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Bertini is a software package for numerically solving systems of polynomial equations. It can find both isolated solutions as well as positive-dimensional solutions.

Bertini is the product of a research program in numerical algebraic geometry. It uses many newly developed methods, including intersecting and projecting algebraic sets, adaptive precision path tracking, methods for treating singular sets, parameter homotopies, user-defined path tracking, and other new research.


Numerical Algebraic Geometry


Polynomial Systems

Numerical Methods

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    Daniel Bates, Andrew Sommese, Jonathan Hauenstein

    Numerical algebraic geometry is the area devoted to the solution and manipulation of polynomial systems by numerical methods, which are mainly based on continuation. Due to the extreme intrinsic parallelism of continuation, polynomial systems may be successfully dealt with that are much larger than is possible with other methods. Singular solutions require special numerical methods called endgames, and the endgames currently used do not take advantage of parallelism. This article gives an ove…

  • Author(s):
    Charles Wampler II, Daniel Bates, Andrew Sommese, Jonathan Hauenstein

    Dedicated to our collaborator, mentor, and friend, Andrew Sommese, by Bates, Hauenstein, and Wampler on the occasion of his sixtieth birthday.

    When numerically tracking implicitly-defined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy fo…

  • Author(s):
    Charles Wampler, Andrew Sommese, Jonathan Hauenstein

    Though numerical methods to find all the isolated solutions of nonlinear systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomial systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operati…

  • Author(s):
    Daniel Bates, Andrew Sommese, Jonathan Hauenstein

    Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for…