A continuation method to solve polynomial systems and its complexity

Abstract

In a recent work Shub (Found. Comput. Math. 9:171-178, 2009), Shub obtained a new upper bound for the number of steps needed to continue a known zero η 0 of a system f 0 , to a zero η T of an input system f T , following the path of pairsHe proved that if one can choose the step-size in an optimal way, then the number of steps is essentially bounded by the length of the path of ( f t , η t ) in the so-called condition metric. However, the proof of that result in Shub (Found. Comput. Math. 9:171-178, 2009) is not constructive. We give an explicit description of an algorithm which attains that complexity bound, including the choice of step-size.

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

“…An actual algorithm was not described in [29]; only its existence was proved. A specific description of the method has been given in [5]. Other descriptions were obtained independently and simultaneously, see [16,19].…”

“…Our algorithm thus uses the two ingredients outlined above (choice of the path f t with randomized (g, ζ 0 ) and the path-following method of [5,29]). We may informally write our main result as follows.…”

“…Here we use an explicit description [5] of this method (there are other explicit descriptions available, see [16,19]). It performs a number of "homotopy steps", each of them being one application of projective Newton's method.…”

Spherical projective path tracking for homotopy continuation methods

“…In our algorithm, we chose the predictor-corrector scheme for the path tracking. There is a similar path tracking algorithm using "Projective Newton's iterations" alone which has been intensively studied theoretically such as in [5], [6], [7], [8], [9], [10], [11], [12], [15], [25], [26], [28], [29], [30], and [31], to list a few. In Section 11, numerical results are presented in comparing these two approaches.…”

“…In particular, as mentioned in [3], one would choose a =x/ x 2 in certain situations; in those cases the resulting Davidenko equation will be exactly the same as Equation (5). Therefore in one sense, Equation (5) looks like a result of a specific choice of an affine chart: we always use x(t)/ x(t) 2 , • C = 0 as the affine chart.…”