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.     

A parallel method for fast and practical high-order newton interpolation

We present parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. Forn+1 given input pairs, the proposed interpolation algorithm requires only 2 [log(n+1)]+2 parallel arithmetic steps and circuit sizeO(n ²), reducing the best known circuit size for parallel interpolation by a factor of logn. The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform. We report on numerical experiments comparing this with other serial and parallel algorithms. The experiments indicate that the method can be very useful for very high-order interpolation, which is made possible for special sets of interpolation nodes.Supported in part by the National Science Foundation under Grant No. NSF DCR-8603722.Supported by the National Science Foundation under Grants No. US NSF MIP-8410110, US NSF DCR85-09970, and US NSF CCR-8717942 and AT&T under Grant AT&T AFFL67Sameh.     

A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

A summability method for the arithmetic Fourier transform

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients, which is suitable for parallel processing and in which there are no multiplications by complex exponentials. This is accomplished by the use of the Möbius function and Möbius inversion. However, the algorithm does require the evaluation of the function at an array of irregularly spaced points. In the case that the function has been sampled at regularly spaced points, interpolation is used at the intermediate points of the array. Generally theAFT is most effective when used to calculate the Fourier cosine coefficients of an even function.In this paper a summability method is used to derive a modification of theAFT algorithm. The proof of the modification is quite independent of theAFT itself and involves a summation by primes. One advantage of the new algorithm is that with a suitable sampling scheme low order Fourier coefficients may be calculated without interpolation.     

Computing border bases

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zero-dimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering σ and hence compute a border basis that contains the reduced σ-Gröbner basis. We show an example in which this computation actually has advantages over Buchberger's algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.     

A reliable modification of the cross rule for rational Hermite interpolation

Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule.In this note we introduce a simple recursive algorithm for computation of the values of neighbours of the singular block. This allows to compute entries in the rational interpolation table along antidiagonals even in the presence of singular blocks. Moreover, in the case of non-square singular blocks, we discuss a facility to monitor the stability.Dedicated to Professor G. Mühlbach on the occasion of his 50th birthday     

Biorthogonal Polynomial Bases and Vandermonde-like Matrices

This article considers a family of Gram matrices of pairs of bases of a finite dimensional vector space of polynomials with respect to certain indefinite inner products. Such a family includes all the generalized confluent Vandermonde matrices relative to any polynomial basis, like the Chebyshev-Vandermonde matrices, for example. Using the biorthogonality of pairs of bases with respect to a divided difference functional, properties of matrices and functionals, as well as interpolation formulas are obtained. I show that the computation of the inverse of a Vandermonde-like matrix is essentially equivalent to the computation of the partial fractions decompositions of a set of rational functions with a common denominator. I also explain why the various Chebyshev-Vandermonde matrices are the simplest generalizations of the classic Vandermonde matrices and describe a simple algorithm for the computation of their inverses, which requires a number of multiplications of the order of 3N².     

Fast computation of determinants of Bézout matrices and application to curve implicitization

When using bivariate polynomial interpolation for computing the implicit equation of a rational plane algebraic curve given by its parametric equations, the generation of the interpolation data is the most costly of the two stages of the process. In this work a new way of generating those interpolation data with less computational cost is presented. The method is based on an efficient computation of the determinants of certain constant Bézout matrices.     

Algorithms for solving Hermite interpolation problems using the Fast Fourier Transform

We present a method for computing the Hermite interpolation polynomial based on equally spaced nodes on the unit circle with an arbitrary number of derivatives in the case of algebraic and Laurent polynomials. It is an adaptation of the method of the Fast Fourier Transform (FFT) for this type of problems with the following characteristics: easy computation, small number of operations and easy implementation.In the second part of the paper we adapt the algorithm for computing the Hermite interpolation polynomial based on the nodes of the Tchebycheff polynomials and we also study Hermite trigonometric interpolation problems.     

Minimum cycle bases of graphs on surfaces

In this paper we study the cycle base structures of embedded graphs on surfaces. We first give a sufficient and necessary condition for a set of facial cycles to be contained in a minimum cycle base (or MCB in short) and then set up a 1-1 correspondence between the set of MCBs and the set of collections of nonseparating cycles which are in general positions on surfaces and are of shortest total length. This provides a way to enumerate MCBs in a graph via nonseparating cycles. In particular, some known results such as P.F. Stadler's work on Halin graphs [Minimum cycle bases of Halin graphs, J. Graph Theory 43 (2003) 150-155] and Leydold and Stadler's results on outer-planar graphs [Minimum cycle bases of outerplanar graphs, Electronic J. Combin. 5(16) (1998) 14] are concluded. As applications, the number of MCBs in some types of graphs embedded in lower surfaces (with arbitrarily high genera) is found. Finally, we present an interpolation theorem for the number of one-sided cycles contained in MCB of an embedded graph.     

Computations in differential and difference modules

Constructive methods based on the Gröbner bases theory have been used many times in commutative algebra over the past 20 years, in particular, they allow the computation of such important invariants of manifolds given by systems of algebraic equations as their Hilbert polynomials. In differential and difference algebra, the analogous roles play characteristic sets.In this paper, algorithms for computations in differential and difference modules, which allow for the computation of characteristic sets (Gröbner bases) in differential, difference, and polynomial modules and differential (difference) dimension polynomials, are described. The algorithms are implemented in the algorithmic language REFAL.     

Computing near-best fixed pole rational interpolants

We study rational interpolation formulas on the interval [−1,1] for a given set of real or complex conjugate poles outside this interval. Interpolation points which are near-best in a Chebyshev sense were derived in earlier work. The present paper discusses several computation aspects of the interpolation points and the corresponding interpolants. We also study a related set of points (that includes the end points), which is more suitable for applications in rational spectral methods. Some examples are given at the end of this paper.     

Bases for kernel-based spaces

Since it is well-known (De Marchi and Schaback (2001) [4]) that standard bases of kernel translates are badly conditioned while the interpolation itself is not unstable in function space, this paper surveys the choices of other bases. All data-dependent bases turn out to be defined via a factorization of the kernel matrix defined by these data, and a discussion of various matrix factorizations (e.g. Cholesky, QR, SVD) provides a variety of different bases with different properties. Special attention is given to duality, stability, orthogonality, adaptivity, and computational efficiency. The “Newton” basis arising from a pivoted Cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the “native” Hilbert space of the kernel. Efficient adaptive algorithms for calculating the Newton basis along the lines of orthogonal matching pursuit conclude the paper.     

On the use of Kronecker's algorithm in the generalized rational interpolation problem

Kronecker's algorithm can be used to solve the generalized rational interpolation problem. In order to present the algorithm, rational forms are used here instead of too restrictive rational fractions. The proposed algorithm is reliable as soon as the functionals that characterize the problem satisfy two precise conditions. These conditions are fulfilled in the modified Hermite rational interpolation problem and, as a consequence, in the special case of the Cauchy problem and of the Padé approximation problem. This reliability covers two properties: on one hand, every rational form resulting from the algorithm is a solution of the problem whereas, on the other hand, every solution of the problem is found by the algorithm (with the exception of a possible reduction of the rational form). However, if the algorithm yields a non-reduced rational form, then the corresponding rational fraction is not a solution of the problem.     

Newton basis for multivariate Birkhoff interpolation

Multivariate Birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far. In this paper, we introduce a special new type of multivariate Birkhoff interpolation and present a Newton paradigm for it. Using the algorithms proposed in this paper, we can construct a Hermite system for any interpolation problem of this type and then obtain a Newton basis for the problem w.r.t. the Hermite system.     

Multivariate polynomial interpolation under projectivities part I: lagrange and newton interpolation formulas

In this note interpolation by real polynomials of several real variables is treated. Existence and unicity of the interpolant for knot systems being the perspective images of certain regular knot systems is discussed. Moreover, for such systems a Newton interpolation formula is derived allowing a recursive computation of the interpolant via multivariate divided differences. A numerical example is given.Partially supported by CICYT Res. Grant PS 87/0060 and by a Europe Travel Grant CAI-CONAI, Spain, 1988.     

A fast algorithm for scalar Nevanlinna-Pick interpolation

Summary In this paper, we derive a fast algorithm for the scalar Nevanlinna-Pick interpolation. Givenn distinct pointsz _i in the unit disk |z|<1 andn complex numbersw _i satisfying the Pick condition for 1in, the new Nevanlinna-Pick interpolation algorithm requires onlyO(n) arithmetic operations to evaluate the interpolatory rational function at a particular value ofz, in contrast to the classical algorithm which requiresO(n ²) arithmetic operations to compute the so-called Fenyves array (which is inherent in the classical algorithm). The new algorithm bypasses the generation of the Fenyves array to speed up the computation, and also yields a parallel scheme requiring onlyO(logn) arithmetic operations on a concurrent-read, exclusive-write parallel random access machine withn processors. We must remark that the rational functionf(z) computed by the new algorithm is one degree higher than the function computed by the classical algorithm.Supported in part by the US Army Research Office Grant No. DAAL03-91-G-0106     

A Criterion for Bases of the Ring of Symmetric Functions

We give a criterion for bases of the ring of symmetric functions in n indeterminates over a commutative ring R with identity. A related algorithm is presented in the last section. Received April 13, 2004     

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

A new formal approach to the rational interpolation problem

Summary An elegant and fast recursive algorithm is developed to solve the rational interpolation problem in a complementary way compared to existing methods. We allow confluent interpolation points, poles, and infinity as one of the interpolation points. Not only one specific solution is given but a nice parametrization of all solutions. We also give a linear algebra interpretation of the problem showing that our algorithm can also be used to handle a specific class of structured matrices.