Lehmer–Schur algorithm

Abstract: Several engineering applications need a robust method to find all the roots of a set of nonlinear equations automatically. The proposed method guarantees monotonous convergence, and it can determine whole submanifolds of the roots if the number of unknowns is larger than the number of equations. The critical steps of the multidimensional bisection method are described and possible solutions are proposed. An efficient computational scheme is introduced. The efficiency of the method is characterized by the box-counting fractal dimension of the evaluated points. The multidimensional bisection method is much more efficient than the brute force method. The proposed method can also be used to determine the fractal dimension of the submanifold of the solutions with satisfactory accuracy.

Abstract: We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Given an oracle that provides approximations of each of the coefficients of $F$ to any absolute error bound and given an arbitrary square $\mathcal{B}$ in the complex plane containing only simple roots of $F$, our algorithm returns disjoint isolating disks for the roots of $F$ in $\mathcal{B}$. Our complexity analysis bounds the absolute error to which the coefficients of $F$ have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square $\mathcal{B}$, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the \emph{benchmark problem}, namely, to isolate all the roots of a polynomial of degree $n$ with integer coefficients of bit size less than $\tau$, our algorithm needs $\tilde O(n^3+n^2\tau)$ bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schonhage's splitting circle technique. Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "soft-test" to count the number of roots in a disk. Using Schroder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.

Abstract: Bisection Method is one of the simplest methods in numerical analysis to find the roots of a non-linear equation. It is based on Intermediate Value Theorem. The algorithm proposed in this paper predicts the optimal interval in which the roots of the function may lie and then applies the bisection method to converge at the root within the tolerance range defined by the user. This algorithm also calculates another root of the equation, if that root lies just outside the range of the interval found.