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Using a low-storage contour-integral eigensolver for nonsymmetric matrices with collocation to compute vibrational
Luca Corneo1, Tucker Carrington1
1Department of Chemistry, Queen's University, Kingston, Ontario K7L 3N6, Canada.
The Journal of Chemical Physics
|July 15, 2026
Summary
We developed a new low-storage eigensolver for computing vibrational spectra. This method efficiently solves non-Hermitian eigenvalue problems, reducing memory requirements for large matrices.
Area of Science:
- Quantum chemistry
- Computational physics
- Spectroscopy
Background:
- Collocation methods offer advantages for computing vibrational spectra by eliminating the need for quadrature.
- Solving the resulting non-Hermitian eigenvalue problem is computationally demanding, especially for large matrices.
- Existing eigensolvers for non-Hermitian matrices require significant memory due to the need to store and orthogonalize numerous vectors.
Purpose of the Study:
- To develop a memory-efficient eigensolver for non-Hermitian matrices arising in collocation methods for vibrational spectra.
- To combine the strengths of the block Sakurai-Sugiura method and multi-shift quasi-minimal residual linear solvers.
- To create a low-storage algorithm that retains the advantages of established methods for symmetric matrices.
Main Methods:
- A novel low-storage eigensolver was proposed, integrating the block Sakurai-Sugiura method.
- The multi-shift quasi-minimal residual linear solver was incorporated to handle the non-Hermitian eigenvalue problem.
- The approach was designed to require storage of only a few vectors, minimizing memory footprint.
Main Results:
- The proposed method successfully computes vibrational spectra using collocation.
- It significantly reduces memory requirements compared to established eigensolvers for non-Hermitian matrices.
- The algorithm demonstrates efficiency comparable to the Cullum-Willoughby Lanczos approach for symmetric matrices.
Conclusions:
- The developed low-storage eigensolver provides an efficient and memory-saving solution for computing vibrational spectra via collocation.
- This method overcomes the limitations of traditional eigensolvers for large, non-Hermitian matrices.
- It offers a practical alternative for large-scale quantum chemistry and computational physics simulations.
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