Given an \(n\) by \(n\) matrix \(A\),
function eigs()
can calculate a specified
number of eigenvalues and eigenvectors of \(A\).
Users can specify the selection criterion 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 the base package. |
dgeMatrix |
General matrix, equivalent to matrix ,
defined in the Matrix package. |
dgCMatrix |
Column oriented sparse matrix, defined in the Matrix package. |
dgRMatrix |
Row oriented sparse matrix, defined in the Matrix package. |
dsyMatrix |
Symmetric matrix, defined in the Matrix package. |
dsCMatrix |
Symmetric column oriented sparse matrix, defined in the Matrix package. |
dsRMatrix |
Symmetric row oriented sparse matrix, defined in the Matrix package. |
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 general results in faster and more stable computation.
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.
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)# S3 method for matrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for dgeMatrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for dsyMatrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for dgCMatrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for dsCMatrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for dgRMatrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for dsRMatrix
eigs(A, k, which = "LM", sigma = NULL, opts = list(), ...)
# S3 method for `function`
eigs(
A,
k,
which = "LM",
sigma = NULL,
opts = list(),
...,
n = NULL,
args = NULL
)
eigs_sym(A, k, which = "LM", sigma = NULL, opts = list(),
lower = TRUE, ...)
# S3 method for `function`
eigs_sym(
A,
k,
which = "LM",
sigma = NULL,
opts = list(),
lower = TRUE,
...,
n = NULL,
args = NULL
)
The matrix whose eigenvalues/vectors are to be computed. It can also be a function which receives a vector \(x\) and calculates \(Ax\). See section Function Interface for details.
Number of eigenvalues requested.
Selection criterion. See Details below.
Shift parameter. See section Shift-And-Invert Mode.
Control parameters related to the computing algorithm. See Details below.
Arguments for specialized S3 function calls, for example
lower
, n
and args
.
Only used when A
is a function, to specify the
dimension of the implicit matrix. See section
Function Interface for details.
Only used when A
is a function. This argument
will be passed to the A
function when it is called.
See section Function Interface for details.
For symmetric matrices, should the lower triangle or upper triangle be used.
A list of converged eigenvalues and eigenvectors.
Computed eigenvalues.
Computed eigenvectors. vectors[, j]
corresponds to values[j]
.
Number of converged eigenvalues.
Number of iterations used in the computation.
Number of matrix operations used in the computation.
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 eigs()
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,
https://docs.scipy.org/doc/scipy/tutorial/arpack.html.
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
.
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. |
eigs()
with matrix types "matrix", "dgeMatrix", "dgCMatrix"
and "dgRMatrix" can use "LM", "SM", "LR", "SR", "LI" and "SI".
eigs_sym()
with all supported matrix types,
and eigs()
with symmetric matrix types
("dsyMatrix", "dsCMatrix", and "dsRMatrix") 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.
initvec
Initial vector of length \(n\) supplied to the
Arnoldi/Lanczos iteration. It may speed up the convergence
if initvec
is close to an eigenvector of \(A\).
# NOT RUN {
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
# }
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