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A new method for finding the optimal smoothing parameter for the abstract smoothing spline. (English) Zbl 1247.41005
Abstract smooth spline $$\sigma_\alpha$$ with smoothing parameter $$\alpha >0$$ is defined as the solution of a certain minimization problem. A typical example is the univariate polynomial spline which smoothes given data. Given an estimate for the noise, a standard choice for the parameter $$\alpha$$ is that $$\sigma_\alpha$$ should satisfy a certain residual equation. In the standard Reinsch algorithm [C. H. Reinsch, Numer. Math. 16, 451–454 (1971; Zbl 1248.65020)], the residual equation is transformed to another equation, which is then solved with Newton’s method. Here the author presents a new method for solving the residual equation. It is observed that in contrast to the Newton’s method which converges quadratically, the convergence rate of the author’s method can be of any specified order.
##### MSC:
 41A15 Spline approximation
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##### References:
 [1] Bezhaev, A.Yu.; Vasilenko, V.A., Variational theory of splines, (2001), Kluwer Academic/Plenum Publishers · Zbl 0979.41007 [2] Demmler, A.; Reinsch, C., Oscillation matrices with spline smoothing, Numer. math., 24, 375-382, (1975) · Zbl 0297.65002 [3] Duchon, J., Splines minimizing rotation-invariant seminorms in Sobolev spaces, Lect. notes math., 571, 85-100, (1977) [4] Gordonova, V.I.; Morozov, V.A., Numerical algorithms of parameter selection in regularization method, Zh. vychisl. mat. mat. fiz., 13, 539-545, (1973), in Russian [5] Laurent, P.-J., Approximation et optimization, (1972), Hermann Paris [6] Reinsch, C.H., Smoothing by spline functions. II, Numer. math., 16, 451-454, (1971) · Zbl 1248.65020 [7] Rozhenko, A.I., On optimal choice of spline-smoothing parameter, (), 79-86 · Zbl 0993.65023 [8] A.I. Rozhenko, Spline software Sdm.net: Splines for Data Mining under dot.net, http://sites.google.com/site/arozhenkosite/sdm [9] Schaback, R.; Wendland, H., Characterization and construction of radial basis functions, (), 1-24 · Zbl 1035.41013 [10] Traub, J.F., Iterative methods for the solution of equations, (1982), Chelsea Publishing Company · Zbl 0472.65040
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