R: Find a Specified Number of Eigenvalues/vectors for Square...
eigs
R Documentation
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:
matrix
The most commonly used matrix type,
defined in base package.
dgeMatrix
General matrix, equivalent to matrix,
defined in Matrix package.
dgCMatrix
Column oriented sparse matrix, defined in
Matrix package.
dgRMatrix
Row oriented sparse matrix, defined in
Matrix package.
dsyMatrix
Symmetrix matrix, defined in Matrix
package.
function
Implicitly specify the matrix through a
function that has the effect of calculating
f(x) = A * x. See section
Function Interface for details.
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.
The matrix whose eigenvalues/vectors are to be computed.
It can also be a function which receives a vector x
and calculates A * x.
See section Function Interface for details.
k
Number of eigenvalues requested.
which
Selection criteria. See Details below.
sigma
Shift parameter. See section Shift-And-Invert Mode.
opts
Control parameters related to the computing
algorithm. See Details below.
lower
For symmetric matrices, should the lower triangle
or upper triangle be used.
...
Additional arguments such as n and args that are
related to the Function Interface. See
eigs() in the RSpectra package.
Details
The which argument is a character string
that specifies the type of eigenvalues to be computed.
Possible values are:
"LM"
The k eigenvalues with largest magnitude. Here the
magnitude means the Euclidean norm of complex numbers.
"SM"
The k eigenvalues with smallest magnitude.
"LR"
The k eigenvalues with largest real part.
"SR"
The k eigenvalues with smallest real part.
"LI"
The k eigenvalues with largest imaginary part.
"SI"
The k eigenvalues with smallest imaginary part.
"LA"
The k largest (algebraic) eigenvalues, considering any
negative sign.
"SA"
The k smallest (algebraic) eigenvalues, considering any
negative sign.
"BE"
Compute k eigenvalues, half from each end of the
spectrum. When k is odd, compute more from the high
and then from the low end.
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 <= ncv <= n, and
for symmetric matrix, the constraint is
k < ncv <= 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.
values
Computed eigenvalues.
vectors
Computed eigenvectors. vectors[, j] corresponds to values[j].
nconv
Number of converged eigenvalues.
niter
Number of iterations used in the computation.
nops
Number of matrix operations used in the computation.
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
1/(λ-σ)
where λ's are the eigenvalues of A. This mode is useful
when user wants to find eigenvalues closest to a given number.
For example, if σ=0, then which = "LM" will select the
largest values of 1/|λ|, 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
A * x, 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.
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