Characterization of strict approximations in subspaces of spline functions

The problem of approximating a given function by spline functions with fixed knots is discussed. Strict approximations which are particular unique best Chebyshev approximations are considered. The chief purpose is to develop a characterization theorem for these strict approximations.     

An algorithm for the computation of strict approximations in subspaces of spline functions

The problem of approximating a given function by spline functions with fixed knots is discussed. Strict approximations which are particular unique best Chebyshev approximations are considered. The chief purpose is to develop a characterization theorem for these strict approximations.     

Chebyshev approximations in subspaces of spline functions

The main purpose of this paper is to consider strict approximations from subspaces of spline functions of degree m-1 with k fixed knots. Rice defines the strict approximation which is a particular unique best Chebyshev approximation for problems defined on a finite set. In order to determine best approximations on an interval I we define a sequence of strict approximations on finite subsets of I where the subsets fill up the interval. It is shown that the sequences always converge if k≤m. In the case k>m the sequences are convergent if we restrict ourselves to problems defined on certain subsets of I. It seems to be natural to denote these limits as strict approximations. To be able to compute these functions we also develop a Remez type algorithm.     

Parametric Cubic Spline Approach to the Solution of a System of Second-Order Boundary-Value Problems

We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.     

The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions

The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used.     

Some new linear relations for even degree polynomial splines on a uniform mesh

This paper is concerned with even degree polynomial splines, degreeN, defined on a uniform mesh. There are well-known linear relations among the spline values atN successive knots and thepth derivatives of the spline at the sameN knots. Some new linear relations are developed which connectN–1 spline values withN–1 derivative values. The leading terms of the truncation errors of these new linear relations are found in terms of Bernoulli numbers.     

Iterative methods based on spline approximations to detect discontinuities from Fourier data

Recently, spline approximations have been proposed for the reconstruction of piecewise smooth functions from Fourier data. That approach makes possible to retrieve the functions from their Fourier coefficients for any given degree of accuracy when the discontinuity points are known. In this paper we present iterative methods based on those spline approximations, for several degrees, to find locations and amplitudes of the jumps of a piecewise smooth function, given its Fourier coefficients. We also present numerical experiments comparing with different previous approaches.     

Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem

In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods.     

Strongly unique spline approximations with free knots

A necessary and a sufficient alternation condition for strongly unique best spline approximations with free knots is given. In the case of simple knots these conditions coincide, and strongly unique best approximations and strongly unique local best approximations are the same. The numerical consequences are discussed.     

Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

We discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: [p(x)y′]′ = f(x, y), α₀y(a) - α₁y′(a) = A, β₀y(b) + β₁y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method.     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

三次多项式样条解一类四阶边值问题

应用三次多项式样条函数解一组四阶单侧、障碍、接触边值问题,取半结点为网格点,并增加了边界方程,应用方法解文献中的数值例子,说明了方法的高效性,数值结果也显示了方法的优越性.     

A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems

Bickley [5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h² uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process. We present a new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems: y″ = f(x, y, y′), a < x < b, α₀y(a) − α₀y′(a) = A, β₀y(b) + β₁y′(b) = B. We generate the solution at the nodal points by a fourth-order method and then use ‘conditions of continuity’ to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution. We show that our method provides order h⁴ uniformly convergent approximations over [a, b]. The fourth order of the presented method is demonstrated computationally by two examples.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.     

A family of optimal fourth-order methods for multiple roots of nonlinear equations

Newton-Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton-type method at the first step. In this paper, we present a new two-step optimal fourth-order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also given to explain the dynamical behavior of the new methods around the multiple roots and decide the best values of the free parameters involved. Finally, we compare our methods with the existing schemes of the same order with a real life application as well as standard test problems. From the numerical results, we find that our methods can be considered as a better alternative for the existing procedures of same order.     

A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems

A family of fourth and second-order accurate numerical schemes is presented for the solution of fifth-order boundary-value problems with two-point-boundary conditions. The non-polynomial sextic spline functions are applied to construct the numerical algorithms. This approach generalizes polynomial spline algorithms, and provides solution at every point of range of integration. Convergence of the methods is discussed through standard convergence analysis. A numerical illustration is given to show the pertinent features of the technique.     

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

End conditions for improved cubic spline derivative approximations

We consider the problem of deriving accurate end conditions for cubic spline interpolation at equally spaced knots. In particular we derive a number of end conditions which lead to derivative approximations of high accuracy.     

On the new fourth-order methods for the simultaneous approximation of polynomial zeros

A new iterative method of the fourth-order for the simultaneous determination of polynomial zeros is proposed. This method is based on a suitable zero-relation derived from the fourth-order method for a single zero belonging to the Schröder basic sequence. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is studied in detail for the proposed method. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. The construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed. Finally, numerical examples and the comparison with existing fourth-order methods are given.     

Generalized spline wavelets

A generalized multiresolution of multiplicityr, generated byr linearly independent spline functions with multiple knots, is introduced. With the help of the autocorrelation symbol and the two-scale symbol of the scaling functions, spline wavelets with multiple knots can be completely characterized. New decomposition and reconstruction algorithms, based on the Fourier technique, are presented.