Fixed- and Free-Knot Univariate Least-Squares Data Approximation ...

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Defense Technical Information Center Compilation Part Notice ADP013744 TITLE: Fixed- and Free-Knot Univariate Least-Squares Data Approximation by Polynomial Splines DISTRIBUTION: Approved for public release, distribution unlimited

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Fixed- and free-knot univariate least-squares data approximation by polynomial splines Maurice Cox, Peter Harris and Paul Kenward National Physical Laboratory, Teddington, Middlesex, TW11 OLW, UK maurice. cox@npl .co.uk, peter. harris@npl. co.uk, paul.kenward@npl .co.uk

Abstract Fixed- and free-knot least-squares data approximation by polynomial splines is considered. Classes of knot-placement algorithms are discussed. A practical example of knot placement is presented, and future possibilities in free-knot spline approximation are addressed.

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Introduction

The representation of univariate polynomial splines in terms of B-splines is reviewed (Section 2), leading to the problem of obtaining fixed- and free-knot f2 spline approximations (Section 3). The accepted approach to the fixed-knot case is recalled (Section 4) and the manner in which spline uncertainties can be evaluated given (Section 5). The importance of families of spline approximants is emphasised (Section 6). The free-knot problem is formulated (Section 7) and several of the established and some lesser-known knot-placement strategies reviewed (Section 8). Conclusions are drawn and future possibilities indicated (Section 9). 2

Univariate polynomial splines

Let I := [Xmin,Xmnax] be an interval of the x-axis, and Xmin = A0 < Al -