alphaCertified swMATH ID: 7351 Software Authors: Jonathan D. Hauenstein; Frank Sottile Description: Algorithm 921: alphaCertified: Certifying solutions to polynomial systems. Smale’s α-theory uses estimates related to the convergence of Newton’s method to certify that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on α-theory to certify solutions of polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements algorithms that certify whether a given point corresponds to a real solution, and algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms. Homepage: http://dl.acm.org/citation.cfm?id=2331136 Related Software: Bertini; Macaulay2; PHCpack; HOM4PS; NumericalAlgebraicGeometry; SINGULAR; ISOLATE; NAG4M2; Magma; MonodromySolver; HomotopyContinuation; INTLAB; PHoM; MixedVol; Paramotopy; NewtonLib; MultRoot; STRINGVACUA; CoCoA; Algorithm 931 Cited in: 56 Publications all top 5 Cited by 88 Authors 22 Hauenstein, Jonathan D. 11 Sottile, Frank 7 Szántó, Ágnes 5 Mehta, Dhagash 4 Hein, Nickolas 4 Timme, Sascha 3 Bates, Daniel J. 3 Cheng, Jinsan 3 Dou, Xiaojie 3 Leykin, Anton 3 Niemerg, Matthew E. 3 Sommese, Andrew John 2 Breiding, Paul 2 Brysiewicz, Taylor 2 del Campo, Abraham Martín 2 Duff, Timothy 2 Harris, Katherine M. 2 He, Yang-Hui Evariste 2 Lee, Kisun 2 Lee, Tsung-Lin 2 Regan, Margaret H. 1 Alaeddini, Adel 1 Ayyildiz Akoglu, Tulay 1 Beltrán, Carlos 1 Bernal, Edgar A. 1 Bozóki, Sándor 1 Brake, Daniel A. 1 Brake, Danielle A. 1 Brustenga i Moncusí, Laura 1 Burr, Michael A. 1 Chen, Tianran 1 Decker, Wolfram 1 Do, Ngoc T. 1 Frohmader, Andrew 1 García Puente, Luis David 1 Griffin, Zachary A. 1 Gross, Elizabeth 1 Heaton, Alexander 1 Helmer, Martin 1 Hill, Cvetelina 1 Hillar, Christopher J. 1 Imbach, Rémi 1 Jensen, Anders Nedergaard 1 Kaluba, Marek 1 Kuchment, Peter A. 1 Kulkarni, Avinash 1 Lairez, Pierre 1 Levandovskyy, Viktor 1 Li, Tien-Yien 1 Li, Zhe 1 Liddell, Alan C. jun. 1 Lorenz, Benjamin 1 Manivel, Laurent 1 Mantzaflaris, Angelos 1 Moroz, Guillaume 1 Mourrain, Bernard 1 Oeding, Luke 1 Ottaviani, Giorgio Maria 1 Pan, Victor Yakovlevich 1 Peterson, Chris 1 Pfister, Gerhard 1 Pouget, Marc 1 Rodriguez, Jose Israel 1 Rónyai, Lajos 1 Ruatta, Olivier 1 Ruffo, James 1 Schreyer, Frank-Olaf 1 Sciabica, Mark 1 Sertöz, Emre Can 1 Shen, Fei 1 Sherman, Samantha N. 1 Shirinkam, Sara 1 Sommars, Jeff 1 Sturmfels, Bernd 1 Sun, Chuang 1 Szendrői, Balázs 1 Tang, Tingting 1 Teitler, Zach 1 Vinzant, Cynthia 1 Wampler, Charles W. II 1 Wan, Baocheng 1 Weinstein, Madeleine 1 Wen, Junyi 1 Wu, Wenyuan 1 Xia, Bican 1 Yahl, Thomas 1 Zhang, Shugong 1 Zhi, Lihong all top 5 Cited in 27 Serials 11 Journal of Symbolic Computation 3 Foundations of Computational Mathematics 2 Applied Mathematics and Computation 2 Numerical Algorithms 2 Notices of the American Mathematical Society 2 Experimental Mathematics 2 Journal of High Energy Physics 2 Journal of Algebra and its Applications 1 IMA Journal of Numerical Analysis 1 Journal of Mathematical Physics 1 ACM Transactions on Mathematical Software 1 Journal of Computational and Applied Mathematics 1 Journal of Pure and Applied Algebra 1 Journal für die Reine und Angewandte Mathematik 1 Le Matematiche 1 Theoretical Computer Science 1 Acta Applicandae Mathematicae 1 Journal of the American Mathematical Society 1 Computational Geometry 1 Advances in Computational Mathematics 1 Vietnam Journal of Mathematics 1 Advances in Geometry 1 Journal of Statistical Mechanics: Theory and Experiment 1 Mathematics in Computer Science 1 Advances in High Energy Physics 1 ACM Communications in Computer Algebra 1 Mathematics all top 5 Cited in 17 Fields 37 Numerical analysis (65-XX) 29 Algebraic geometry (14-XX) 16 Computer science (68-XX) 13 Commutative algebra (13-XX) 3 Quantum theory (81-XX) 2 Field theory and polynomials (12-XX) 2 Relativity and gravitational theory (83-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Special functions (33-XX) 1 Convex and discrete geometry (52-XX) 1 Statistics (62-XX) 1 Mechanics of particles and systems (70-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Operations research, mathematical programming (90-XX) Citations by Year