Diagonalization of large matrices: a new parallel algorithm.

Area of Science:

• Numerical Analysis
• Quantum Chemistry
• Computational Physics

Background:

• Finding eigenvalues/eigenvectors of large matrices is computationally intensive.
• Existing methods like LAPACK routines can be slow for large-scale problems.
• The dressed matrices formalism offers a new perspective for matrix diagonalization.

Purpose of the Study:

• To develop a new, efficient algorithm for computing the lowest eigenvalue and eigenvector of large real symmetric matrices.
• To implement and test both sequential and parallel versions of the proposed algorithm.
• To compare the algorithm's performance against established methods like LAPACK and the Davidson method.

Main Methods:

• The algorithm diagonalizes (N-1)x2 dressed matrices.
• Sequential and parallel (MPI) versions were implemented.
• Performance was tested on Hilbert matrices and for molecular electronic structure calculations (MRCI).

Main Results:

• The algorithm is up to 340 times faster than LAPACK for N=10^4.
• It shows a 10% speed improvement over the Davidson method.
• Parallel versions demonstrate near-linear speed-up on up to 512 cores for large matrices (N=10^6, N=10^7).
• Identical correlation energies and wave functions were obtained for MRCI calculations compared to the Davidson algorithm.

Conclusions:

• The new dressed matrices algorithm provides a significant computational advantage for large eigenvalue problems.
• Its parallel implementation scales effectively, making it suitable for high-performance computing.
• The algorithm is accurate and efficient for both general matrices and specific quantum chemistry applications.