NEWTON CORRECTION METHODS FOR COMPUTING REAL EIGENPAIRS OF SYMMETRIC TENSORS

Area of Science:

• Numerical Analysis
• Linear Algebra
• Tensor Computations

Background:

• Real eigenpairs of symmetric tensors are crucial in various scientific and engineering applications.
• Existing iterative methods for computing these eigenpairs have limitations in convergence and scope.

Purpose of the Study:

• To propose and analyze a novel, fast iterative Newton-based method for computing real eigenpairs of symmetric tensors.
• To establish conditions for method stability and convergence rate.

Main Methods:

• Development of a Newton-based iterative algorithm.
• Derivation of sufficient conditions for stable fixed points.
• Analysis of local convergence properties, proving quadratic convergence rates.

Main Results:

• The proposed method exhibits quadratic convergence for real eigenpairs of symmetric tensors.
• Empirical results show convergence to a greater number of eigenpairs compared to existing methods.
• The method typically identifies all real eigenpairs with sufficient random initializations.

Conclusions:

• The developed Newton-based method is efficient and effective for computing real eigenpairs of symmetric tensors.
• Sufficient conditions for local convergence are shown to hold for all real eigenpairs of generic symmetric tensors.
• This advancement offers a robust tool for applications requiring accurate tensor eigenpair computation.