Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra
نویسندگان
چکیده
Let g: E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ F, the image under g of the closed ball about x ∈ E with radius ε. When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x, ε)) are ε-pseudosolutions. Our aim is to investigate the size of the set g(B(x, ε)). We derive upper and lower bounds of the following form: g(x)+ Dg(x)(B(0, c1ε)) ⊆ g(B(x, ε)) ⊆ g(x)+ Dg(x)(B(0, c2ε)), where Dg(x) denotes the derivative of g at x . We consider both the case where g is given explicitly and the case where g is given implicitly. We apply our results to the Date received: November 28, 2001. Final version received: July 1, 2002. Communicated by Arieh Iserles. Online publication: November 22, 2002. AMS classification: 65F15, 65H10, 65Y20. 2 J.-P. Dedieu, M.-H. Kim, M. Shub, and F. Tisseur implicit function associated with the evaluation map, namely the solution map, and to the polynomial eigenvalue problem. Our results are stated in terms of an invariant γ which has been extensively used by various authors in the study of Newton’s method. The main tool used here is an implicit γ theorem, which estimates the γ of an implicit function in terms of the γ of the function defining it.
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ورودعنوان ژورنال:
- Foundations of Computational Mathematics
دوره 3 شماره
صفحات -
تاریخ انتشار 2003