felipez / eigenvalues Goto Github PK

The search space matrix V is doubled in each iteration, forcing a restart after just a few steps. Instead, if we add to the search space just the number of eigenpairs nvalues we speedup the algorithm at the cos of some small precision in each iteration.

Once a given eigenvalue/eigenvector has converged it should be deflect to improve the convergence of the to other eigenvalues/eigenvectors see this paper

Generalized the algorithm to other numerical types

Feature

Use PyO3 to provide a Python interface for the library.

Replace the call to the QR method by a modified Gram-Schmidt orthogonalization.

TODO:

• implement modified Gram-Schmidt
• Replace QR by Gram-Scmidt
• Orthogonalize only the new correction vectors against the old basis
• Make a block update for the Davidson search space

The eigenvalues/eigenvectors pairs computed by nalgebra::linalg::SymmetricEigen are not sorted

Implement the Hermitian Lanczos algorithm

Use the residues computed during the convergency checking to compute the new correction

Generate the orthonormal subspace different from the identity matrix when the diagonal values are not sorted.

Tell us what would be a nice feature to make the eigenvalues library even better!

Implement the solution for the general eigenvalue problem

Allow the user to decide for the Davidson method if compute the lowest, highest or somewhere in the middle (close to a given value) eigenvalues.
Currently, only the lowest eigenvalues are computed.
TODO:

• Lowest
• Highest
• In the middle

TODO:

• Vectorize the modified Gram-Schmidt.
• Implement a 2 iterations modified Gram-Schimdt, see: http://stoppels.blog/posts/orthogonalization-performance
• Normalize only the new vectors.

Call directly the nalgebra-lapack interface

When I build and cargo run, I met the case that the computed is not equal to expected. I can repeat it once in several runs.

Computed eigenvalues:

┌ ┐
│ 265.16850862887065 │
│ 231.55356256737628 │
│ 200.46232307251603 │
└ ┘

Expected eigenvalues:

┌ ┐
│ 231.5535625673764 │
│ 199.51034256887814 │
│ 198.36444600986013 │
└ ┘