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Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence
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Published on: June 24, 2016

A geometric Newton method for Oja's vector field.

P A Absil1, M Ishteva, L De Lathauwer

  • 1Department of Mathematical Engineering, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium. absil@inma.ucl.ac.be

Neural Computation
|November 21, 2008
PubMed
Summary
This summary is machine-generated.

Newton's method for solving a matrix equation fails due to symmetry issues. A new geometric Newton algorithm, using differential-geometric techniques, overcomes this degeneracy to find the matrix equation's zeros.

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Published on: June 24, 2016

Area of Science:

  • Numerical Analysis
  • Linear Algebra
  • Differential Geometry

Background:

  • Newton's method is a standard iterative technique for finding roots of equations.
  • Solving matrix equations is crucial in various scientific and engineering fields.
  • The matrix equation F(X) = AX - XX(T)AX = 0 presents unique challenges due to symmetry.

Purpose of the Study:

  • To address the non-isolated zeros issue in Newton's method for the specific matrix equation.
  • To develop a modified Newton's method that is robust against symmetry-induced degeneracies.
  • To leverage differential-geometric principles for a more effective numerical algorithm.

Main Methods:

  • Analysis of the symmetry of the matrix function F(X) under the orthogonal group.
  • Application of differential-geometric techniques to remove the problematic symmetry.
  • Development and implementation of a "geometric" Newton algorithm.

Main Results:

  • Identified symmetry as the cause of non-isolated zeros for the original Newton's method.
  • Successfully removed the symmetry using differential-geometric methods.
  • The proposed geometric Newton algorithm effectively finds the zeros of F(X) without degeneracy issues.

Conclusions:

  • Differential-geometric techniques offer a powerful approach to resolve degeneracies in numerical methods.
  • The geometric Newton algorithm provides a robust alternative for solving the studied matrix equation.
  • This work advances the application of geometric principles in numerical analysis for matrix problems.