Least Square Polynomial Spline Approximation

Patent, Paul D.

doi:10.1016/b978-0-12-068650-6.50039-3

Cited by 2 publications

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“…The following theorem provides us with bounds for the errors in interpolatory quadratures and will enable us to bound the error in an arbitrary composite quadrature scheme based on interpolatory formulas. The proof depends on Peano's kernel theorem and can be found in [13]. This result is quite general and sharper bounds can be derived for certain interpolatory schemes such as Gaussian and Newton-Cotes formulas.…”

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confidence: 98%

“…Numerical results (cf. [13]) did not reflect this degradation in the order of accuracy for the L 00 -norm. Indeed, Schultz [19] has derived the sharper result.…”

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confidence: 99%

“…[13]) indicated that the exponent of K might be increased by 1/2 in some cases. Indeed, a slight modification in the derivation of (3.9) in the proof of the preceding theorem leads to the inequality We proceed to define the concept of the consistency of quadrature schemes for the approximate solution of the least square problem (cf.…”

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confidence: 99%

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The Effect of Quadrature Errors in the Computation of $L^2 $ Piecewise Polynomial Approximations

Patent¹

1976

SIAM J. Numer. Anal.

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Abstract. In this paper we investigate the L 2 piecewise polynomial approximation problem. L 2 bounds for the derivatives of the error in approximating sufficiently smooth functions by polynomial splines follow immediately from the analogous results for polynomial spline interpolation. We derive L 2 bounds for the errors introduced by the use of two types of quadrature rules for the numerical computation of L 2 piecewise polynomial approximations. These bounds enable us to present some asymptotic results and to examine the consistent convergence of appropriately chosen sequences of such approximations. Some numerical results are also included.1. Introduction. Interpolation, approximation, and smoothing techniques employing piecewise polynomials have received considerable attention in the literature for many years now. The reader is referred to [21] for an extensive bibliography of the spline literature through 1966 including much of the research into the theory and use of the univariate splines. In particular, the excellent approximation properties of polynomial spline interpolates, coupled with their computability, make their use as approximations quite attractive; cf.[18] and [8]. There are, however, several situations where the use of these interpolation techniques is inappropriate. First of all, these methods should not be used when the data are contaminated by errors since errors cause unwanted inflections in polynomial spline interpolates. Secondly, they become computationally unattractive when there are large quantities of data because of the corresponding increase in the dimensionality of the spline spaces involved. Reinsch [14] and Ritter [15] have considered generalized interpolation techniques for the purpose of data smoothing which, just like interpolation, suffer from the drawback of being computationally unattractive for large data sets.In [22] and [23], respectively, Schumaker presents an overview of the theory of approximation by spline functions and discusses methods for computing such approximations. An examination of the papers which Schumaker references indicates that the emphasis has been on L oo spline approximation. However, Smith [24] as well as de Boor and Rice [6] have documented the implementation of discrete least squares techniques using cubic polynomial splines. Smith approached the problem through the normal equations for the cardinal spline representation introduced by Schoenberg in his early work on splines. The normal matrix was found to be ill-conditioned in all but the simplest cases. Smith recognized that the B-spline representation for the spline spaces might circumvent these numerical difficulties. The approach taken by de Boor and Rice is based on the GramSchmidt orthogonalization process and is motivated by their desire to generate an iterative algorithm with which the nonlinear least squares cubic spline approximation problem can be approached (cf. [7]).

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confidence: 98%

“…Numerical results (cf. [13]) did not reflect this degradation in the order of accuracy for the L 00 -norm. Indeed, Schultz [19] has derived the sharper result.…”

mentioning

confidence: 99%