iVI: An iterative vector interaction method for large eigenvalue problems.

Area of Science:

• Numerical Analysis
• Computational Chemistry
• Linear Algebra

Background:

• Computing eigenpairs of large matrices is crucial in many scientific disciplines.
• Existing methods may face challenges with convergence or computational cost for multiple eigenpairs.
• The static-dynamic-static framework provides a basis for developing new algorithms.

Purpose of the Study:

• To introduce a novel iterative Vector Interaction (iVI) method.
• To compute multiple exterior or interior eigenpairs of large symmetric/Hermitian matrices.
• To demonstrate the efficacy of the iVI method.

Main Methods:

• The proposed iterative Vector Interaction (iVI) method is based on the static-dynamic-static framework.
• The iVI method operates within a fixed-dimensional search subspace.
• The method iteratively refines approximations to converge to eigenpairs.

Main Results:

• The iVI method demonstrates fast and monotonic convergence.
• The method accurately computes multiple exterior and interior eigenpairs.
• The efficacy was validated using both mathematical and physical matrices.

Conclusions:

• The iterative Vector Interaction (iVI) method is an effective approach for computing multiple eigenpairs of large matrices.
• iVI offers a robust alternative to existing methods, particularly for large-scale problems.
• The method's performance highlights its potential in various computational fields.