Sparse polynomial interpolation in nonstandard bases. Here one computes the gcd modulo p where p is chosen to be a machine size prime. Closely related to sparse interpolation are two decision problems. Sparse interpolation of multivariate rational functions. We will discuss this further along with approximation in the future. 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. It is important to note that even if the goal polynomial for algorithm dis very sparse the intermediate results can be completely dense. 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.
Beyond the multivariate chebyshev polynomial obtained as tensor products of univariate chebyshev polynomials, the theory of. 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. 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. Number of points needed to determine a sparse univariate. Spielman gave the rst deterministic polynomial time algorithms for sparse interpolation over nite elds with large characteristic. Instead of a univariate in terpolation with the newton algorithm, we now employ the racing algorithm presented in refs. A new algorithm for sparse multivariate polynomial interpolation is presented. The benortiwari algorithm is a sparse polynomial interpolation algorithm and. We are interested in algorithms whose computational complexity is polynomial in t. The basic polynomial algebra subprograms bpas library provides support for arithmetic operations with polynomials on modern computer architectures, in particular hardware accelerators. Subalgorlthms for termwise sparse and dense interpolation are given in figures 2 and 3, respectively.
We discuss two algorithms for sparse interpolation, one due to benor. Modular algorithm for sparse multivariate polynomial. Parallel sparse polynomial interpolation over finite fields cecm. Therefore, the semidefinite programming approach initiated by.
Highdimensional adaptive sparse polynomial interpolation. 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. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the schur basis by barvinok and fomin advances in applied mathematics, 183. Improved sparse multivariate polynomial interpolation. 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. 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. Welcome to the bpas library exposed bpas ring classes.
Fits a spline y spl x of degree k to the provided x, y data. Sparse polynomial interpolation cecm simon fraser university. Slp, with cost softlylinear in the sparsity of the interpolant. The class of linear interpolation also contains spline interpolation. Introduction in this paper, we study a family of decision problems arising from sparse univariate polynomial interpolation through a given set of points. 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. Finally, since both rx and qx are degree m 1 polynomials, and they agree on m points, they are in fact the same polynomial. 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. In general, interpolation problem is essential to solve the following equation.
It races the dense newton interpolation against the benortiwari algorithm4. Lobo2 1department of mathematics, north carolina state university, raleigh, north carolina 276958205. The problem of multivariate polynomial interpolation is hardly new, with early work going back at least to kronecker 17. Canada michael monagan department of mathematics, simon fraser university, burnaby, b. Let f be an nvariate polynomial given by a straightline program, which has a total degree bound d and a term bound t. We present an errorcorrecting interpolation algorithm for a univariate blackbox polynomial that has a sparse representation using chebyshev polynomials as a term basis. The problem of univariate interpolation is the following simple one. A new randomization that improves the kronecker substitution trick by reducing the degree.
Numerically robust interpolation methods for sparse polynomials are important for speed and reliability. 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. While techniques for interpolating dense polynomials have been known for a long time e. For sparse interpolation in terms of multivariate chebyshev polynomials of. Parallel sparse polynomial interpolation over finite fields seyed mohammad mahdi javadi school of computing science, simon fraser university, burnaby, b. Early termination in sparse interpolation algorithms. Symbolicnumeric sparse interpolation of multivariate polynomials. We can construct polynomialfx from its support points by interpolation method. Sparse polynomial interpolation codes and their decoding. Multivariate sparse interpolation using randomized kronecker.
Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. An informationtheoretic lower bound on the complexity of univariate sparse interpolation is. In fact, our interpolation algorithm is general enough to accommodate. Sparse multivariate polynomial interpolation in the basis.
Sparse polynomial interpolation and testing semantic scholar. Revisit sparse polynomial interpolation based on randomized. Parallelization of the sparse modular gcd algorithm for. High dimensional polynomial interpolation on sparse. Early termination in benortiwari sparse interpolation and a hybrid of zippels algorithm erich kaltofen1 wenshin lee1 austin a. In their monte carlo version, it reduces a multivariate 2. To iiluminat e the behavior of the algorithm we describe the steps taken to compute the gcd of two polynomials having three variables. A polynomial ax do aix i is said to be t sparse if at most t of its coefficients ao, al. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. Univariate integration and interpolation operators can be formed on nested grids like chebyshev points u ij. Robust computation methods for sparse interpolation of. We present probabilistic algorithms for the sparse interpolation of an.
Symbolicnumeric sparse interpolation of multivariate. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities in most cases when combining with various univariate. A new deterministic algorithm for sparse multivariate. All other input formats return a multivariate polynomial ring. Experiences with sparse grids and smolyaktype approximations. We use the following notion of sparsity throughout the paper. In 2009, giesbrecht, labahn and lee in 9 presented two new algorithms for sparse interpolation for polynomials with oating point coe cients. Polynomial interpolation is a classical numerical method. Parallel sparse polynomial interpolation over finite fields. Multivariate sparse interpolation using randomized kronecker substitutions andrew arnold cheriton school of computer science university of waterloo waterloo, ontario, canada daniel s.
These methods are generally termed blackbox sparse interpolation. High dimensional polynomial interpolation on sparse grids. Pdf fast parallel algorithms for sparse multivariate. We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. A polynomial with two indeterminates is called a bivariate polynomial. The sparse interpolation problem has received considerable interest over. But by using multivariate sparse polynomial interpolation to recover. Sparse polynomial identity testing is the problem of testing whether a sparse polynomial fis zero from its evaluations.
To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate gcd computation and univariate polynomial interpolation. Sparse polynomial interpolation consists in recovering of a sparse representation of a polynomial p given by. In the dense case, fast fourier transform, interpolation, and divide and conquer techniques can at least asymptoti. Sparse interpolation has applications to learning theory, signal processing, errorcorrecting codes, and symbolic computation. Onedimensional smoothing spline fit to a given set of data points. We consider the problem of interpolating sparse multivariate polynomials from their values. 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. Multiple formulae for polynomial interpolation have been given, notably those of newton and lagrange 1. The benortiwari algorithm is a sparse polynomialinterpolation algorithm and.
The polynomials are assumed to be given by black boxes, i. 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. Using an discrepancy set we can test whether a sparse multivariate polynomial is identically zero, which is a major task in any multivariate interpolation algorithm. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of zippels method and benortiwari algorithm, respectively.
On the decidability of sparse univariate polynomial. 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. A deterministic algorithm for sparse multivariate polynomial interpolation. Sparse interpolation in terms of multivariate chebyshev.
Trigonometric interpolation is used extensively for the numerical fourier analysis of time series and cyclic phenomena in general. Based on the initial work by comer, kaltofen and pernet proc. Typical operations are polynomial multiplication, multipoint evaluation and interpolation, real root isolation for both univariate and multivariate systems. 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. Interpolation terpolatesubpackage for objects used in interpolation. Sparse univariate interpolation over large finite fields. Early termination in benortiwari sparse interpolation.
Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. D the following theorem asserts that the sparsity given by theorem 3. Comput complexity 1 1991 sparse polynomial interpolation 73. Our main application for discrepancy sets is the derandomization of the interpolation algorithm of man92. In practice p is typically represented by dags directed acyclic graphs or spls straight line programs. The method of solution involves the polynomial enumeration techniques of d. 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. In this paper, we study a family of decision problems arising from sparse univariate polynomial interpolation through a given set of points. Early termination in sparse interpolation algorithms erich kaltofena, wenshin leeb adepartment of mathematics, north carolina state university raleigh, north carolina 276958205, u. They reduce the multivariate interpolation problem to the univariate interpolation problem. Issac 2012, we define the sparse polynomial interpolation codes. The authors consider the problem of reconstructing i. Early termination in benortiwari sparse interpolation and a.
In this paper, we propose new deterministic and monte carlo interpolation algorithms for sparse multivariate polynomials represented by straightline programs. In section3we show how multivariate sparse interpolation can be reduced to retrieving the support of certain linear forms on a laurent polynomial ring. Keywords sparse polynomial, interpolation, arithmetic, factorization. For example, if we use the early termination benortiwari algorithm 16 or its. For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical benortiwari algorithm by m.
Sparse computations there is little theoretical knowledge about sparse polynomial algorithms. A simple expression for multivariate lagrange interpolation. Roche computer science department united states naval academy annapolis, maryland, usa issac 2014 kobe, japan july 24, 2014. 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. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. On the decidability of sparse univariate polynomial interpolation.
Interpolation of dense and sparse rational functions and. For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical benortiwari algorithm by. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm. 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 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. We describe efficient new algorithms for these problems. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials. 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. It is studied very well for univariate polynomials. 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. Sparse interpolation is a key part of many algorithms in computer algebra such as polynomial gcd computation 17, 7, over z. The main idea in our algorithm comes from the well.
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. Polynomial interpolation will be addressed shortly in length. 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. This bound is nearly met by pronys 26 algorithm as adapted to the polynomial setting. These solutions are then interpolated to recover the variables eliminated in the simplified problems.
First, we examine a new approach to the standard sparse interpolation problem over large. Fast parallel algorithms for sparse multivariate polynomial. Canada abstract we present a probabilistic algorithm to interpolate a sparse multivariate polynomial. Faster sparse multivariate polynomial interpolation 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. An informationtheoretic lower bound on the complexity of univariate sparse interpolation is, the number of bits used to encode f in 1. Computing sparse gcd of multivariate polynomials via. 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. Multivariate sparse interpolation using randomized. 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. Sparse interpolation of multivariate rational functions core. We show that the sparse polynomial interpolation problem reduces to a discrete superresolution problem on the ndimensional torus. Interpolation and approximation of sparse multivariate polynomials over gf2 ron m.
Efficient computation with sparse and dense polynomials. The rst is a modi cation of the benortiwari algorithm that uses ot probes. 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. Maple, magma, macsyma, singular and mathematica all implement wangs multivariate hensel lifting, which, for sparse polynomials, can be exponential in n. 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.
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