lobpcg.md

[Locally optimal block preconditioned conjugate gradient (LOBPCG)](@id LOBPCG)

Solves the generalized eigenproblem $Ax = λBx$ approximately where $A$ and $B$ are Hermitian linear maps, and $B$ is positive definite. $B$ is taken to be the identity by default. It can find the smallest (or largest) k eigenvalues and their corresponding eigenvectors which are B-orthonormal. It also admits a preconditioner and a "constraints" matrix C, such that the algorithm returns the smallest (or largest) eigenvalues associated with the eigenvectors in the nullspace of C'B.

Usage

lobpcg
lobpcg!

Implementation Details

A LOBPCGIterator is created to pre-allocate all the memory required by the method using the constructor LOBPCGIterator(A, B, largest, X, P, C) where A and B are the matrices from the generalized eigenvalue problem, largest indicates if the problem is a maximum or minimum eigenvalue problem, X is the initial eigenbasis, randomly sampled if not input, where size(X, 2) is the block size bs. P is the preconditioner, nothing by default, and C is the constraints matrix. The desired k eigenvalues are found bs at a time.

A deterministic seed is used for generating pseudo-random initial data for the algorithm; this can be controlled by passing a different pseudorandom number generator (an AbstractRNG) via the rng keyword argument.

References

Implementation is based on ¹ and ².

Footnotes

1. Andrew V. Knyazev. "Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method" SIAM Journal on Scientific Computing, 23(2):517–541 2001. ↩

2. See Scipy LOBPCG implementation ↩