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This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newtons method and subdivision for this problem. Our algorithm is intended only for nondegenerate cases, in which case the solution is a 1-dimensional curve. Our first main contribution is a definition of a condition number measuring reciprocal distance to degeneracy that can distinguish poor and well conditioned instances of this problem. (Degenerate problems would be infinitely ill conditioned in our framework.) Our second contribution, which is the main novelty of our algorithm, is an analysis showing that its running time is bounded in terms of the condition number of the problem instance as well as $n$ and the polynomial degrees.
In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list by reducing the task to an eigenvalue problem in a considerably faster and simpler w
We propose an iterative improvement method for the Harrow-Hassidim-Lloyd (HHL) algorithm to solve a linear system of equations. This is a quantum-classical hybrid algorithm. The accuracy is essential to solve the linear system of equations. However,
Based on the geometric {it Triangle Algorithm} for testing membership of a point in a convex set, we present a novel iterative algorithm for testing the solvability of a real linear system $Ax=b$, where $A$ is an $m times n$ matrix of arbitrary rank.
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of $m$ linear inequalities in $n$ variables $(P): A^{top}x le u$ when its set of solutions has positive volume. However, when $(P)$ is infeasible, the ellipsoid
In this paper, we study the quantum dynamics of a one degree-of-freedom (DOF) Hamiltonian that is a normal form for a saddle node bifurcation of equilibrium points in phase space. The Hamiltonian has the form of the sum of kinetic energy and potentia