Newton's method on Graßmann manifolds

2007/09/14 by Uwe Helmke, Helmke, Uwe, Knut Hüper +3 · 1 citation
Computer Science · Engineering · Mathematics · #15A18 #49M15 #53B20 #65F15 #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Statistical and numerical algorithms

paper · pdf · doi:10.48550/arxiv.0709.2205

openalex publication_date 2007/09/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergence results for Newton's method on a manifold. Using special choices of the coordinates, new numerical algorithms are derived for principal component analysis and invariant subspace computations with improved computational complexity properties.

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• Numerical algorithms on the affine Grassmannian

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