Multivariate polynomial interpolation is a component in recent approximate multivariate factorization algorithms see 8, 12, 31 and in the decomposition of approximately speci ed polynomial systems 32, 33. It is studied very well for univariate polynomials. Slp, with cost softlylinear in the sparsity of the interpolant. A polynomial with two indeterminates is called a bivariate polynomial. The sparse interpolation problem has received considerable interest over. Revisit sparse polynomial interpolation based on randomized. On the decidability of sparse univariate polynomial. D the following theorem asserts that the sparsity given by theorem 3. Early termination in benortiwari sparse interpolation and a hybrid of zippels algorithm erich kaltofen1 wenshin lee1 austin a. The problem of multivariate polynomial interpolation is hardly new, with early work going back at least to kronecker 17. Let f be an nvariate polynomial given by a straightline program, which has a total degree bound d and a term bound t. In the dense case, fast fourier transform, interpolation, and divide and conquer techniques can at least asymptoti. Implementations and illustrated examples the combination of different dense algorithms for univariate rational interpolation and sparse polynomial interpolation leads to a class of sparse multivariate rational interpolation algorithms.
All other input formats return a multivariate polynomial ring. Sparsity is considered in a di erent, bitcomplexity model, using arbitrary precision arithmetic by mansour 28, who presents a randomized algorithm for interpolating a sparse integer polynomial from limited precision interpolation points wherein bits of guaranteed accuracy can be extracted at. Request pdf sparse interpolation in terms of multivariate chebyshev polynomials sparse interpolation refers to the exact recovery of a function as a short linear combination of basis. Computing sparse gcd of multivariate polynomials via.
It is important to note that even if the goal polynomial for algorithm dis very sparse the intermediate results can be completely dense. The benortiwari algorithm is a sparse polynomial interpolation algorithm and. While techniques for interpolating dense polynomials have been known for a long time e. Sparse interpolation of multivariate rational functions core. In section3we show how multivariate sparse interpolation can be reduced to retrieving the support of certain linear forms on a laurent polynomial ring. In 2009, giesbrecht, labahn and lee in 9 presented two new algorithms for sparse interpolation for polynomials with oating point coe cients.
High dimensional polynomial interpolation on sparse. Multivariate sparse interpolation using randomized. Closely related to sparse interpolation are two decision problems. Sparse interpolation is a key part of many algorithms in computer algebra such as polynomial gcd computation 17, 7, over z. Roche computer science department united states naval academy annapolis, maryland, usa issac 2014 kobe, japan july 24, 2014. Univariate integration and interpolation operators can be formed on nested grids like chebyshev points u ij. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. In their monte carlo version, it reduces a multivariate 2. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials.
Sparse multivariate polynomial interpolation in the basis. It is a multimodular extension of the benor and tiwari algorithm, and is designed to be a practical method to construct symbolic formulas from numeric data produced by vector or massivelyparallel processors. Parallel sparse polynomial interpolation over finite fields cecm. Faster interpolation algorithms for sparse multivariate. We consider the problem of interpolating sparse multivariate polynomials from their values. To iiluminat e the behavior of the algorithm we describe the steps taken to compute the gcd of two polynomials having three variables.
We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. A new randomization that improves the kronecker substitution trick by reducing the degree. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the schur basis by barvinok and fomin advances in applied mathematics, 183. Maple, magma, macsyma, singular and mathematica all implement wangs multivariate hensel lifting, which, for sparse polynomials, can be exponential in n. A simple expression for multivariate lagrange interpolation. An informationtheoretic lower bound on the complexity of univariate sparse interpolation is. Sparse computations there is little theoretical knowledge about sparse polynomial algorithms. We present an errorcorrecting interpolation algorithm for a univariate blackbox polynomial that has a sparse representation using chebyshev polynomials as a term basis.
A new algorithm for sparse multivariate polynomial interpolation is presented. We can construct polynomialfx from its support points by interpolation method. A new deterministic algorithm for sparse multivariate. This bound is nearly met by pronys 26 algorithm as adapted to the polynomial setting. For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical benortiwari algorithm by. Polynomial interpolation is an important component in approximate multivariate factorization algorithms 7, and sparse interpolation can often be used to speed up the procedure when there are more than two variables. Efficient computation with sparse and dense polynomials. Based on the initial work by comer, kaltofen and pernet proc.
Here one computes the gcd modulo p where p is chosen to be a machine size prime. In 10, lakshman and saunders solved the interpolation problem from multiple derivatives for the case of univariate polynomials that admit a sparse representation in the standard power basis. In fact, our interpolation algorithm is general enough to accommodate. Subalgorlthms for termwise sparse and dense interpolation are given in figures 2 and 3, respectively. We use the following notion of sparsity throughout the paper. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities in most cases when combining with various univariate. Interpolation of dense and sparse rational functions and. The full sparse modular algorithm alter nates between stages of dense interpolations using algorithm d abovc, and stages of sparse interpolation in algorithm s below.
For sparse interpolation in terms of multivariate chebyshev polynomials of. For sparse interpolation in terms of multivariate chebyshev polynomials of the rst and second kind, we show how we can consider the restriction of the linear form to the ring of in. Parallelization of the sparse modular gcd algorithm for. The resulting univariate polynomial often has a significantly lower degree than the kronecker substitution polynomial, at the expense of a small number of term collisions. Sparse univariate interpolation over large finite fields. Highdimensional adaptive sparse polynomial interpolation and. The polynomials are assumed to be given by black boxes, i. We consider the problem of determining whether or not there exists a sparse univariate polynomial that interpolates a given setsx i,y i of points. An interpolation algorithm for sparse polynomials over. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Early termination in benortiwari sparse interpolation. Symbolicnumeric sparse interpolation of multivariate. We discuss two algorithms for sparse interpolation, one due to benor. Sparse interpolation the next two chapters examine methods to determine the sparse representation of an unknown polynomial, given only a way to evaluate the polynomial at chosen points.
When numerator and denominator are at most of degree d, then the number of possible terms in f is odn and explodes exponentially as the number of variables increases. Polynomial interpolation is a classical numerical method. Pdf fast parallel algorithms for sparse multivariate. For example, if we use the early termination benortiwari algorithm 16 or its. A somewhat related paper is what can and cant we do with sparse polynomials, which references on the bitcomplexity of sparse polynomial and series multiplication. Multivariate sparse interpolation using randomized kronecker substitutions andrew arnold cheriton school of computer science university of waterloo waterloo, ontario, canada daniel s. Combining our reduction method and the modified univariate benor and tiwari algorithm, we give a monte carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of blackbox polynomial over finite fields. Early termination in sparse interpolation algorithms.
The rst is a modi cation of the benortiwari algorithm that uses ot probes. Sparse polynomial identity testing is the problem of testing whether a sparse polynomial fis zero from its evaluations. For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical benortiwari algorithm by m. As listed below, this subpackage contains spline functions and classes, onedimensional and multidimensional univariate and multivariate interpolation classes, lagrange and taylor polynomial interpolators, and wrappers for fitpack and dfitpack functions. Polynomial interpolation will be addressed shortly in length.
Welcome to the bpas library exposed bpas ring classes. Sparse polynomial interpolation consists in recovering of a sparse representation of a polynomial p given by a blackbox program which computes values of p at as many points as necessary. In practice p is typically represented by dags directed acyclic graphs or spls straight line programs. Robust computation methods for sparse interpolation of. We present probabilistic algorithms for the sparse interpolation of an. But by using multivariate sparse polynomial interpolation to recover. Early termination in benortiwari sparse interpolation and a. Sparse interpolation in terms of multivariate chebyshev polynomials 1 introduction the goal of sparse interpolation is the exact recovery of a function as a short linear combination of elements in a speci c set of functions, usually of in nite cardinality, from a limited number of evaluations, or other functional values. Parallel sparse polynomial interpolation over finite fields seyed mohammad mahdi javadi school of computing science, simon fraser university, burnaby, b.
The basic polynomial algebra subprograms bpas library provides support for arithmetic operations with polynomials on modern computer architectures, in particular hardware accelerators. In this paper, we study a family of decision problems arising from sparse univariate polynomial interpolation through a given set of points. We are interested in algorithms whose computational complexity is polynomial in t. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of zippels method and benortiwari algorithm, respectively. The method of solution involves the polynomial enumeration techniques of d. Multivariate sparse interpolation using randomized kronecker. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate gcd computation and univariate polynomial interpolation. Its not identical because the support is known, and because in this context i dont need to actually do the interpolation, only to prove that there is a unique solution. In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zeros that are sparse in a the pochhammer basis, or b the chebyshev basis. The benortiwari algorithm is a sparse polynomialinterpolation algorithm and. The classical pronys method for exponential sums from 1795 which can be regarded as the genesis of sparse interpolation was later applied to sparse interpolation over the integers 4. These methods are generally termed blackbox sparse interpolation. Comput complexity 1 1991 sparse polynomial interpolation 73.
Modular algorithm for sparse multivariate polynomial. Multiple formulae for polynomial interpolation have been given, notably those of newton and lagrange 1. Onedimensional smoothing spline fit to a given set of data points. Sparse interpolation has applications to learning theory, signal processing, errorcorrecting codes, and symbolic computation. On the decidability of sparse univariate polynomial interpolation. As in the rest of this paper, define n, t, and d to be respectively the number of variables and known bounds on the number of terms and degree of the unknown polynomial. First, we examine a new approach to the standard sparse interpolation problem over large.
Canada michael monagan department of mathematics, simon fraser university, burnaby, b. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm. In general, interpolation problem is essential to solve the following equation. Sparse interpolation in terms of multivariate chebyshev. Canada abstract we present a probabilistic algorithm to interpolate a sparse multivariate polynomial. High dimensional polynomial interpolation on sparse grids. Trigonometric interpolation is used extensively for the numerical fourier analysis of time series and cyclic phenomena in general. An interpolation algorithm for sparse polynomials over z m kai werther august 9, 1994 abstract we give a quasi polynomial time algorithm for the problem of interpolating sparse polynomials over integer residue class rings z m from their values given by a black box. Experiences with sparse grids and smolyaktype approximations. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial.
Finally, since both rx and qx are degree m 1 polynomials, and they agree on m points, they are in fact the same polynomial. Beyond the multivariate chebyshev polynomial obtained as tensor products of univariate chebyshev polynomials, the theory of. Sparse polynomial interpolation codes and their decoding. Our main application for discrepancy sets is the derandomization of the interpolation algorithm of man92. Fast parallel algorithms for sparse multivariate polynomial.
In this section, we consider the interpolation of a univariate polynomial f with deg f sparse univariate polynomial that interpolates a given setsx i,y i of points. Fits a spline y spl x of degree k to the provided x, y data. Interpolation terpolatesubpackage for objects used in interpolation. Highdimensional adaptive sparse polynomial interpolation and applications to parametric pdes abdellah chkifa, albert cohen, christoph schwab august 2, 2012 abstract we consider the problem of lagrange polynomial interpolation in high or countably in nite dimension, motivated by the fast computation of solution to parametricstochastic pdes. We describe efficient new algorithms for these problems. An informationtheoretic lower bound on the complexity of univariate sparse interpolation is, the number of bits used to encode f in 1.
They reduce the multivariate interpolation problem to the univariate interpolation problem. Highdimensional adaptive sparse polynomial interpolation. Parallel sparse polynomial interpolation over finite fields. Introduction in this paper, we study a family of decision problems arising from sparse univariate polynomial interpolation through a given set of points.
The authors consider the problem of reconstructing i. We will discuss this further along with approximation in the future. Sparse polynomial interpolation and testing semantic scholar. Sparse polynomial interpolation consists in recovering of a sparse representation of a polynomial p given by. Symbolicnumeric sparse interpolation of multivariate polynomials.
Issac 2012, we define the sparse polynomial interpolation codes. In contrast, the interpolation of rational functions is a younger eld of research dating back to thieles formula from the beginning of the 20th century1. Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. A deterministic algorithm for sparse multivariate polynomial interpolation. The problem of univariate interpolation is the following simple one. Lobo2 1department of mathematics, north carolina state university, raleigh, north carolina 276958205. The main idea in our algorithm comes from the well. Instead of a univariate in terpolation with the newton algorithm, we now employ the racing algorithm presented in refs. Interpolation and approximation of sparse multivariate polynomials over gf2 ron m. We show that the sparse polynomial interpolation problem reduces to a discrete superresolution problem on the ndimensional torus. Improved sparse multivariate polynomial interpolation. For a tsparse and nvariate polynomial f with a degree bound d and a term bound t, we show that at least half of the terms of f can be recovered from the univariate polynomials obtained from f by. It races the dense newton interpolation against the benortiwari algorithm4.
Spielman gave the rst deterministic polynomial time algorithms for sparse interpolation over nite elds with large characteristic. Algorithm, decidability, interpolation, sparse polynomial, complexity. Sparse polynomial interpolation in nonstandard bases. Typical operations are polynomial multiplication, multipoint evaluation and interpolation, real root isolation for both univariate and multivariate systems. Number of points needed to determine a sparse univariate. Therefore, the semidefinite programming approach initiated by.
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