Can We Approximate Zeros of Functions with Non-zero Topological Degree?
The bisection method provides an affirmative answer for scalar functions. We show that the answer is negative for bivariate functions. This means, in particular, that an arbitrary continuation method cannot approximate a zero of every smooth bivariate function with non-zero topological degree
For which error criteria can we solve nonlinear equations?
AbstractFor which error criteria can we solve a nonlinear scalar equation f (x) = 0, where f is a real function on the interval [a, b]? The information on f consists of adaptive evaluations of arbitrary linear functionals and an algorithm is any mapping based on these evaluations. The error of an algorithm is defined by its worst performance. For the root criterion we prove there does not exist an algorithm to find a point x such that ∥x−α∥≤ϵ, where α is a zero of f and ϵ < (b−a)2. This holds for rbitrary information and for the class of infinitely many times differentiable functions with all simple zeros. We do not assume that f(a)f(b) ≤ 0. For the residual criterion we exhibit almost optimal information and algorithm. More precisely, we prove that if x is the value computed by our algorithm using n function values then f(x) = O(n−r), where r measures the smoothness of the class of functions f. Finally a general error criterion is introduced and some of our results are generalized