# bibliography

@article{ArnoldiFit_ZhangSuLi,
title = {Accurate polynomial fitting and evaluation via Arnoldi},
journal = {Numerical Algebra, Control and Optimization},
volume = {14},
number = {3},
pages = {526-546}
year = {2024},
issn = {2155-3289},
doi = {10.3934/naco.2023002},
url = {https://www.aimsciences.org/article/id/63c52c4cb5351f4889afcfd4},
author = {Lei-Hong Zhang and Yangfeng Su and Ren-Cang Li},
keywords = {Rounding error analysis, polynomial interpolation, Lanczos process, Arnoldi process, SOAR, least-squares, Vandermonde matrix}
}

@article{ArnoldiFit_BrubeckNakatsukasaTefethen,
author = {Brubeck, Pablo D. and Nakatsukasa, Yuji and Trefethen, Lloyd N.},
title = {Vandermonde with Arnoldi},
journal = {SIAM Review},
volume = {63},
number = {2},
pages = {405-415},
year = {2021},
doi = {10.1137/19M130100X},
URL = {https://doi.org/10.1137/19M130100X},
eprint = {https://doi.org/10.1137/19M130100X},
abstract = { Vandermonde matrices are exponentially ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem. This amounts to on-the-fly construction of discrete orthogonal polynomials by Stieltjes orthogonalization. }
}



@article{AAA_NakatsukasaSetTrefethen,
author = {Nakatsukasa, Yuji and S{\`e}te, Olivier and Trefethen, Lloyd N.},
title = {The AAA Algorithm for Rational Approximation},
journal = {SIAM Journal on Scientific Computing},
volume = {40},
number = {3},
pages = {A1494-A1522},
year = {2018},
doi = {10.1137/16M1106122},
URL = {https://doi.org/10.1137/16M1106122},
eprint = {https://doi.org/10.1137/16M1106122},
abstract = { We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of MATLAB and requiring no user input parameters. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points and (2) greedy selection of the support points to avoid exponential instabilities. The name AAA stands for “adaptive Antoulas--Anderson” in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a MATLAB code and nine applications and describe variants targeted at problems of different kinds. Comparisons are made with vector fitting, RKFIT, and other existing methods for rational approximation. }
}


@Article{narep108,
  author = 	 {Higham, Nicholas},
  title = 	 {Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems.},
  journal = 	 {Numerische Mathematik},
  year = 	 1987,
  volume = 	 50,
  number = 	 5,
  pages = 	 613,
  doi = {10.1007/BF01408579}
}

@InProceedings{langlois_et_al:DSP:2006:442,
author = {Philippe Langlois and Stef Graillat and Nicolas Louvet},
title = {Compensated Horner Scheme},
booktitle = {Algebraic and Numerical Algorithms and Computer-assisted Proofs},
year = {2006},
editor = {Bruno Buchberger and Shin'ichi Oishi and Michael Plum and Sigfried M. Rump},
number = {05391},
series = {Dagstuhl Seminar Proceedings},
ISSN = {1862-4405},
publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2006/442},
annote = {Keywords: Polynomial evaluation, Horner scheme, error-free transformation, floating point arithmetic, accuracy} }

@inproceedings{10.1145/860854.860907,
author = {Zeng, Zhonggang},
title = {A method computing multiple roots of inexact polynomials},
year = {2003},
isbn = {1581136412},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/860854.860907},
doi = {10.1145/860854.860907},
abstract = {We present a combination of two novel algorithms that accurately calculate multiple roots of general polynomials. For a given multiplicity structure and initial root estimates, Algorithm I transforms the singular root-finding into a nonsingular least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm I, we design a numerical GCD-finder, including a partial singular value decomposition and an iterative refinement, as the main engine for Algorithm II that calculates the multiplicity structure and the initial root approximation. The combined method calculates multiple roots with high forward accuracy without using multiprecision arithmetic, even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the true sensitivity of the multiple roots, a pejorative condition number is proposed and error bounds are given. Extensive computational experiments are presented. The error analysis and numerical results confirm that a polynomial being ill-conditioned in conventional sense can be well conditioned pejoratively. In those cases, the multiple roots can be computed with remarkable accuracy.},
booktitle = {Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation},
pages = {266–272},
numpages = {7},
keywords = {polynomial root-finding},
location = {Philadelphia, PA, USA},
series = {ISSAC '03}
}

@Article{LeeLiZeng,
  author = 	 {Lee, Tsung-Lin and Li, Tien-Yien and Zeng, Zhonggang},
  title = 	 {RankRev: aMatlab package for computing the numerical rank and updating/downdating},
  journal = 	 {Numerical Algorithms},
  year = 	 2018,
  key = 	 77,
  pages = 	 {559-576},
  doi = {10.1007/s11075-017-0328-7},
}

@InBook{ZengNGCD,
  author = 	 {Zhonggang Zeng },
  title = 	 {Randomization, Relaxation, and Complexity in Polynomial Equation Solving},
  chapter = 	 {The numerical greatest common divisor of univariate polynomials},
  publisher = 	 {AMS},
  year = 	 2011,
  pages = 	 {187-217}}


@Article{lejaorder,
  author = 	 {Calvetti, D. and Reichel, L.},
  title = 	 {On the Evaluation of Polynomial Coefficients},
  journal = 	 {Numerical Algorithms },
  year = 	 2003,
  volume = 	 33,
  pages = 	 {153-161}}
