Solvers for Large Scale Eigenvalue and SVD Problems

Introduction

rARPACK was originally an R wrapper of the ARPACK library to solve large scale eigen value/vector problems. From version 0.8-0 it changed the backend to the Spectra library. This R package is typically used to compute a few eigen values/vectors of an n by n matrix, e.g., the k largest eigen values, which is usually more efficient than eigen() if k << n.

Currently this package provides function eigs() for eigenvalue/eigenvector problems, and svds() for truncated SVD. Different matrix types in R, including sparse matrices, are supported. Below is a list of implemented ones:

• matrix (defined in base R)
• dgeMatrix (defined in Matrix package, for general matrices)
• dsyMatrix (defined in Matrix package, for symmetric matrices)
• dgCMatrix (defined in Matrix package, for column oriented sparse matrices)
• dgRMatrix (defined in Matrix package, for row oriented sparse matrices)
• function (implicitly specify the matrix by providing a function that calculates matrix product A %*% x)

Example

We first generate some matrices:

library(Matrix)
n = 20
k = 5

set.seed(111)
A1 = matrix(rnorm(n^2), n)  ## class "matrix"
A2 = Matrix(A1)             ## class "dgeMatrix"

General matrices have complex eigenvalues:

eigs(A1, k)
eigs(A2, k, opts = list(retvec = FALSE))  ## eigenvalues only

rARPACK also works on sparse matrices:

A1[sample(n^2, n^2 / 2)] = 0
A3 = as(A1, "dgCMatrix")
A4 = as(A1, "dgRMatrix")

eigs(A3, k)
eigs(A4, k)

Function interface is also supported:

f = function(x, args)
{
    as.numeric(args %*% x)
}
eigs(f, k, n = n, args = A3)

Symmetric matrices have real eigenvalues.

A5 = crossprod(A1)
eigs_sym(A5, k)

To find the smallest (in absolute value) k eigenvalues of A5, we have two approaches:

eigs_sym(A5, k, which = "SM")
eigs_sym(A5, k, sigma = 0)

The results should be the same, but the latter method is far more stable on large matrices.

For SVD problems, you can specify the number of singular values (k), number of left singular vectors (nu) and number of right singular vectors(nv).

m = 100
n = 20
k = 5
set.seed(111)
A = matrix(rnorm(m * n), m)

svds(A, k)
svds(t(A), k, nu = 0, nv = 3)

Similar to eigs(), svds() supports sparse matrices:

A[sample(m * n, m * n / 2)] = 0
Asp1 = as(A, "dgCMatrix")
Asp2 = as(A, "dgRMatrix")

svds(Asp1, k)
svds(Asp2, k, nu = 0, nv = 0)