Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

Area of Science:

• Numerical Analysis
• Linear Algebra
• Scientific Computing

Background:

• Computing eigenvectors of symmetric block tridiagonal matrices is crucial in various scientific and engineering fields.
• Existing methods, often based on tridiagonalization, can be computationally intensive for large matrices.

Purpose of the Study:

• To develop novel, efficient algorithms for computing eigenvectors of symmetric block tridiagonal matrices.
• To leverage twisted block factorizations for improved computational performance.

Main Methods:

• Exploration of new methods based on twisted block factorizations.
• Review of the relationship between block factorizations and eigenvectors.
• Design of algorithmic strategies utilizing inverse iteration with starting vectors derived from factorizations.

Main Results:

• Proposed algorithms demonstrate substantial reductions in runtime compared to state-of-the-art methods, particularly for large matrices and small bandwidths.
• Numerical accuracy, measured by residuals, is generally comparable to existing techniques.
• A minor loss in eigenvector orthogonality was observed in specific cases with clustered eigenvalues.

Conclusions:

• The novel algorithms based on twisted block factorizations provide an efficient approach for eigenvector computation.
• Future research will focus on addressing the observed orthogonality issues in cases of clustered eigenvalues.