Adaptive Multiprecision Path Tracking

Abstract

Abstract. 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 re-run 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.

Isosingular Sets and Deflation

“…For our computations, we used the software package Bertini v1.2 [6], choosing the regenerative cascade [17] with adaptive precision tracking [4,5,7] to compute the numerical irreducible decompositions. The serial computations used a 2.4 GHz Opteron 250 processor with 64-bit Linux.…”

Polynomial homotopies on multicore workstations

“…Often it occurs that there are a couple of solution paths that require extra care, for which the use multiprecision arithmetic is necessary [7]. The paper [5] surveys applications of high-precision computations, The use of double doubles in numerical linear algebra is considered in [35].…”

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

“…Ill-conditioned segments of the paths are short-lived, but they do show up, especially in larger problems. Thus, adaptive multiple-precision techniques [BHSW08,BHSW09] are essential. These methods are the key to Bertini's reliability in this regard.…”

“…Due to the high-multiplicity, numerical methods must use adaptive-precision tracking [BHSW08,BHSW09] which increases the cost. Again this example can be analyzed without a computer by rewriting the system (by subtracting two equations and factoring).…”