Accurate simple zeros of polynomials in floating point arithmetic
Computers & Mathematics with Applications
Sweeping algebraic curves for singular solutions
Journal of Computational and Applied Mathematics
Polynomial homotopies on multicore workstations
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Sampling algebraic sets in local intrinsic coordinates
Computers & Mathematics with Applications
Efficient path tracking methods
Numerical Algorithms
Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems
ACM Transactions on Mathematical Software (TOMS)
Journal of Computational and Applied Mathematics
Continuation Along Bifurcation Branches for a Tumor Model with a Necrotic Core
Journal of Scientific Computing
Journal of Computational Physics
Numerically Computing Real Points on Algebraic Sets
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Cell Cycle Control and Bifurcation for a Free Boundary Problem Modeling Tissue Growth
Journal of Scientific Computing
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This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.