
@online{github,
  author = {<NAME>},
  title = {multivar_horner},
  year = 2018,
  url = {https://github.com/MrMinimal64/multivar_horner},
  urldate = {2020-05-28},
}

@article{numpy,
 author = {<NAME> and <NAME> <NAME>},
 title = {The NumPy Array: A Structure for Efficient Numerical Computation},
 journal = {Computing in Science and Engg.},
 issue_date = {March 2011},
 volume = {13},
 number = {2},
 month = mar,
 year = {2011},
 issn = {1521-9615},
 pages = {22--30},
 numpages = {9},
 url = {http://dx.doi.org/10.1109/MCSE.2011.37},
 doi = {10.1109/MCSE.2011.37},
 acmid = {1957466},
 publisher = {IEEE Educational Activities Department},
 address = {Piscataway, NJ, USA},
 keywords = {NumPy, Python, Python, NumPy, scientific programming, numerical computations, programming libraries, numerical computations, programming libraries, scientific programming},
}

@inproceedings{numba,
author = {<NAME> and <NAME> and <NAME>},
title = {Numba: A LLVM-Based Python JIT Compiler},
year = {2015},
isbn = {9781450340052},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/2833157.2833162},
doi = {10.1145/2833157.2833162},
booktitle = {Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC},
articleno = {Article 7},
numpages = {6},
keywords = {compiler, LLVM, Python},
location = {Austin, Texas},
series = {LLVM ’15}
}


@article{horner1819xxi,
  title={XXI. A new method of solving numerical equations of all orders, by continuous approximation},
  author={<NAME>},
  journal={Philosophical Transactions of the Royal Society of London},
  number={109},
  pages={308--335},
  year={1819},
  publisher={The Royal Society London}
}

@article{hart1968formal,
  title={A formal basis for the heuristic determination of minimum cost paths},
  author={<NAME> and Nilsson, <NAME> and <NAME>},
  journal={IEEE transactions on Systems Science and Cybernetics},
  volume={4},
  number={2},
  pages={100--107},
  year={1968},
  publisher={IEEE}
}

@book{neumaier2001introduction,
  title={Introduction to numerical analysis},
  author={<NAME>},
  year={2001},
  publisher={Cambridge University Press}
}


@article{michelfeitresponse,
  title={A Response Surface Method based on Multivariate Newton Interpolation},
  author={<NAME>},
  year={2019},
  url = {https://mosaic.mpi-cbg.de/docs/Michelfeit2019.pdf},
  urldate = {2020-05-28},
}

@article{Hecht1,
  title={Multivariate {N}ewton Interpolation},
  author={<NAME> Hoffmann, <NAME> and Cheeseman, <NAME> and Sbalzarini, <NAME>},
  journal={arXiv preprint arXiv:1812.04256},
  year={2018},
}


@inproceedings{Hecht2,
	Address = {London, UK},
	Author = {<NAME> Sbalzarini, <NAME>.},
	Booktitle = {Intelligent Computing. Proc. 2018 IEEE Computing Conf., Vol. 2,},
	Date-Added = {2018-11-05 11:04:48 +0100},
	Date-Modified = {2018-11-05 11:11:10 +0100},
	Editor = {<NAME>. and <NAME>. and <NAME>.},
	Keywords = {interpolation, Fourier transform, high dimensions},
	Pages = {53--75},
	Publisher = {Springer Nature},
	Series = {Advances in Intelligent Systems and Computing},
	Title = {Fast Interpolation and {F}ourier Transform in High-Dimensional Spaces},
	Volume = {857},
	Year = {2018},
}

@article{trefethen2017multivariate,
  title={Multivariate polynomial approximation in the hypercube},
  author={<NAME>},
  journal={Proceedings of the American Mathematical Society},
  volume={145},
  number={11},
  pages={4837--4844},
  year={2017}
}

@article{greedyHorner,
  author    = {<NAME> and <NAME>},
  title     = {Greedy algorithms for optimizing multivariate {H}orner schemes},
  year      = {2004},
  publisher = {Association for Computing Machinery (ACM)},
}


@article{pena2000multivariate,
  title={On the multivariate {H}orner scheme},
  author={Pe{\~n}a, <NAME> and <NAME>},
  journal={SIAM journal on numerical analysis},
  volume={37},
  number={4},
  pages={1186--1197},
  year={2000},
  publisher={SIAM},
  abstract={We present an efficient version of the Horner scheme for the evaluation of multivariate
polynomials and study its stability properties. In particular, we show that the backward error of
evaluation is bounded by a quantity that is linear in the total degree of the polynomial, which itself
is usually significantly smaller than the number of operations involved in the evaluation process.
Moreover, we show that the same type of estimates holds also for the forward error, provided that
the polynomial basis is chosen suitably.},
}

@Article{pena2000multivariate2,
  author    = {Pe{\~n}a, <NAME> and <NAME>},
  title     = {On the multivariate {H}orner scheme {II}: running error analysis},
  journal   = {Computing},
  year      = {2000},
  volume    = {65},
  number    = {4},
  pages     = {313--322},
  abstract  = {As an extension of our previous paper [10] which gave for-ward and backward error estimates, we perform a running error analysis ofthe multivariate Horner scheme.  This leads to a modified algorithm whichcomputes the value of the polynomial together with an error estimate.},
  publisher = {Springer},
}


@article{kojima2008efficient,
  title={Efficient evaluation of polynomials and their partial derivatives in homotopy continuation methods},
  author={<NAME>},
  journal={Journal of the Operations Research Society of Japan},
  volume={51},
  number={1},
  pages={29--54},
  year={2008},
  publisher={The Operations Research Society of Japan},
}



@article{carnicer1990evaluation,
  title={Evaluation of multivariate polynomials and their derivatives},
  author={<NAME> and <NAME>},
  journal={Mathematics of Computation},
  volume={54},
  number={189},
  pages={231--243},
  year={1990},
  abstract={An extension of Homer's algorithm to the evaluation of m-variate
polynomials and their derivatives is obtained. The schemes of computation are
represented by trees because this type of graph describes exactly in which order
the computations must be done. Some examples of algorithms for one and two
variables are given.},
}


@article{kuipers2013improving,
  title={Improving multivariate {H}orner schemes with {M}onte {C}arlo tree search},
  author={<NAME> <NAME> and Vermaseren, JAM and <NAME>, <NAME>},
  journal={Computer Physics Communications},
  volume={184},
  number={11},
  pages={2391--2395},
  year={2013},
  publisher={Elsevier},
    abstract={Optimizing the cost of evaluating a polynomial is a classic problem in computer science. For polynomials in one variable, Horner’s method provides a scheme for producing a computationally
efficient form. For multivariate polynomials it is possible to generalize Horner’s method, but this
leaves freedom in the order of the variables. Traditionally, greedy schemes like most-occurring vari-
able first are used. This simple textbook algorithm has given remarkably efficient results. Finding
better algorithms has proved difficult. In trying to improve upon the greedy scheme we have im-
plemented Monte Carlo tree search, a recent search method from the field of artificial intelligence.
This results in better Horner schemes and reduces the cost of evaluating polynomials, sometimes by
factors up to two.},
}

@PhdThesis{LeeFactorization2013,
  author      = {<NAME>},
  title       = {Factorization of multivariate polynomials},
  institution = {Technische Universität Kaiserslautern},
  year        = {2013},
  date        = {2013-06-13},
}

@inproceedings{leiserson2010efficient,
  title={Efficient evaluation of large polynomials},
  author={Leiserson, <NAME> and <NAME> and Maza, <NAME> and <NAME>},
  booktitle={International Congress on Mathematical Software},
  pages={342--353},
  year={2010},
  organization={Springer}
}

# applications polynomials

@book{prasolov2009polynomials,
  title={Polynomials},
  author={<NAME>},
  volume={11},
  year={2009},
  publisher={Springer Science \& Business Media}
}

@phdthesis{boumova2002applications,
  title={Applications of polynomials to spherical codes and designs},
  author={<NAME>},
  year={2002},
  school={Technische Universiteit Eindhoven}
}

@article{cools2002advances,
  title={Advances in multidimensional integration},
  author={<NAME>},
  journal={Journal of computational and applied mathematics},
  volume={149},
  number={1},
  pages={1--12},
  year={2002},
  publisher={Elsevier}
}


@book{akritas1989elements,
  title={Elements of computer algebra with applications},
  author={<NAME>},
  year={1989},
  publisher={John Wiley \& Sons, Inc.}
}

