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Show 9 propriate value for p. Either a heur:istic must be used [2], or else p must be modified interactively until a satisfactory fit is found . Also, the resulting curve has a knot for every data point, which lessens its utility in applications where compactness of representation is important, and the computation of the approximation is subject to reduced numerical accuracy [3]. De Boor and Rice [4] developed an algorithm for a least squares C2 explicit cubic spline approximation using a small number of carefully placed knots. The nonlinear approximation is produced by varying not only the coefficients of the basis functions, as in linear least squares, but also varying the functions themselves by modifying the the knot positions. By permitting the knot loca-tions to vary, fewer knots are usually needed to approximate data to within a user-specified tolerance. In this respect, it performs much better than the smoothing spline. Golub and Pereyra have developed another variable knot method that uses a more sophisticated model. Because the model more accurately reflects the variable nature of the knots and the B-spline coefficients, the convergence is much faster. Use of this model will be discussed in greater detail in sections 3.4 and 3.5 . Jupp [14] proposes a transform that can be used with Golub and Pereyra's model. By applying a transform which reduces the tendency of knots to coalesce (since they are unlikely to separate again once this has happened, and many of these are undesirable solutions), the likelihood of convergence to a global minimum is improved. This does, however, also make corners quite unlikely in the final result, which seems inconsistent with our needs. All three of these variable knot algorithms are nonlinear; no matter which one is used, the initial knot set should be chosen with care to avoid local minima which are not also global [14], and to increase the speed of convergence. A criterion by Powell [20] is used in an approach that automatically fits a piecewise cubic curve to data [13]. It is assumed that all of the information in the data has been captured by the explicit curve f when the errors e. = f(x.) - y. I I I |
