Sparse interpolation of multivariate rational functions

Cuyt, Annie; Lee, Wen-shin

doi:10.1016/j.tcs.2010.11.050

Cited by 34 publications

(52 citation statements)

References 12 publications

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“…However we found that the technique proposed in ref. [26] for sparse rational functions can be easily adapted to the dense case, by combining it with the other techniques we mentioned for univariate rational functions and multivariate polynomials. The resulting algorithm is capable of efficiently reconstructing functions with many non-vanishing terms and depending on several variables, as we will show in the examples.…”

Section: Jhep12(2016)030mentioning

confidence: 99%

Scattering amplitudes over finite fields and multivariate functional reconstruction

Peraro

2016

J. High Energ. Phys.

222

288

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Several problems in computer algebra can be efficiently solved by reducing them to calculations over finite fields. In this paper, we describe an algorithm for the reconstruction of multivariate polynomials and rational functions from their evaluation over finite fields. Calculations over finite fields can in turn be efficiently performed using machine-size integers in statically-typed languages. We then discuss the application of the algorithm to several techniques related to the computation of scattering amplitudes, such as the four-and six-dimensional spinor-helicity formalism, tree-level recursion relations, and multi-loop integrand reduction via generalized unitarity. The method has good efficiency and scales well with the number of variables and the complexity of the problem. As an example combining these techniques, we present the calculation of full analytic expressions for the two-loop five-point on-shell integrands of the maximal cuts of the planar penta-box and the non-planar double-pentagon topologies in Yang-Mills theory, for a complete set of independent helicity configurations.

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Section: Jhep12(2016)030mentioning

confidence: 99%

Scattering amplitudes over finite fields and multivariate functional reconstruction

Peraro

2016

J. High Energ. Phys.

222

288

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“…[24]). One may also reconstruct the fraction using Strassen's removal of divisions approach: see [5] (cf. [11, end of Section 7] and [13, Section 4]).…”

Section: Error-correcting Multivariate Rational Function Interpolationmentioning

confidence: 99%

“…[11, end of Section 7] and [13, Section 4]). [5] recovers the sparse homogeneous parts from highest to lowest degree. Since their algorithm and Algorithm Black Box Numerator and Denominator in [15] are based on univariate Cauchy interpolation, any black box error rate 1/q < 1/2 can be handled by those methods.…”

Section: Error-correcting Multivariate Rational Function Interpolationmentioning

confidence: 99%

Sparse multivariate function recovery from values with noise and outlier errors

Kaltofen

Yang

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). The generalization of the BerlekampWelch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012 [to be submitted]. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers ("cleans up data") through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.

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“…In recent years, many advanced interpolation algorithms have been designed and developed by different researchers [1][2][3][4][5]. Jakobsson et al developed a technique for interpolation with quotients of two radial basis function expansions to approximate functions with poles [6].…”

Section: Introductionmentioning

confidence: 99%

“…Goodman and Meek presented a planar interpolation method using a pair of rational spirals to solve planar and two-point G 2 Hermite interpolation problem [16]. Hussain and Sarfraz used a C 1 piecewise rational cubic function to visualize the data arranged over a rectangular grid [17].…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Interpolation and Other Interpolation Methods Used in Robot Calibrations

Bai¹,

Guo²,

Agbegha³

2012

Journal of Robotics

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A novel interpolation algorithm, fuzzy interpolation, is presented and compared with other popular interpolation methods widely implemented in industrial robots calibrations and manufacturing applications. Different interpolation algorithms have been developed, reported, and implemented in many industrial robot calibrations and manufacturing processes in recent years. Most of them are based on looking for the optimal interpolation trajectories based on some known values on given points around a workspace. However, it is rare to build an optimal interpolation results based on some random noises, and this is one of the most popular topics in industrial testing and measurement applications. The fuzzy interpolation algorithm (FIA) reported in this paper provides a convenient and simple way to solve this problem and offers more accurate interpolation results based on given position or orientation errors that are randomly distributed in real time. This method can be implemented in many industrial applications, such as manipulators measurements and calibrations, industrial automations, and semiconductor manufacturing processes.

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Sparse interpolation of multivariate rational functions

Cited by 34 publications

References 12 publications

Scattering amplitudes over finite fields and multivariate functional reconstruction

Scattering amplitudes over finite fields and multivariate functional reconstruction

Sparse multivariate function recovery from values with noise and outlier errors

Fuzzy Interpolation and Other Interpolation Methods Used in Robot Calibrations

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