On parametrization of interpolating curves

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.     

On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.     

Interpolation Methods for Curve Construction

This paper surveys a wide selection of the interpolation algorithms that are in use in financial markets for construction of curves such as forward curves, basis curves, and most importantly, yield curves. In the case of yield curves the issue of bootstrapping is reviewed and how the interpolation algorithm should be intimately connected to the bootstrap itself is discussed. The criterion for inclusion in this survey is that the method has been implemented by a software vendor (or indeed an inhouse developer) as a viable option for yield curve interpolation. As will be seen, many of these methods suffer from problems: they posit unreasonable expections, or are not even necessarily arbitrage free. Moreover, many methods lead one to derive hedging strategies that are not intuitively reasonable. In the last sections, two new interpolation methods (the monotone convex method and the minimal method) are introduced, which it is believed overcome many of the problems highlighted with the other methods discussed in the earlier sections.     

Boundary interpolation in the ball

We solve a boundary interpolation problem in the reproducing kernel Hilbert space of functions analytic in the unit ball of with reproducing kernel 1/(1−∑₁^Nz_kw_k^*). We introduce the notion of Brune factor (or Blaschke–Potapov factor of the third kind) in this setting.     

Constrained variational refinement

A non-uniform, variational refinement scheme is presented for computing piecewise linear curves that minimize a certain discrete energy functional subject to convex constraints on the error from interpolation. Optimality conditions are derived for both the fixed and free-knot problems. These conditions are expressed in terms of jumps in certain (discrete) derivatives. A computational algorithm is given that applies to constraints whose boundaries are either piecewise linear or spherical. The results are applied to closed periodic curves, open curves with various boundary conditions, and (approximate) Hermite interpolation.     

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices.

Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both – systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed.

A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix.     

Equivalence constants for certain matrix norms II

We give an almost complete solution of a problem posed by Klaus and Li [A.-L. Klaus, C.-K. Li, Isometries for the vector (p, q) norm and the induced (p, q) norm, Linear and Multilinear Algebra 38 (1995) 315–332]. Klaus and Li’s problem, which arose during their investigations of isometries, was to relate the Frobenius (or Hilbert–Schmidt) norm of a matrix to various operator norms of that matrix. Our methods are based on earlier work of Feng [B.Q. Feng, Equivalence constants for certain matrix norms, Linear Algebra Appl. 374 (2003) 247–253] and Tonge [A. Tonge, Equivalence constants for matrix norms: a problem of Goldberg, Linear Algebra Appl. 306 (2000) 1–13], but introduce as a new ingredient some techniques developed by Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Bilinear forms bounded in space [p, q], Quart. J. Math. (Oxford) 5 (1934) 241–254].     

曲边三角形及曲边四边形上的混合函数插值格式

一、引言 二元函数在标准三角形上的混合函数插值格式在许多文献,例如,Birkhofft,Barnhill,Gordon及Gregory等的文章中都有讨论。在三角形周边上对高阶偏导数进行插值,而且计算比较简单的是J.A.Gregory的文章中所给出的一种混合函数插值格式。这种格式是由简单函数的线性组合所构成的,而且格式是对称的,因此计算比较简便。但是J.A.Gregory只是对直边三角形给出了格式。本文企图推广Gregory的格式,给出曲边三角形上对高阶偏导数进行插值的插值格式。我们还进一步给出了曲边四边形上     

On Multivariate Interpolation

A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of noncommutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established.     

Cubic spline solution of singularly perturbed boundary value problems with significant first derivatives

We develop a numerical technique for a class of singularly perturbed two-point singular boundary value problems on an uniform mesh using polynomial cubic spline. The scheme derived in this paper is second-order accurate. The resulting linear system of equations has been solved by using a tri-diagonal solver. Numerical results are provided to illustrate the proposed method and to compared with the methods in [R.K. Mohanty, Urvashi Arora, A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives, Appl. Math. Comput., 172 (2006) 531–544; M.K. Kadalbajoo, V.K. Aggarwal, Fitted mesh B-spline method for solving a class of singular singularly perturbed boundary value problems, Int. J. Comput. Math. 82 (2005) 67–76].     

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G¹ Hermite data; however, one could also obtain higher order algorithms.     

Interpolation approximations based on Gauss–Lobatto–Legendre–Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss–Lobatto–Legendre–Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss–Lobatto quadrature for PDEs with Dirichlet boundary conditions.     

On the Convergence of Polynomial Approximation of Rational Functions

This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-point Hermite interpolation converges if all eigenvalue moduli of a certainr×rmatrix are less than 2, whereris the degree of the rational function (or curve), and where the elements of the matrix are expressions involving only the denominator polynomial coefficients (weights) of the rational function (or curve). As a corollary for the special case ofr=1, a necessary and sufficient condition for convergence is also obtained which only involves the roots of the denominator of the rational function and which is shown to be superior to the condition obtained by the traditional remainder theory for polynomial interpolation. For the low degree cases (r=1, 2, and 3), concrete conditions are derived. Application to rational Bernstein–Bézier curves is discussed.     

Bézier曲线降多阶逼近的一种方法

文献[1,2]讨论了Bezier曲线一次降多阶逼近问题,得到了很好的结果.文献[1]利用广义逆矩阵得到不保端点插值的降多阶逼近曲线的控制顶点的表达式.但却没有得到带端点任意阶插值条件的降多阶逼近曲线的控制顶点的表达式.文献[2]得到了带端点任意阶插值的降多阶逼近曲线的控制顶点的解析表达式.本文首先给出两Bezier曲线间距离的定义;然后根据降阶曲线与原曲线间的距离最小,分别得到了用矩阵表示的不保端点插值和保端点任意阶插值的降多阶逼近曲线的控制顶点的显示表达式.所给数值例子显示,用本文方法得到的降多阶逼近曲线对原曲线有很好的逼近效果.     

A general framework for high-accuracy parametric interpolation

In this paper we establish a general framework for so-called parametric, polynomial, interpolation methods for parametric curves. In contrast to traditional methods, which typically approximate the components of the curve separately, parametric methods utilize geometric information (which depends on all the components) about the curve to generate the interpolant. The general framework suggests a multitude of interpolation methods in all space dimensions, and some of these have been studied by other authors as independent methods of approximation. Since the approximation methods are nonlinear, questions of solvability and stability have to be considered. As a special case of a general result, we prove that four points on a planar curve can be interpolated by a quadratic with fourth-order accuracy, if the points are sufficiently close to a point with nonvanishing curvature. We also find that six points on a planar curve can be interpolated by a cubic, with sixth-order accuracy, provided the points are sufficiently close to a point where the curvature does not have a double zero. In space it turns out that five points sufficiently close to a point with nonvanishing torsion can be interpolated by a cubic, with fifth-order accuracy.

     

Constructing C1 triangular patch of degree six by the Boolean sum of an approximation operator and an interpolation operator

A new method to construct C¹ triangular patches which satisfy the given boundary curves and cross-boundary slopes is presented. The Boolean sum of an approximation operator and an interpolation operator is employed to construct the triangular patch. The approximation operator is used to construct a polynomial patch of degree six. The polynomial of degree six affords more freedoms, which makes the approximation operator not only approximate the given boundary interpolation conditions but also have a better approximation precision in the interior of the triangle, so that the triangular patch has a better precision on both the boundary and the interior of the triangular domain. The interpolation operator is utilized to build an interpolation patch which satisfies the given boundary conditions. The Boolean sum of the approximation and interpolation patches forms the triangular patch. Comparison results of the new method with other three methods are given.     

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.Original Russian Text Copyright © 2005 Lipchinskii A. G.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 822–833, July–August, 2005.     

On the theory of differential games in systems with aftereffect : PMM vol. 42, no. 6, 1978 pp.969–977

An encounter-evasion differential game is studied for control systems with aftereffect [1–4]. A feature of the system being analyzed is that it has a time-lag effect with respect to the controls which provides the system with important new peculiarities. Using the investigations in [1–4], conditions for the solvability of the problem are indicated and the required control procedures are constructed.     

Interpolation of Limited and Weakly Compact Operators on Families of Banach Spaces

In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos–Peetre methods of interpolation for finite families.     

Multilevel quasi-interpolation on a sparse grid with the Gaussian

Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in Georgoulis et al. (SIAM J. Sci. Comput. 35, 815–832, 2013), we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time when compared with classical quasi-interpolation. Also, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional positive-weight quadrature formula based on Q-MuSIKSapproximation of the integrand. We present the results of numerical experimentation for both quasi-interpolation and quadrature in high dimensions.