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Register a new userYet another eigenvalue algorithm for solving polynomial systems
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Added by
Mat\\'ias R. Bender
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
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.
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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.
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