LOBPCG

# Locally optimal block preconditioned conjugate gradient (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

The Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)

Finds the nev extremal eigenvalues and their corresponding eigenvectors satisfying AX = λBX.

A and B may be generic types but Base.mul!(C, AorB, X) must be defined for vectors and strided matrices X and C. size(A, i::Int) and eltype(A) must also be defined for A.

lobpcg(A, [B,] largest, nev; kwargs...) -> results

Arguments

• A: linear operator;
• B: linear operator;
• largest: true if largest eigenvalues are desired and false if smallest;
• nev: number of eigenvalues desired.

Keywords

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• P: preconditioner of residual vectors, must overload ldiv!;

• C: constraint to deflate the residual and solution vectors orthogonal to a subspace; must overload mul!;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source
lobpcg(A, [B,] largest, X0; kwargs...) -> results

Arguments

• A: linear operator;
• B: linear operator;
• largest: true if largest eigenvalues are desired and false if smallest;
• X0: Initial guess, will not be modified. The number of columns is the number of eigenvectors desired.

Keywords

• not_zeros: default is false. If true, X0 will be assumed to not have any all-zeros column.

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• P: preconditioner of residual vectors, must overload ldiv!;

• C: constraint to deflate the residual and solution vectors orthogonal to a subspace; must overload mul!;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source

lobpcg(A, [B,] largest, X0, nev; kwargs...) -> results

Arguments

• A: linear operator;
• B: linear operator;
• largest: true if largest eigenvalues are desired and false if smallest;
• X0: block vectors such that the eigenvalues will be found size(X0, 2) at a time; the columns are also used to initialize the first batch of Ritz vectors;
• nev: number of eigenvalues desired.

Keywords

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• P: preconditioner of residual vectors, must overload ldiv!;

• C: constraint to deflate the residual and solution vectors orthogonal to a subspace; must overload mul!;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source
lobpcg!(iterator::LOBPCGIterator; kwargs...) -> results

Arguments

• iterator::LOBPCGIterator: a struct having all the variables required for the LOBPCG algorithm.

Keywords

• not_zeros: default is false. If true, the initial Ritz vectors will be assumed to not have any all-zeros column.

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source

## 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.

## References

Implementation is based on [Knyazev1993] and [Scipy].

[Knyazev1993]

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.