Birkhoff Interpolation Problem Research Articles

Solving a Birkhoff Interpolation Problem for (0,1,5) data

Abstract: The following research work deals with a special type of Birkhoff interpolation problem in which we have 3 set of data prescribed in the unit interval I = [0, 1]. Data values are the function value, first derivatives and fifth derivatives prescribed at nodes of the unit interval. We obtained a unique spline interpolating the given data along with the convergence problem.

New fractional pseudospectral methods with accurate convergence rates for fractional differential equations

The main purpose of this paper is to introduce generalized fractional pseudospectral integration and differentiation matrices using a family of fractional interpolants, called fractional Lagrange interpolants. We develop novel approaches to the numerical solution of fractional differential equations with a singular behavior at an end-point. To achieve this goal, we present efficient and stable methods based on three-term recurrence relations, generalized barycentric representations, and Jacobi-Gauss quadrature rules to evaluate the corresponding matrices. In a special case, we prove the equivalence of the proposed fractional pseudospectral methods using a suitable fractional Birkhoff interpolation problem. In fact, the fractional integration matrix yields the stable inverse of the fractional differentiation matrix, and the resulting system is well-conditioned. We develop efficient implementation procedures for providing optimal error estimates with accurate convergence rates for the interpolation operators and the proposed schemes in the $L^{2}$-norm. Some numerical results are given to illustrate the accuracy and performance of the algorithms and the convergence rates.

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

A Class of Birkhoff Type Interpolation and Applications

In this paper a class of Birkhoff type interpolation problem on arbitrary knots is studied. This class of Birkhoff interpolation is a generalization of a Birkhoff problem, which is included in one of the articles in the referenced list. Because the behavior of first and second order derivatives are more important than the higher order, therefore, for a basic function, a better approximation will be obtained than higher order derivatives. The low error value calculated for some functions using this class represents the relative importance of this class of Birkhoff interpolation problem. One of the uses of this class from the Birkhoff polynomials is to review and propose a quadrature formula for this class. One of the ways to reduce the error for interpolation polynomials of this class of Birkhoff interpolation problem is to select the Chebyshev polynomial zeros as nodal points, which can be used to solve numerical differential equations.

Reconstruction of a function from Hermite–Birkhoff data

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about some function and its derivatives. A Birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space. In this paper we split up the initial problem in subproblems having a unique polynomial solution and use multinode rational basis functions in order to obtain a global interpolant.

Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds

The Hermite–Birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered. Each proposed interpolant is expressed as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions depend on the geodesic distance, are orthonormal with respect to the point-evaluation functionals, and have all derivatives equal zero up to a certain order at the interpolation points. A remarkable feature of such interpolants, which belong to the class of partition of unity methods, is that their construction does not require solving linear systems. Numerical tests are given to show the interpolation performance.

English

In a previous paper, we derived necessary and sufficient conditions for the invertibility of square submatrices of the Pascal upper triangular matrix. To do so, we established a connection with the two-point Birkhoff interpolation problem. In this paper, we extend this result by deriving a formula for the rank of submatrices of the Pascal matrix. Our formula works for both square and non-square submatrices. We also provide bases for the row and column spaces of these submatrices. Further, we apply our result to one-point lacunary polynomial approximation.

A New Collocation Scheme Using Non-polynomial Basis Functions

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: \({\mathscr {L}}_\lambda ={d^2}/{dx^2}-\lambda ^2 \) with constant \(\lambda .\) With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator \({\mathscr {L}}_\lambda \) being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.

Characterizing the Codimension of Zero Singularities for Time-Delay Systems

The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author's knowledge, the bounds of such a multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters' algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.

On the unique minimal monomial basis of Birkhoff interpolation problem

This paper studies the minimal monomial basis of the n-variable Birkhoff interpolation problem. First, the authors give a fast B-Lex algorithm which has an explicit geometric interpretation to compute the minimal monomial interpolation basis under lexicographic order and the algorithm is in fact a generalization of lex game algorithm. In practice, people usually desire the lowest degree interpolation polynomial, so the interpolation problems need to be solved under, for example, graded monomial order instead of lexicographic order. However, there barely exist fast algorithms for the nonlexicographic order problem. Hence, the authors in addition provide a criterion to determine whether an n-variable Birkhoff interpolation problem has unique minimal monomial basis, which means it owns the same minimal monomial basis w.r.t. arbitrary monomial order. Thus, for problems in this case, the authors can easily get the minimal monomial basis with little computation cost w.r.t. arbitrary monomial order by using our fast B-Lex algorithm.

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order µ ? ( 0 , 1 ) to compute that of any order k + µ with integer k ? 0 , while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order µ ? ( 0 , 1 ) . Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at interior JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can be solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, is essential for our algorithm development.

Stable monomial basis for multivariate Birkhoff interpolation problems

For a given node set Z and the corresponding interpolation conditions, one of the key problems in multivariate Birkhoff interpolation is determining a monomial basis that spans the interpolation space. However, a slight perturbation of the node set will result in a different monomial basis. We propose a more general multivariate Birkhoff interpolation scheme and provide a numerical algorithm that can determine the stable monomial basis to ensure that the interpolation polynomial always exists in the space spanned by the monomial basis for any perturbation of the node set within a given error ε.

Spline methods for a Birkhoff interpolation problem

In this paper, we study two new spline methods for the Birkhoff interpolation problem proposed in Costabile and Longo (Calcolo 47:49---63, 2010). Our methods are based on cubic spline and quintic spline respectively. The new methods are easy to be implemented. They are effective not only in approximating the original function but also in approximating its derivatives. The respective remainders and error bounds are given. Numerical tests are performed. Numerical results confirm the theoretical analysis and show that our methods are very practical.

Solution of a Birkhoff Interpolation Problem by a Special Spline Function

In this paper we have discussed a special lacunary interpolation problem in which the function values, first derivatives at the nodes and the third derivatives at any point λ (0 in between the nodes are prescribed. We have solved the unique existence and convergence problems, using spline functions. As this holds for any λ (0 we named it a generalized problem. General Terms Your general terms must be any term which can be used for general classification of the submitted material such as Pattern Recognition, Security, Algorithms et. al.

A fast algorithm for the multivariate Birkhoff interpolation problem

Multivariate Birkhoff interpolation is the most complicated polynomial interpolation problem and the theory about it is far from systematic and complete. In this paper we derive an Algorithm B-MB (Birkhoff-Monomial Basis) and prove B-MB giving the minimal interpolation monomial basis w.r.t. the lexicographical order of the multivariate Birkhoff problem. This algorithm is the generalization of Algorithm MB in [L. Cerlinco, M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math. 139 (1995) 73–87] which is a well known fast algorithm used to compute the interpolation monomial basis of the Hermite interpolation problem.

The proper interpolation space for multivariate Birkhoff interpolation

Multivariate Birkhoff interpolation problem has many important applications, such as in finite element method. In this paper two algorithms are given to compute the basis of the minimal interpolation space and the lower interpolation space respectively for an arbitrary given node set and the corresponding interpolation conditions on each node. We can get the monomial basis, Newton-type basis as well as Lagrange-type basis. The interpolation polynomial can be derived from the basis directly.

A Birkhoff interpolation problem and application

A Birkhoff interpolation problem is studied. The explicit solution and estimation of remainder is given. The cases of equidistant and Chebyshev nodes are considered. This problem is of interest in collocation methods of boundary value problem (Costabile and Napoli in Far East J. Appl. Math. 25(3):289–305, 2006; Costabile and Longo in Applied and Industrial Mathematics in Italy III, 2009, in press). In addition an application to quadrature is given.

Order regularity of two-node Birkhoff interpolation with lacunary polynomials

In this short work we study the existence and uniqueness of solution for some Birkhoff interpolation problems with lacunary polynomials. First we solve the one-node problem; next we solve the two-node problem in the restricted case where one of the nodes is null.

Hierarchical Threshold Secret Sharing

We consider the problem of threshold secret sharing in groups with hierarchical structure. In such settings, the secret is shared among a group of participants that is partitioned into levels. The access structure is then determined by a sequence of threshold requirements: a subset of participants is authorized if it has at least k0 0 members from the highest level, as well as at least k1 > k0 members from the two highest levels and so forth. Such problems may occur in settings where the participants differ in their authority or level of confidence and the presence of higher level participants is imperative to allow the recovery of the common secret. Even though secret sharing in hierarchical groups has been studied extensively in the past, none of the existing solutions addresses the simple setting where, say, a bank transfer should be signed by three employees, at least one of whom must be a department manager. We present a perfect secret sharing scheme for this problem that, unlike most secret sharing schemes that are suitable for hierarchical structures, is ideal. As in Shamir's scheme, the secret is represented as the free coefficient of some polynomial. The novelty of our scheme is the usage of polynomial derivatives in order to generate lesser shares for participants of lower levels. Consequently, our scheme uses Birkhoff interpolation, i.e., the construction of a polynomial according to an unstructured set of point and derivative values. A substantial part of our discussion is dedicated to the question of how to assign identities to the participants from the underlying finite field so that the resulting Birkhoff interpolation problem will be well posed. In addition, we devise an ideal and efficient secret sharing scheme for the closely related hierarchical threshold access structures that were studied by Simmons and Brickell.

A Regularized Solution to the Birkhoff Interpolation Problem

what combination of initial and terminal values suffice to construct a unique solution. This generalizes the concepts of interpolation that were then current. Lagrange interpolation (more properly Warring interpolation, [15, 6]), and the most general Hermite interpolation are special cases if only two time points are considered. See [3] for the most readable account of several different interpolation schemes. The most notable thing about the problem posed by Birkhoff is that there is not always a solution. For example, if n = 2 and we are given u′(0) = 1 and u′(1) = 2 then there is no solution. The most general version of the problem, as posed by Birkhoff, is the following: Let 0 = t0 < t1 < · · · < tN = 1. What combination of n functional values and higher derivatives at the points ti suffices to give a unique solution to the problem. It is this problem that has been studied in the literature for almost one hundred years. Primary references are [12, 8]. Another problem that has been studied and will be shown to be related to Birkhoff interpolation is the study of optimal digital to analog conversion using linear system theory. Two versions of this have been studied, interpolation and smoothing, but