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Orthogonal complex bases of spline, which are orthogonal to each other, are proposed in this paper. The proposed orthogonal spline bases make up spline function space of periodic zone. A new multi-resolution theory based on orthogonal spline base is proposed in the paper. Other than bases of wavelet theory, orthogonal spline bases are totally orthogonal to each other, therefore there is no loss or getting in signal when we decompose the signal. The theory is applied to signal processing, the result showed potential application in signal processing.

We use a quartic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of third-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.

Arc splines are G^1 continuous curves made of circular arcs and straight-line segments. They have the advantages that the curvature of an arc spline is known and controlled at all but a finite number of points, and that the offset curve of an arc spline is another arc spline. Arc splines are used by computer-controlled machines as a natural curve along which to cut and are used by highway route planners as a natural curve along which to plan the centre line of a road. In this paper, it is shown how to increase the smoothness of a planar arc spline by replacing parts of it and thus to create a G^2 continuous curve. The replacement parts are low-degree NURBS curves: cubic Bezier curves and quadratic rational Bezier curves.

A three-dimensional (3-D) shape optimization algorithm is developed by combining the geometric parameterization of the design surface using spline technique and design sensitivity analysis. The design surface is parameterized using Bezier spline or B-spline, and the control points of the spline are taken as the design variables. Considering nonlinearity of the material, the design sensitivities for the nodal points on the design surface and the adjoint variable are computed based on 3-D nonlinear finite-element method. The parameterized design sensitivities for the control points of the spline are computed from those for the nodal points. Through a numerical implementation, the developed algorithm is proved to guarantee a smooth design surface.

This paper presents a parameterization and an interpolation method for quintic splines, which result in a smooth and consistent feed rate profile. The discrepancy between the spline parameter and the actual arc length leads to undesirable feed fluctuations and discontinuity, which elicit themselves as high frequency acceleration and jerk harmonics, causing unwanted structural vibrations and excessive tracking error. Two different approaches are presented that alleviate this problem. The first approach is based on modifying the spline tool path so that it is optimally parameterized with respect to its arc length, which allows it to be accurately interpolated in real-time with minimal complexity. The second approach is based on scheduling the spline parameter to accurately yield the desired arc displacement (hence feed rate), either by approximation of the relationship between the arc length and the spline parameter with a feed correction polynomial, or by solving the spline parameter iteratively in real-time at each interpolation step. This approach is particularly suited for predetermined spline tool paths, which are not arc-length parameterized and cannot be modified. The proposed methods have been compared to approximately arc-length C3 quintic spline parameterization (Wang, F.-C., Wright, P. K., Barsky, B. A., and Yang, D. C. H., 1999, “Approximately Arc-Length Parameterized C3 Quintic Interpolatory Splines,” ASME J. Mech. Des, 121, No. 3., pp. 430–439) and first- and second-order Taylor series interpolation techniques (Huang, J.-T., and Yang, D. C. H., 1992, “Precision Command Generation for Computer Controlled Machines,” Precision Machining: Technology and Machine Development and Improvement, ASME-PED 58, pp. 89–104; Lin, R.-S. 2000, “Real-Time Surface Interpolator for 3-D Parametric Surface Machining on 3-Axis Machine Tools,” Intl. J. Mach. Tools Manuf., 40, No.10, pp. 1513–1526) in terms of feed rate consistency, computational efficiency, and experimental contouring accuracy.

For given monotone data we propose the construction of a histopolating linear/linear rational spline of class C1. The uniqueness and existence of this spline is proved. The method is implemented via the representation with histogram heights and first derivatives of the spline. The use of Newton's method and ordinary iterations are discussed. Numerical tests support completely the theoretical results.

In this paper we are interested in the approximation of continuous surface texture profiles defined only at discretely obtained points. A continuous representation is required in order to be able to apply filtration methods that assume uniform data spacing to practical data that in general is not uniformly spaced. By reconstructing surface profiles as natural cubic spline interpolants accurately using numerically stable fitting algorithms, we can provide a mathematically sound basis on which to compute surface texture profile parameters. As an important direct benefit, parameters involving the integrals of surface profiles can be calculated directly from the spline interpolant. Examples are given to illustrate the advantages of using this reconstruction method.

In the earlier application of the spline finite strip method(FSM), the uniform B3-spline functions were used in the longitudinal direction while the conventional interpolation functions were used in the transverse direction to construct the displacement filed in a strip. To overcome the shortcoming of the uniform B3-spline, non-periodic B-spline was developed as the displacement function. The use of non-periodic B3-spline function requires no tangential vectors at both ends to interpolate the geometry of shell and the Kronecker delta property is also satisfied at the end boundaries. Recently, non-periodic spline FSM which was modified to have a multiple knots at the boundary was developed for the shell analysis and applied to the analysis of bridges. In the formulation of a non-symmetric spline finite strip method, the concepts of non-periodic B3-spline and a stress-resultant finite strip with drilling degrees of freedom for a shell are used. The introduction of non-symmetrically spaced knots in the longitudinal direction allows the selective local refinement to improve the accuracy of solution at the connections or at the location of concentrated load. A number of numerical tests were performed to prove the accuracy and efficiency of the present study.

The representation of signals in spline signal spaces has several advantages compared to the general approach of working with band-limited functions. Among them are the finite support of B-splines, simple manipulations like differentiation, integration, etc. In this paper, the deterministic treatment of spline signals is extended to a stochastic analysis. The main focus is on the analysis of second-order characteristics of spline signals, i.e. auto-correlation and cross-correlation functions and sequences are calculated. The signal analysis as well as synthesis steps are analyzed and an algorithm for estimating the auto-correlation and cross-correlation functions and sequences is derived.

In this paper, we use parametric cubic spline function to develop a numerical method, which is fourth order for a specific choice of the parameter, for computing smooth approximations to the solution for second order boundary value problems. Some numerical evidence is also included to demonstrate the superiority of our method.

This paper presents an extension of the Modified Spline Technique (MST) formulation for data fitting named power spline, which gives consistent results for sets of data of concave and convex curves. This method is based on a technique which couples an implicit formulation of the maximum likelihood principle to the spline method, making the method suitable to fit data when no physical model is available. The MST method proved to be better than to the spline method and the extended spline fit technique (EST), because it provided accurate results for the first and second derivatives in sets of data where the EST solution developed inaccuracies. The EST method is a formulation that couples Least Squares to the spline method. There are, however, some sets of data where the MST method would show inconsistent solutions for the first and second derivatives. The power spline method eliminates these problems for concave or convex curves. Another improvement on the method is a more flexible choice for the interval boundaries. 1

In this article, using spline in compression, we discuss three difference schemes for the numerical solution of singularly perturbed two-point singular boundary-value problems. The proposed schemes are second-order accurate and applicable to problems both in singular and non-singular cases. Convergence analysis of a difference scheme is discussed and numerical results are given to illustrate the utility of the proposed methods.

A non-uniform mesh difference scheme using cubic spline in tension is presented to solve a class of non-turning point singularly perturbed two point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative subject to Dirichlet-type boundary conditions. To demonstrate the applicability of the proposed method, two numerical examples have been solved and the results are presented along with their comparison with those obtained with and without variable mesh. This paper is a continuation of the previous work [Aziz, T. and Khan, A. (2002). A spline method for second order singularly-perturbed boundary-value problems. J. Comput. Appl. Math., 147(2), 445–452.] given for uniform mesh case.

The existence, uniqueness and stability for the functional-differential equation of neutral type using spline of deficiency 3 with stepsize 3h spline function of degree four are presented in Ref. [1]. In this paper, we extend the study to the convergence of our proposed spline method. We prove that, if the local error is of order p, then the global error is of order p as well. Numerical examples are presented to illustrate the convergence of the method.

Some exponentially fitted difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression. These schemes are second-order accurate. Numerical examples are given in support of the theoretical results.

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.

In this paper, spline collocation method (SCM) is successfully extended to solve the generalized problems of beam structures. The spline functions in SCM are re-formulated by finite difference method in a systematical way that is easily understood by engineers. The manipulation of SCM is further simplified by the introduction of quintic table so that the matrix-vector governing equation can be easily formulated to solve the weighting coefficients. SCM is first examined by the problems of a generalized single-span beam undergoing various types of loadings and boundary conditions, and it is then extended to the problems of continuous beam with multiple spans. By comparing with available analytical results, differential quadrature method (DQM), if any, excellent accuracy in deflection is achieved.

The existence, uniqueness and stability for delay dynamic system using spline of deficiency 3 with stepsize 3 h spline function of degree four are presented in El-Safty, et al . (Existence, uniqueness and stability of the spline function for delay controlled dynamic system - Intern. J. Computer Math. , 77, 629-640, 2000). In this paper we extend the study to the convergence of our proposed spline method. We prove that, if the local error is of order p , then the global error is of order p as well and is a five order accurate. Numerical examples are presented to illustrative the convergence of the method.

A number of numerical methods based on a piecewise polynomial approximation have been proposed for the estimation of the term structure of interest rates. Some drawbacks have been pointed out, such as a possible non monotonic estimated discount function and a highly fluctuating spot and forward rates. In order to overcome these kind of problems, we study the feasibility of an adaptive regression spline technique which use a monotone basis together with two alternative knot location procedures: a deterministic greedy algorithm and its randomized version in a simulated annealing framework. The features of the proposed method are tested on a set of data.

Some difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression on non-uniform mesh. These schemes are second order accurate. Numerical examples are given in support of the theoretical results.

In this paper we present a novel spline strip kernel particle method (SSKPM) that has been developed for solving a class of two-dimensional (2D) elasticity problems. This new approach combines the concepts of the mesh-free methods and the spline strip method. For the interpolation of the assumed displacement field, we employed the kernel particle shape functions in the transverse direction, and the B3-spline function in the longitudinal direction. The formulation is validated on several beam and semi-infinite plate problems. The numerical results of these test problems are then compared with the existing solutions obtained by the exact or numerical methods. From this study we conclude that the SSKPM is a potential alternative to the classical finite strip method (FSM). Copyright © 2003 John Wiley & Sons, Ltd.

A numerical method based on cubic spline with adaptive grid is given for the self-adjoint singularly perturbed two point boundary value problems. The scheme derived in this method is second order accurate. Numerical examples are given to support the predicted theory.

Some difference schemes for singularly perturbed two-point boundary value problems are derived using spline in tension. These schemes are second-order accurate. Numerical examples are given in support of the theoretical results.

Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique. This paper considers the design problem for specific types of SS-ANOVA models. As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior. Three examples for additive and interaction spline models are provided for illustration. A comparison of the asymptotic designs, the 2d factorial designs, and the glp designs is given by numerical computation.

One-dimensional singularly-perturbed two-point boundary-value problems arising in various fields of science and engineering (for instance, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics, etc.) are treated. Either these problems exhibits boundary layer(s) at one or both ends of the underlying interval or they possess oscillatory behavior depending on the nature of the coefficient of the first derivative term. Some spline difference schemes are derived for these problems using splines in compression and splines in tension. Second-order uniform convergence is achieved for both kind of schemes. By making use of the continuity of the first-order derivative of the spline function, a tridiagonal system is obtained which can be solved efficiently by well-known algorithms. Numerical examples are given to illustrate the theory.

Discrete-time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions. Each problem is written as a Schrodinger-type system, which is then approximated by Crank--Nicolson and/or alternating direction implicit orthogonal spline collocation schemes. These schemes are shown to be second-order accurate in time and of optimal order accuracy in space in the Hm-norm, m=1,2.

This paper presents a class of numerical method for the approximate solution for the functional-differential equation of neutral type. These methods are essentially based on the spline functions. The study of existence and uniqueness are considered of 3h-step spline function of degree m =4. Numerical examples and comparisons with other methods are given.

In this paper we present a subdivision algorithm for the evaluation of the trigonometric spline curves with uniform knots.

We obtain estimates for errors of interpolation on a nonuniform grid for a parabolic nonperiodic spline of defect 1.

In this paper we consider integration rules based on products of one-dimensional nodal spline quadratures for the numerical evaluation of 2-D singular integrals defined in the Hadamard finite part sense. We derive convergence results and error bounds. Some numerical tests are also presented.

A numerical method based on spline in tension is given for the self-adjoint singularly perturbed two point boundary value problems. The schemes derived in this method are second order accurate. One numerical example is given to support the predicted theory.

A significant curve modeling technique has been introduced with a view to its applications in geometric modeling, computer graphics and computer-aided design. It is a new spline method using quadratic trigonometric functions with well-controlled shape influences of parameters introduced through geometric continuity of order two. The proposed curve model owns the best possible geometric properties such as convex hull, partition of unity, affine invariance and variation diminishing. A quadratic normally has three control points giving lesser flexibility to one piece of curve. However, a cubic has four control points giving higher flexibility to one piece of curve. In the proposed scheme, we have introduced a quadratic trigonometric with four control points. Thus, the proposed quadratic trigonometric has embedded geometric features of cubic/cubic trigonometric. The proposed spline method, constrained with Nu spline like GC2 smoothness, produces a quadratic trigonometric Nu spline (QTNS) with interesting shape control locally and globally. The method is helpful for a variety of shape effects, like point tension, interval tension or global tension. It also produces a quadratic trigonometric alternative to cubic/cubic trigonometric spline because of having four control points in its piecewise description.

Spline functions are an approach to the smoothing of bivariate data (a spline is a term for a flexible strip of metal or rubber used by a draftsman to draw curves). Splines are polynomials within intervals of the x-variable that are connected across different values of x. In this article, the theory behind such functions is described. Keywords: nonparametric regression; polynomial regression; roughness penalty approach

This paper develops the method to approximate a parametric curve by an arc spline with a feature extraction process using Hough transform. This has been done to make the curve compatible with devices and applications which have limitations of parametric inputs. The proposed algorithm uses Hough transform technique to detect linear and circular-arc parts on the curve prioritizing the maximum arc length. Arc spline segments are then generated corresponding to the detected shapes with C1 continuity. This method is different from other arc spline approximation schemes which subdivide the curve according to time or error data. This work also develops a representation method to describe the shape or pattern of the curve using arc spline information. Experimental results show that similarity of any two curves can be detected and measured by comparing their arc spline data. In application, this method can be used to generate vector graphics and tool paths for numerical control machines with higher compatibility than a parametric description.

The choice of nonlinear description used in Volterra based models strongly influences the robustness of the parameter's extraction. Traditionally, monomials are used as the nonlinear description. However, it has been shown that other polynomials provide easier extraction. Volterra-based models can also be described using spline interpolated look-up tables (LUTs). Splines have increased locality which improves conditioning, converging to polynomials as the spline order-1 tends to the number of LUT points. In this paper we first show how to describe Volterra models in terms of splines and then study the conditioning and error for increasing spline order in a practical example.

In this paper, we investigate the use of suitable spline functions of polynomial form to approximate the solution of fractional delay differen- tial equations. The description of the proposed approximation method is first introduced. The error analysis and stability of the method are theoretically investigated. The proposed spline function is an extension and generalization to the polynomial spline functions used in (15). Numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.

Based on discretization methods for solving semi-infinite programming problems, this paper presents a spline smoothing Newton method for semi-infinite minimax problems. The spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; that is, it introduces an active set technique, so it is more efficient for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. Numerical tests show that the new method is very efficient.

This paper discussed the local control of interpolating function by using rational cubic spline (cubic/quadratic) with three parameters originally proposed by the authors. The rational spline hasC1continuity. The bounded properties of the rational cubic interpolants and shape controls of the rational interpolants are discussed in detail. The value control, inflection point control, and convexity control at a point by using the proposed rational cubic spline are constructed. Several numerical results are presented to show the capability of the method. Numerical comparisons with the existing scheme are also further elaborated. From the results, it was indicated that the scheme works well and it is comparable with the established existing scheme.

We present a concept for representation of motion-capture data where information from the animation data is used to generate a blending-type spline curve using local curves of Hermite type.

A numerical method based on cubic spline with nonuniform grid is given for the singularly perturbed two-point boundary value problems having turning point. The scheme derived in this paper is second-order accurate. Numerical examples are given to support the predicted theory.

The aim of this paper is to give a numerical method for the solution of evolution equations in Banach spaces. This method is based on a spline approximation of the exact solution which is a generalisation and continuation of earlier results [1–4]. It is considered to be a first-order evolutionary problem, given the numerical method and proven the convergence of the approximate solution.

In this paper we investigate the approximation power of local bivariate quadratic C1 quasi-interpolating (q-i) spline operators with a four-directional mesh. In particular, we show that they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds both for the errors and for the spline partial derivatives, in the case the spline is more differentiable than the function. Then such general results are applied to prove new properties of two interesting q-i spline operators, proposed and partially studied in Chui and Wang (Sci. Sinica XXVII (1984) 1129–1142).

Background : A new realistic geometry (RG) spline Laplacian estimation technique has been developed for high-resolution EEG imaging. Methods : Estimation of the parameters associated with the spline Laplacian is formulated by seeking the general inverse of a transfer matrix. The number of spline parameters, which need to be determined through regularization, is reduced to one in the present approach, thus enabling easy implementation of the RG spline Laplacian estimator. Results : Computer simulation studies have been conducted to test the feasibility of the new approach in a 3-concentric-sphere head model. The new technique has also been applied to human visual evoked potential data with a RG head model. Conclusions : The present numerical and experimental results demonstrate the feasibility of the new approach and indicate that the RG spline Laplacian can be estimated easily from the surface potentials and the scalp geometry.

This paper proposes a method for multi‐terminal resistor optimal design based on a stochastic optimization algorithm coupled with a high accuracy boundary element method numerical analysis module using spline boundary elements. The aim of the optimization process is to find out the optimal shape of the resistor so that the values of the partial resistances between its terminals can be globally reduced. In the first part of the paper, a short introduction to the boundary element method formulation using spline interpolation of the boundary unknowns is performed. Then, the genetic algorithm optimization method is briefly described, emphasising the numerical implementation particularities related to the actual application. In the second part of the paper, three numerical examples are proposed and the final conclusions are outlined.

A self-contained mathematical derivation of the smoothing spline with weighted roughness measure is offered, along with a discussion of the computational details required for its implementation. The introduction of a weighted roughness measure, even if only piecewise constant, provides additional flexibility in the shaping of the smoothing spline, while not materially increasing the computational cost of its construction or use.

In this work, a new approach is proposed for constructing splines with tension. The basic idea is in the use of distributions theory, which allows us to define suitable Hilbert spaces in which the tension spline minimizes some energy functional. Classical orthogonal conditions and characterizations of the spline in terms of a fundamental solution of a differential operator are provided. An explicit representation of the tension spline is given. The tension spline can be computed by solving a linear system. Some numerical examples are given to illustrate this approach.

Some difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression. These schemes are second order accurate. Numerical examples are given in support of the theoretical results.