اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدYet another eigenvalue algorithm for solving polynomial systems
168
0
0.0
(
0
)
نشر من قبل
Mat\\'ias R. Bender
تاريخ النشر
2021
مجال البحث
الهندسة المعلوماتية
والبحث باللغة
English
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 way than in previous methods. This results in an algorithm which solves systems with isolated solutions in a reliable and efficient way, outperforming homotopy methods in overdetermined cases. We provide an implementation in the proof-of-concept Julia package EigenvalueSolver.jl.
قيم البحث
اقرأ أيضاً
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.
Let $C_{A,r}$ be the ellipsoid determined as the image of the Euclidean ball of radius $r$ under the linear map $A$. The basic procedure in our algorithm computes a point in $C_{A,r}$ that is either within $varepsilon$ distance to $b$, or acts as a certificate proving $b ot in C_{A,r}$. Each iteration takes $O(mn)$ operations and when $b$ is well-situated in $C_{A,r}$, the number of iterations is proportional to $log{(1/varepsilon)}$. If $Ax=b$ is solvable the algorithm computes an approximate solution or the minimum-norm solution. Otherwise, it computes a certificate to unsolvability, or the minimum-norm least-squares solution. It is also applicable to complex input. In a computational comparison with the state-of-the-art algorithm BiCGSTAB ({it Bi-conjugate gradient method stabilized}), the Triangle Algorithm is very competitive. In fact, when the iterates of BiCGSTAB do not converge, our algorithm can verify $Ax=b$ is unsolvable and approximate the minimum-norm least-squares solution. The Triangle Algorithm is robust, simple to implement, and requires no preconditioner, making it attractive to practitioners, as well as researchers and educators.
We propose a deterministic Kaczmarz method for solving linear systems $Ax=b$ with $A$ nonsingular. Instead of using orthogonal projections, we use reflections in the original Kaczmarz iterative method. This generates a series of points on an $n$-sphe
re $S$ centered at the solution $x_*=A^{-1}b$. We show that these points are nicely distributed on $S$. Taking the average of several points will lead to an effective approximation to the solution. We will show how to choose these points efficiently. The numerical tests show that in practice this deterministic scheme converges much faster than we expected and can beat the (block) randomized Kaczmarz methods.
We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact, complex toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which in practice might arise from h
omogenizing a sparse polynomial system. We prove a new eigenvalue theorem in the toric compact setting, which leads to a novel, robust numerical approach for solving this problem. Our method works in particular for systems having isolated solutions with arbitrary multiplicities. It depends on the multigraded regularity properties of $I$. We study these properties and provide bounds on the size of the matrices involved in our approach in the case where $I$ is a complete intersection.
A novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multi-column gradient such that earlier columns are
decoupled from later ones. Global convergence to eigenvectors instead of eigenspace is guaranteed almost surely. Locally, algorithms converge linearly with convergence rate depending on eigengaps. Momentum acceleration, exact linesearch, and column locking are incorporated to further accelerate both algorithms and reduce their computational costs. We demonstrate the efficiency of both algorithms on several random matrices with different spectrum distribution and matrices from computational chemistry.
The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this package may be used to exploit this structure, with examples.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
