Interpolants for Runge-Kutta formulas
ACM Transactions on Mathematical Software (TOMS)
Computing all solutions to polynomial systems using homotopy continuation
Applied Mathematics and Computation
A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides
ACM Transactions on Mathematical Software (TOMS)
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
Adaptive Multiprecision Path Tracking
SIAM Journal on Numerical Analysis
Solving Polynominal Systems Using Continuation for Engineering and Scientific Problems
Solving Polynominal Systems Using Continuation for Engineering and Scientific Problems
Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems
ACM Transactions on Mathematical Software (TOMS)
Numerically Computing Real Points on Algebraic Sets
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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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 using Runge-Kutta methods in conjunction with adaptive precision are provided. These methods have been implemented in the Bertini software package, and several examples are described.