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Backward Error and Condition of Structured Linear Systems
 SIMAX
, 1992
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Choice of Norms for Data Fitting and Function Approximation
, 2000
"... This article is, however, not concerned with interpolation, and thus in the data fitting context, it will be assumed that the data can be modelled by a function containing a number of free parameters, and minimizing a norm is appropriate. Perhaps the most commonly occurring criterion in such cases i ..."
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This article is, however, not concerned with interpolation, and thus in the data fitting context, it will be assumed that the data can be modelled by a function containing a number of free parameters, and minimizing a norm is appropriate. Perhaps the most commonly occurring criterion in such cases is the least squares norm. Its use has a long and distinguished history, it is relatively well understood, and there are good algorithms available. Yet there are often situations where it is not ideal. For example, a statistical justification for least squares requires certain assumptions about the error pattern in the data, and if these are not satisfied there may be bias in the estimate
Approximation in Normed Linear Spaces
, 2000
"... A historical account is given of the development of methods for solving approximation problems set in normed linear spaces. Approximation of both real functions and real data is considered, with particular reference to L p (or l p ) and Chebyshev norms. As well as coverage of methods for the usu ..."
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A historical account is given of the development of methods for solving approximation problems set in normed linear spaces. Approximation of both real functions and real data is considered, with particular reference to L p (or l p ) and Chebyshev norms. As well as coverage of methods for the usual linear problems, an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters, and special attention is paid to some particular nonlinear approximating families. 1 Introduction The purpose of this paper is to give a historical account of the development of numerical methods for a range of problems in best approximation, that is problems which involve the minimization of a norm. A treatment is given of approximation of both real functions and data. For the approximation of functions, the emphasis is on the use of the Chebyshev norm, while for data approximation, we consider a wider range of criteria, including the other l ...