End Conditions for Interpolatory Quintic Splines

Present address: Department of Mathematics, University of Tabriz, Tabriz, Iran. Accurate end conditions are derived for quintic spline interpolationat equally spaced knots. These conditions are in terms of availablefunction values at the knots and lead to O(h⁶) convergence uniformlyon the interval of interpolation.     

Efficient Solution of the Complex Quadratic Tridiagonal System for C2 PH Quintic Splines

The construction of C ² Pythagorean-hodograph (PH) quintic spline curves that interpolate a sequence of points p ₀,...,p _N and satisfy prescribed end conditions incurs a tridiagonal system of N quadratic equations in N complex unknowns. Albrecht and Farouki [1] invoke the homotopy method to compute all 2 ^N+k solutions to this system, among which there is a unique good PH spline that is free of undesired loops and extreme curvature variations (k{–1,0,+1} depends on the adopted end conditions). However, the homotopy method becomes prohibitively expensive when N10, and efficient methods to construct the good spline only are desirable. The use of iterative solution methods is described herein, with starting approximations derived from ordinary C ² cubic splines. The system Jacobian satisfies a global Lipschitz condition in C ^N , yielding a simple closed-form expression of the Kantorovich condition for convergence of Newton–Raphson iterations, that can be evaluated with O(N ²) cost. These methods are also generalized to the case of non-uniform knots.     

拟贯穿剖分上的二元样条Lagrange插值

分片代数曲线足经典代数曲线的推广.利用沿分片代数曲线插值以及分片代数曲线的Nother型定理,给出了一类构造拟贯穿剖分上的二元样条Lagrange插值适定结点组的一种方法,并给出具体算法与实例.     

A Numerical Algorithm for Cubature by Bivariate Splines on Nonuniform Partitions

We have studied the numerical integration 2D based on bivariate C¹ local polynomial splines with a criss–cross triangulation of nonuniform rectangular partition. We have constructed the cubature formula and proved the convergence properties and error bounds. The paper includes some numerical tests that illustrate the performance of the corresponding algorithm. In the appendix there are explicit expressions of the quadratic polynomial restrictions of the B-splines related to every triangular cell.     

Nonobtuse Tetrahedral Partitions that Refine Locally Towards Fichera-Like Corners

Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain. The first author was supported by the Swedish Foundation for Strategic Research, the second author was supported by Grant No. 49051 of the Academy of Finland, the third author was supported by Grant No. A 1019201 of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AV0Z 10190503.     

Lagrange Interpolation by C 1 Cubic Splines on Triangulated Quadrangulations

We describe local Lagrange interpolation methods based on C ¹ cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.     

A Posteriori Corrections for Non-periodic Cubic and Quintic Interpolating Splines at Equally Spaced Knots

The method proposed recently by Lucas [13], for the a posterioricorrection of odd-degree interpolating periodic splines is extendedto non-periodic cubic and quintic splines.     

Three-Convex Approximation by Free Knot Splines in C [0, 1]

A Jackson-type estimate is obtained for the approximation of 3 -convex functions by 3 -convex splines with free knots. The order of approximation is the same as for the Jackson-type estimate for unconstrained approximation by splines with free knots. Shape-preserving free knot spline approximation of k -convex functions, k > 3 , is also considered. January 15, 1996. Date revised: December 9, 1996.     

Shape-Preserving Smoothing of 3-Convex Splines of Degree 4

For every 3-convex piecewise-polynomial function s of degree ≤4 with n equidistant knots on [0, 1] we construct a 3-convex spline s ₁ (s ₁ ∈ C ⁽³⁾) of degree ≤4 with the same knots that satisfies the inequality

Cubic Splines on Curved Spaces

We consider a second-order problem in the calculus of variations,with an application to robotics in mind. The analysis is carriedout on a general Riemannian manifold M and then specializedto the case where M is the Lie group SO(3) of rotations in R³.For SO(3), the Euler-Lagrange equations reduce to interestingnonlinear systems of ordinary differential equations in R³.     

Splines on generalized quasi-cross-cut partitions

A new method of obtaining the dimensions of the polynomial splines of degree k and smoothness r f- 1 (r ⩾ 1, and k ⩾ r f- 1 + ξ_Δ) on generalized quasi-cross-cut partitions is presented, where ξ_Δ is a number depending on the structure of the partition.     

BestL 1-approximations of classesW 1 r by Splines fromW 1 r

We obtain the exact values of the bestL ₁-approximations of the classesW ₁ ^r of periodic functions by periodic polynomial splines of degreer and defect 1 with equidistant knots that belong to the classW ₁ ^r .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1410–1413, October, 1994.     

Splines on Triangulations with Hanging Vertices

Polynomial spline spaces defined on triangulations with hanging vertices are studied. In addition to dimension formulae, explicit basis functions are constructed, and their supports and stability are discussed. The approximation power of the spaces is also treated.     

二元C1四次样条插值

本文在较一般的平面三角剖分激造了一种C¹四次样条插值格式.这种格式仅用到被插函数的函数值与一阶导数值信息,并得出插值样条的递推计算格式.     

Exponential Box-Like Splines on Nonuniform Grids

We generalize the exponential box spline by allowing it to have arbitrarily spaced knots in any of its directions and derive the corresponding recurrence and differentiation rules. The corresponding spline space is spanned by the shifts of finitely many such splines and contains the usual family of exponential polynomials. The (local) linear independence of the spanning set is equivalent to a geometric condition closely related to unimodularity. January 10, 1996. Date revised: December 9, 1997. Date accepted: March 18, 1998.     

A Note on Binary Plane Partitions

   Abstract. This paper considers binary space partition s (BSP for short) for disjoint line segments in the plane. The BSP for a disjoint set of objects is a scheme dividing the space recursively by hyperplanes until the resulting fragments of objects are separated. The size of a BSP is the number of resulting fragments of the objects. We show that the minimal size of a BSP for n disjoint line segments in the plane is Ω (n log n /log log n) in the worst case.     

An Efficient Algorithm for Generating Univariate Cubic L 1 Splines

An active set based algorithm for calculating the coefficients of univariate cubic L ₁ splines is developed. It decomposes the original problem in a geometric-programming setting into independent optimization problems of smaller sizes. This algorithm requires only simple algebraic operations to obtain an exact optimal solution in a finite number of iterations. In stability and computational efficiency, the algorithm outperforms a currently widely used discretization-based primal affine algorithm.     

A Note on Binary Plane Partitions

Abstract. This paper considers binary space partition s (BSP for short) for disjoint line segments in the plane. The BSP for a disjoint set of objects is a scheme dividing the space recursively by hyperplanes until the resulting fragments of objects are separated. The size of a BSP is the number of resulting fragments of the objects. We show that the minimal size of a BSP for n disjoint line segments in the plane is Ω (n log n /log log n) in the worst case.