Given a family of curves or surfaces in R s , an important problem is that of finding a member of the family which gives a "best" fit to m given data points. There are many application areas, for example metrology, computer graphics, pattern recognition, and the most commonly used criterion is the least squares norm. However, there may be wild points in the data, and a more robust estimator such as the l1 norm may be more appropriate. On the other hand, the object of modelling the data may be to assess the quality of a manufactured part, so that accept/reject decisions may be required, and this suggests the use of the Chebyshev norm.We consider here the use of the l1 and l∞ norms in the context of fitting to data curves and surfaces defined parametrically. There are different ways to formulate the problems, and we review here formulations, theory and methods which generalize in a natural way those available for least squares. As well as considering methods which apply in general, some attention is given to a fundamental fitting problem, that of lines in three dimensions.
In this paper, a new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations is presented and analyzed. This new family of iterative methods is obtained by composing an iterative method proposed by Chun [3] with Newton's method and approximating the first-appeared derivative in the last step by a combination of already evaluated function values. The proposed family is optimal since its efficiency index is 8 1/4 ≈ 1.6818. The convergence analysis of the new family is studied in this paper. Several numerical examples are presented to illustrate the efficiency and accuracy of the family.
A formulation for the fractional Legendre functions is constructed to find the solution of the fractional Riccati equation. The fractional derivative is described in the Caputo sense. The method is based on the Tau Legendre and path following methods. Theoretical and numerical results are presented. Analysis for the presented method is given.
For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of least squares by the Chebyshev norm. For example, this criterion may be a more appropriate one in the context of accept/reject decisions for manufactured parts. The resulting problem has some interesting features: it has much structure which can be exploited, but generally the solution is not unique. We consider a method of Gauss-Newton type and show that if the non-uniqueness is resolved in a way which is consistent with a particular way of exploiting the structure in the linear subproblem, this can not only allow the method to be properly defined, but can permit a second order rate of convergence. Numerical examples are given to illustrate this. (2000): 65D10, 65K05.
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