Shape-preserving Univariate Cubic Andhigher-degree L_1 Splines With Function-value-based Andrnmultistep Minimization Principles

摘要

We investigate univariate C~2 quintic L_1 splines and C~3 seventh-degree L_1 splines and revisit C~1 cubic L_1 splines. We first investigate these L_1 splines when they are calculated by minimizing integrals of absolute values of expressions involving various levels of derivatives from zeroth derivatives (function values) to fourth derivatives and compare these L_1 splines with conventional "L_2 splines" calculated by minimizing analogous integrals involving squares of such expressions. The L_2 splines do not preserve the shape of irregular data well. The quintic and seventh-degree L_1 splines also do not preserve shape well, although they do so better than quintic and seventh-degree L_2 splines. Consistent with previously known results, the cubic L_1 splines do preserve shape well. For both L_1 and L_2 splines, the lower the level of the derivative in the minimization principle, the better the shape preservation is. Function-value-based cubic L_1 splines preserve shape well for all situations tested except the one in which an "S-curve" occurs when a flatter representation might be expected. A multi-step procedure for calculating the coefficients of quintic and seventh-degree L_1 splines is proposed. As a basis for this procedure, the function-value-based cubic L_1 spline is calculated. The quintic L_1 spline is calculated by fixing the first derivatives at the nodes to be those of the cubic L_1 spline and calculating the second derivatives at the nodes by minimizing a second-derivative-based quintic L_1 spline functional. The seventh-degree L_1 spline is calculated by fixing the first and second derivatives at the nodes to be those of the quintic L_1 spline and calculating the third derivatives at the nodes by minimizing a second-derivative-based seventh-degree L_1 spline functional. Computational results indicate that C~2 quintic L_1 splines and C~3 seventh-degree L_1 splines calculated in this manner preserve shape well for all situations tested except one in which an "S-curve" occurs when a flatter representation might be expected. To ensure that the parameters of cubic and higher-degree L_1 splines depend continuously on the positions of the data, weights that depend on the local interval length need to be used in the integrals in the minimization principles of these splines.