Module sparse_eigen 

Unified sparse eigenvalue interface

This module provides a single entry point for computing eigenvalues and eigenvectors of large sparse matrices, automatically selecting the best algorithm based on matrix properties.

Supported Backends

• LOBPCG: Locally Optimal Block Preconditioned Conjugate Gradient. Best for SPD matrices when the smallest/largest eigenvalues are needed.
• IRAM: Implicitly Restarted Arnoldi Method. Best for general (non-symmetric) matrices or when interior eigenvalues are needed.
• Thick-Restart Lanczos: Best for symmetric matrices with moderate dimension Krylov subspace.

Shift-and-Invert

For computing eigenvalues near a given target sigma, the module provides a shift-and-invert wrapper that transforms the problem so that eigenvalues near sigma become the dominant (largest magnitude) eigenvalues of the shifted-and-inverted operator (A - sigma I)^{-1}.

Spectral Transformations

• Standard: Compute eigenvalues of A directly
• Shift-Invert: Compute eigenvalues of (A - sigma I)^{-1}
• Buckling: Compute eigenvalues of (K - sigma KG)^{-1} K
• Cayley: Compute eigenvalues of (A - sigma I)^{-1}(A + sigma I)

References

• Lehoucq, R.B., Sorensen, D.C., & Yang, C. (1998). “ARPACK Users’ Guide”. SIAM.
• Knyazev, A.V. (2001). “Toward the optimal preconditioned eigensolver: LOBPCG”. SIAM J. Sci. Comput.

Structs

SparseEigenConfig

Configuration for the unified sparse eigenvalue solver.

SparseEigenResult

Unified eigenvalue decomposition result.

Enums

EigenMethod

Which eigenvalue algorithm to use.

EigenvalueTarget

Which eigenvalues to compute.

SpectralTransform

Spectral transformation to apply.

Functions

cayley_transform_matvec

Apply Cayley transformation: (A - sigma I)^{-1} * (A + sigma I) * v.

check_eigenpairs

Check if all eigenpairs satisfy the residual tolerance.

compute_residuals

Compute residual norms ||A*v_i - lambda_i * v_i|| for each eigenpair.

shift_invert_eig

Compute eigenvalues nearest to sigma using shift-and-invert.

sparse_eig

Compute eigenvalues and eigenvectors of a sparse matrix.

sparse_eigs

Convenience function: compute eigenvalues of a general (possibly non-symmetric) matrix.

sparse_eigsh

Convenience function: compute eigenvalues of a symmetric matrix.