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 Sequentially localizable functionals.
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Sequentially localizable functionals. Booth, Raymond Sydney
Abstract
A standard way of finding the unique zero on (0, 1) of a continuous decreasing functions with f(0) f(1)< 0, is to test the sign of f(1/2), then the sign of f(1/4) (if f(1/2)< 0) or the sign of f(3/4) (if f(1/2) >0), etc. In this way, the zero of f is localized in n steps to an interval of length 2⁻ⁿ. The unique maximum of a unimodal function on [0,1] can be similarly localized, but the unique maximum of a unimodal function on the unit square cannot. We start by generalizing these problems: let A be a compact subset of Eⁿ, let F be a set of realvalued functions on A, and for each f in F let S(f) be a point in the Cartesian product Aᵏ; S(f) is called a functional on F. Examples of such functionals are zeros, extrema, inflexion points, saddle points, etc., as well as sets of these. A test function T is a function of m real variables, which takes up only a finite number (≥2) of distinct values. An abscissa set Xᵢ is an ordered mtuple (xlᵢ,...xmᵢ) ,, with each Xji in A. A sequential strategy is a way of selecting abscissa sets X₁, X₂,...,where the knowledge of T(f (xlᵢ, . ., f (xmᵢ)) is used to determine Xi+i. The Nth set of indeterminacy for S(f) is the largest subset of in which S(f) can lies consistent with the results of the first N1 tests. A functional S(f) is sequentially localizable if a test function T and a sequential strategy exist, such that for every f in F the sets of indeterminacy shrink to a point (which must then be S(f) itself). First, several conditions are given to ensure the sequential localizability of a functional, these are presented in terms of certain topologies induced on Aᵏ and in terms of contraction maps. It is then shown that if a functional is localizable, there exists an optimal strategy under which the sets of indeterminacy converge fastest; further, the speed of localization is always exponential. Next, the concept of a random strategy and of random localizability is introduced, and it is shown that in many cases random localizability and sequential localizability are equivalent. Also, the speed of the former is not too much worse than the speed of the latter. Finally, optimal and nearoptimal strategies are worked out for some functionals of interest.
Item Metadata
Title 
Sequentially localizable functionals.

Creator  
Publisher 
University of British Columbia

Date Issued 
1965

Description 
A standard way of finding the unique zero on (0, 1) of a continuous decreasing functions with f(0) f(1)< 0, is to test the sign of f(1/2), then the sign of f(1/4) (if f(1/2)< 0) or the sign of f(3/4) (if f(1/2) >0), etc. In this way, the zero of f is localized in n steps to an interval of length 2⁻ⁿ. The unique maximum of a unimodal function on [0,1] can be similarly localized, but the unique maximum of a unimodal function on the unit square cannot. We start by generalizing these problems: let A be a compact subset of Eⁿ, let F be a set of realvalued functions on A, and for each f in F let S(f) be a point in the Cartesian product Aᵏ; S(f) is called a functional on F. Examples of such functionals are zeros, extrema, inflexion points, saddle points, etc., as well as sets of these. A test function T is a function of m real variables, which takes up only a finite number (≥2) of distinct values. An abscissa set Xᵢ is an ordered mtuple (xlᵢ,...xmᵢ) ,, with each Xji in A. A sequential strategy is a way of selecting abscissa sets X₁, X₂,...,where the knowledge of T(f (xlᵢ, . ., f (xmᵢ)) is used to determine Xi+i. The Nth set of indeterminacy for S(f) is the largest subset of in which S(f) can lies consistent with the results of the first N1 tests. A functional S(f) is sequentially localizable if a test function T and a sequential strategy exist, such that for every f in F the sets of indeterminacy shrink to a point (which must then be S(f) itself).
First, several conditions are given to ensure the sequential localizability of a functional, these are presented in terms of certain topologies induced on Aᵏ and in terms of contraction maps. It is then shown that if a functional is localizable, there exists an optimal strategy under which the sets of indeterminacy converge fastest; further, the speed of localization is always exponential. Next, the concept of a random strategy and of random localizability is introduced, and it is shown that in many cases random localizability and sequential localizability are equivalent. Also, the speed of the former is not too much worse than the speed of the latter. Finally, optimal and nearoptimal strategies are worked out for some functionals of interest.

Genre  
Type  
Language 
eng

Date Available 
20110815

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080573

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.