# The NumericalCertification package in Macaulay2

@article{Lee2022TheNP, title={The NumericalCertification package in Macaulay2}, author={Kisun Lee}, journal={ArXiv}, year={2022}, volume={abs/2208.01784} }

The package NumericalCertification implements methods for certifying numerical approximations of solutions for a given system of polynomial equations. For certifying regular solutions, the package implements Smale’s α -theory and Krawczyk method. For a singular solution, we implement soft veriﬁcation using the iterative deﬂation method. We demonstrate the functionalities of the package focusing on interaction with current numerical solvers in Macaulay2 .

## 2 Citations

### Homotopy techniques for analytic combinatorics in several variables

- MathematicsArXiv
- 2022

We combine tools from homotopy continuation solvers with the methods of analytic combinatorics in several variables to give the first practical algorithm and implementation for the asymptotics of…

### On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials

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## References

SHOWING 1-10 OF 27 REFERENCES

### Macaulay2

- a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/,
- 2002

### Certifying zeros of polynomial systems using interval arithmetic

- Mathematics, Computer ScienceACM Transactions on Mathematical Software
- 2023

The software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations, and it is demonstrated that it dramatically outperforms earlier approaches to certification.

### Ma

- Political Science, ArtBest New African Poets 2019 Anthology
- 2020

The relation between Hirnov ooil signals and the current perturbation on the rational surface is examined analytically by using the approximate Green's function for the case of large aspect ratio…

### Certifying approximate solutions to polynomial systems on Macaulay2

- Computer Science, MathematicsACCA
- 2019

We present the Macaulay2 package NumericalCertification for certifying roots of square polynomial systems. It employs the interval Krawczyk method and α-theory as main methods for certification. The…

### Newton’s Method Estimates from Data at One Point

- Mathematics
- 1986

Newton’s method and its modifications have long played a central role in finding solutions of non-linear equations and systems. The work of Kantorovich has been seminal in extending and codifying…

### Journal of Software for Algebra and Geometry

- Mathematics
- 2015

StdPairs, a SageMath library to compute standard pairs of a monomial ideal over a pointed (nonnormal) afﬁne semigroup ring, provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of amonomial ideal.

### Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation

- Computer ScienceISSAC
- 2019

The 2009 International Symposium on Symbolic and Algebraic Computation (ISSAC 2009), which is jointly sponsored by ACM/SIGSAM and the Korea Institute for Advanced Study (KIAS), will be held at KIAS…

### Effective Certification of Approximate Solutions to Systems of Equations Involving Analytic Functions

- Mathematics, Computer ScienceISSAC
- 2019

Algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions are developed based on the existence of oracles for evaluating basic data about the input analytic functions.

### A Test for Existence of Solutions to Nonlinear Systems

- Mathematics
- 1977

Computationally verifiable sufficient conditions are given for the existence of a solution to a system of nonlinear equations using an interval version of Newton’s method given by R. Krawczyk. A si...

### HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia

- Computer ScienceICMS
- 2018

This work presents the Julia package HomotopyContinuation.jl, which provides an algorithmic framework for solving polynomial systems by numerical homotopy continuation by motivating the choice of Julia and how its features allow to improve upon existing software packages with respect to usability, modularity and performance.