Abstract
We consider a family of polynomial systems which arises in the analysis of the stationary solutions of a standard discretization of certain semi-linear second-order parabolic partial differential equations. We prove that this family is well-conditioned from the numeric point of view, and ill-conditioned from the symbolic point of view. We exhibit a polynomial-time numeric algorithm solving any member of this family, which significantly contrasts the exponential behavior of all known symbolic algorithms solving a generic instance of this family of systems.
Highlights
Several scientific and technical problems require the solution of polynomial systems over the real or complex numbers
Numeric and symbolic methods for computing all solutions of a given zero-dimensional polynomial system usually rely on deformation techniques, based on a perturbation of the original system and a subsequent path-following method
There are several variants of homotopy algorithms which profit from special features of the input system, such as sparsity patterns or the existence of suitable low-degree projections
Summary
Several scientific and technical problems require the solution of polynomial systems over the real or complex numbers (see e.g. [43,48]). The usual numerical approach to this problem consists of considering a second-order finite difference discretization in the variable x, with a uniform mesh, keeping the variable t continuous (see [2,9]) This semi-discretization in space leads to the following initial value problem: u1 = 2(n − 1) f (u2) − f (u1) − g(u1), uk = (n − 1) f (uk+1) − 2f (uk) + f (uk−1) − g(uk), (2 k n−1). This allows us to exhibit an algorithm approximating the only positive solution x∗ of (4) by an homotopy continuation method. We see the significant contrast between the exponential complexity behavior of all symbolic methods solving any instance of (4) and the polynomial complexity behavior of our numeric method
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