Generalized Eigenvalue Decomposition — geneig • multivarious

Computes the generalized eigenvalues and eigenvectors for the problem: A x = λ B x. Supports multiple dense and iterative solvers with a unified eigenpair selection interface.

Usage

geneig(
  A = NULL,
  B = NULL,
  ncomp = 2,
  preproc = prep(pass()),
  method = c("robust", "sdiag", "geigen", "primme", "rspectra", "subspace"),
  which = "LA",
  ...
)

Arguments

A

The left-hand side square matrix.

B

The right-hand side square matrix, same dimension as A.

ncomp

Number of eigenpairs to return.

preproc

A preprocessing function to apply to the matrices before solving the generalized eigenvalue problem.

method

One of:

"robust": Uses a stable decomposition via a whitening transform (B must be symmetric PD).
"sdiag": Uses a spectral decomposition of B (must be symmetric PD). Requires A to be symmetric for meaningful results.
"geigen": Uses the geigen package for a general solution (A and B can be non-symmetric).
"primme": Uses the PRIMME package for large/sparse symmetric problems (A and B must be symmetric).
"rspectra": Uses RSpectra; if B is SPD it calls eigs_sym(A, B, ...) directly, otherwise it applies a reciprocal transform to support all targets.
"subspace": Block subspace iteration for symmetric pairs with SPD B (iterative, no external package required).

which

Which eigenpairs to return. One of "LA" (largest algebraic), "SA" (smallest algebraic), "LM" (largest magnitude), or "SM" (smallest magnitude). Aliases: "top"/"largest" → "LA", "bottom"/"smallest" → "SA". Dense backends select eigenpairs post hoc; "primme" supports "LA", "SA", "SM" (not "LM"); "rspectra" honors all four options. Default is "LA".

...

Additional arguments to pass to the underlying solver.

Value

A projector object with generalized eigenvectors and eigenvalues.

References

Golub, G. H. & Van Loan, C. F. (2013) Matrix Computations, 4th ed., § 8.7 – textbook derivation for the "robust" (Cholesky) and "sdiag" (spectral) transforms.

Moler, C. & Stewart, G. (1973) "An Algorithm for Generalized Matrix Eigenvalue Problems". SIAM J. Numer. Anal., 10 (2): 241‑256 – the QZ algorithm behind the geigen backend.

Stathopoulos, A. & McCombs, J. R. (2010) "PRIMME: PReconditioned Iterative Multi‑Method Eigensolver". ACM TOMS 37 (2): 21:1‑21:30 – the algorithmic core of the primme backend.

See also the geigen (CRAN) and PRIMME documentation.

Examples

# \donttest{
# Simulate two matrices
set.seed(123)
A <- matrix(rnorm(50 * 50), 50, 50)
B <- matrix(rnorm(50 * 50), 50, 50)
A <- A %*% t(A) # Make A symmetric
B <- B %*% t(B) + diag(50) * 0.1 # Make B symmetric positive definite

# Solve generalized eigenvalue problem
result <- geneig(A = A, B = B, ncomp = 3)
# }