arnoldi {sanic}

Krylov Subspace Spectral Decomposition

Description

Arnoldi iteration and Lanczos method to iteratively approximate the Hessenberg or tridiagonal form of a matrix A and find its eigenvalues.

Usage

arnoldi(
  a,
  b,
  symmetric,
  iter = nrow(a),
  tol = .Machine$double.eps,
  eigen = TRUE,
  orthogonalise = TRUE
)

lanczos(
  a,
  b,
  iter = nrow(a),
  tol = .Machine$double.eps,
  eigen = TRUE,
  orthogonalise = TRUE
)

Arguments

a

Square numeric matrix.

b

Arbitrary numeric non-zero vector used to construct the basis.

symmetric

Logical scalar indicating whether 'a' is symmetric. By default symmetry is checked up to machine precision, which may take a long time for symmetric matrices.

iter

Integer scalar with the maximum number of iterations. Defaults to the theoretical maximum, i.e. the number of columns in 'a'.

tol

Numeric scalar with the desired tolerance. Defaults to the machine precision.

eigen

Logical scalar indicating whether to compute eigenvalues from the decomposition.

orthogonalise

Logical scalar indicating whether to use plain Lanczos or full reorthogonalisation. Defaults to reorthogonalisation.

Value

Returns a list with slots "H" for the Hessenberg form of 'a' or slots "diagonal" and "subdiagonal" for its triangular form, slot "Q" with the orthonormal basis, and, if requested, eigenvalues in the slot "values".

Examples

set.seed(42)
# Compute Hessenberg of a square matrix
A <- matrix(rnorm(9), nrow = 3, ncol = 3)
ks <- arnoldi(A, symmetric = FALSE)

# Compute tridiagonal of a symmetric matrix
A <- crossprod(matrix(rnorm(9), nrow = 3, ncol = 3))
ks <- lanczos(A)
ks <- arnoldi(A, symmetric = TRUE) # Short-hand