Институт вычислительной математики и математической геофизики СОРАН

Abstracts

Function approximation and cubic formulas

In construction of a smoothing spline, it is important to correctly choose the smoothing parameter, for example, with the residual principle, in which the smoothing parameter gets out so that the root-mean-square error of approximation (the residual) is within an error in the initial data. Algorithms of finding the smoothing parameter by the residual principle are reduced to solving a nonlinear equation, for example, by Newton's method.

Earlier authors had offered the algorithm based on family of bilateral monotonous approximations of the residual of the smoothing spline. Bilateral approximations are constructed on the base of decompositions of the residual operator in power series. By means of these approximations, it is possible to construct an algorithm with any given order of convergence.

Efficiency of the algorithm depends on the number of iterations of finding the optimal smoothing parameter, because it is necessary to build on iteration a new matrix of system of linear equations for solving the smoothing problem with the next value of the parameter. Construction and matrix decomposition is thus carried out for O(N^3) operations, and solving of one smoothing problem is then carried out for O(N^2) operations. Therefore, it is important in this algorithm to reduce the number of iterations, so it is necessary to approximate the residual function on iteration as more precise as possible. Accuracy of the residual approximation is connected with the number of solved problems of construction of a smoothing spline on each step. Using solutions of n problems, the approximation providing convergence of the algorithm with nth degree could be constructed.

In this work, the modification of this algorithm is proposed in which all constructed residual approximations are used, but not just the last most exact one. Residual value thus is specified by means of asymptotic estimations. As a result, the speed of convergence of the algorithm essentially accelerates without additional expenses.

Note. Abstracts are published in author's edition

Mail to Webmaster

|Home Page| |English Part|

Go to Home