Find a Specified Number of Eigenvalues/vectors for Square Matrix

Description

This function is a simple wrapper of the eigs() function in the RSpectra package. Also see the documentation there.

Given an n by n matrix A, function eigs() can calculate a limited number of eigenvalues and eigenvectors of A. Users can specify the selection criteria by argument which, e.g., choosing the k largest or smallest eigenvalues and the corresponding eigenvectors.

Currently eigs() supports matrices of the following classes:

eigs_sym() assumes the matrix is symmetric, and only the lower triangle (or upper triangle, which is controlled by the argument lower) is used for computation, which guarantees that the eigenvalues and eigenvectors are real, and in some cases reduces the workload. One exception is when A is a function, in which case the user is responsible for the symmetry of the operator.

eigs_sym() supports "matrix", "dgeMatrix", "dgCMatrix", "dgRMatrix" and "function" typed matrices.

Usage

eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)

eigs_sym(A, k, which = "LM", sigma = NULL, opts = list(),
   lower = TRUE, ...)

Arguments

Details

The which argument is a character string that specifies the type of eigenvalues to be computed. Possible values are:

eigs() with matrix type "matrix", "dgeMatrix", "dgCMatrix" and "dgRMatrix" can use "LM", "SM", "LR", "SR", "LI" and "SI".

eigs_sym(), and eigs() with matrix type "dsyMatrix" can use "LM", "SM", "LA", "SA" and "BE".

The opts argument is a list that can supply any of the following parameters:

ncv

Number of Lanzcos basis vectors to use. More vectors will result in faster convergence, but with greater memory use. For general matrix, ncv must satisfy k+2\le ncv \le n, and for symmetric matrix, the constraint is k < ncv \le n. Default is min(n, max(2*k+1, 20)).

tol

Precision parameter. Default is 1e-10.

maxitr

Maximum number of iterations. Default is 1000.

retvec

Whether to compute eigenvectors. If FALSE, only calculate and return eigenvalues.

Value

A list of converged eigenvalues and eigenvectors.

Shift-And-Invert Mode

The sigma argument is used in the shift-and-invert mode.

When sigma is not NULL, the selection criteria specified by argument which will apply to

\frac{1}{\lambda-\sigma}

where \lambda's are the eigenvalues of A. This mode is useful when user wants to find eigenvalues closest to a given number. For example, if \sigma=0, then which = "LM" will select the largest values of 1/|\lambda|, which turns out to select eigenvalues of A that have the smallest magnitude. The result of using which = "LM", sigma = 0 will be the same as which = "SM", but the former one is preferable in that ARPACK is good at finding large eigenvalues rather than small ones. More explanation of the shift-and-invert mode can be found in the SciPy document, http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html.

Function Interface

The matrix A can be specified through a function with the definition

function(x, args)
{
    ## should return A %*% x
}

which receives a vector x as an argument and returns a vector of the same length. The function should have the effect of calculating Ax, and extra arguments can be passed in through the args parameter. In eigs(), user should also provide the dimension of the implicit matrix through the argument n.

Author(s)

Yixuan Qiu http://statr.me

Jiali Mei vermouthmjl@gmail.com

See Also

eigen(), svd(), svds()

Examples

library(Matrix)
n = 20
k = 5

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

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

## 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
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)

## find the smallest (in absolute value) k eigenvalues of A5
eigs_sym(A5, k, which = "SM")

## another way to do this: use the sigma argument
eigs_sym(A5, k, sigma = 0)

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

Find the Largest k Singular Values/Vectors of a Matrix

Description

This function is a simple wrapper of the svds() function in the RSpectra package. Also see the documentation there.

Given an m by n matrix A, function svds() can find its largest k singular values and the corresponding singular vectors. It is also called the Truncated Singular Value Decomposition since it only contains a subset of the whole singular triplets.

Currently svds() supports matrices of the following classes:

Note that when A is symmetric, SVD reduces to eigen decomposition, so you may consider using eigs() instead.

Usage

svds(A, k, nu = k, nv = k, opts = list(), ...)

Arguments

Details

The opts argument is a list that can supply any of the following parameters:

ncv

Number of Lanzcos basis vectors to use. More vectors will result in faster convergence, but with greater memory use. ncv must be satisfy k < ncv \le p where p = min(m, n). Default is min(p, max(2*k+1, 20)).

tol

Precision parameter. Default is 1e-10.

maxitr

Maximum number of iterations. Default is 1000.

Value

A list with the following components:

Author(s)

Yixuan Qiu <http://statr.me>

See Also

eigen(), svd(), eigs().

Examples

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)

## Sparse matrices
library(Matrix)
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)