arpack

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ARPACK eigenvector calculation

Interface to the ARPACK library for calculating eigenvectors of sparse matrices

Keywords
graphs
Usage
arpack(func, extra = NULL, sym = FALSE, options = igraph.arpack.default,
env = parent.frame(), complex=!sym)
arpack.unpack.complex(vectors, values, nev)
Arguments
func
The function to perform the matrix-vector multiplication. ARPACK requires to perform these by the user. The function gets the vector $x$ as the first argument, and it should return $Ax$, where $A$ is the input matrix. (The
extra
Extra argument to supply to func.
sym
Logical scalar, whether the input matrix is symmetric. Always supply TRUE here if it is, since it can speed up the computation.
options
Options to ARPACK, a named list to overwrite some of the default option values. See details below.
env
The environment in which func will be evaluated.
complex
Whether to convert the eigenvectors returned by ARPACK into R complex vectors. By default this is not done for symmetric problems (these only have real eigenvectors/values), but only non-symmetric ones. If you have a non-symmetric problem, but
vectors
Eigenvectors, as returned by ARPACK.
values
Eigenvalues, as returned by ARPACK
nev
The number of eigenvectors/values to extract. This can be less than or equal to the number of eigenvalues requested in the original ARPACK call.
Details

ARPACK is a library for solving large scale eigenvalue problems. The package is designed to compute a few eigenvalues and corresponding eigenvectors of a general $n$ by $n$ matrix $A$. It is most appropriate for large sparse or structured matrices $A$ where structured means that a matrix-vector product w <- Av requires order $n$ rather than the usual order $n^2$ floating point operations. Please see http://www.caam.rice.edu/software/ARPACK/ for details.

This function is an interface to ARPACK. igraph does not contain all ARPACK routines, only the ones dealing with symmetric and non-symmetric eigenvalue problems using double precision real numbers.

The eigenvalue calculation in ARPACK (in the simplest case) involves the calculation of the $Av$ product where $A$ is the matrix we work with and $v$ is an arbitrary vector. The function supplied in the fun argument is expected to perform this product. If the product can be done efficiently, e.g. if the matrix is sparse, then arpack is usually able to calculate the eigenvalues very quickly.

The options argument specifies what kind of calculation to perform. It is a list with the following members, they correspond directly to ARPACK parameters. On input it has the following fields:

• bmat
{Character constant, possible values: I, stadard eigenvalue problem, $Ax=\lambda x$; and G, generalized eigenvalue problem, $Ax=\lambda B x$. Currently only I is supported.} n{Numeric scalar. The dimension of the eigenproblem. You only need to set this if you call arpack directly. (I.e. not needed for evcent, page.rank, etc.)} which{Specify which eigenvalues/vectors to compute, character constant with exactly two characters.

Possible values for symmetric input matrices:

• LA
{Compute nev largest (algebraic) eigenvalues.} SA{Compute nev smallest (algebraic) eigenvalues.} LM{Compute nev largest (in magnitude) eigenvalues.} SM{Compute nev smallest (in magnitude) eigenvalues.} BE{Compute nev eigenvalues, half from each end of the spectrum. When nev is odd, compute one more from the high end than from the low end.} }

Possible values for non-symmetric input matrices:

• LM
{Compute nev eigenvalues of largest magnitude.} SM{Compute nev eigenvalues of smallest magnitude.} LR{Compute nev eigenvalues of largest real part.} SR{Compute nev eigenvalues of smallest real part.} LI{Compute nev eigenvalues of largest imaginary part.} SI{Compute nev eigenvalues of smallest imaginary part.}

Value

• A named list with the following members:
• valuesNumeric vector, the desired eigenvalues.
• vectorsNumeric matrix, the desired eigenvectors as columns. If complex=TRUE (the default for non-symmetric problems), then the matrix is complex.
• optionsA named list with the supplied options and some information about the performed calculation, including an ARPACK exit code. See the details above.

• Eigenvalues
• Eigenvectors
• ARPACK

arpack

LM

• nev
• tol
• ncv
• ldv
• ishift
• maxiter
• nb
• mode
• 2
• 3
• 4
• 5
• 2
• 3
• 4
• start
• sigma
• sigmai
• 1
• 3
• iter
• nconv
• numop
• numopb
• numreo

eqn

$M$

• 0

converged

References

D.C. Sorensen, Implicit Application of Polynomial Filters in a k-Step Arnoldi Method. SIAM J. Matr. Anal. Apps., 13 (1992), pp 357-385. R.B. Lehoucq, Analysis and Implementation of an Implicitly Restarted Arnoldi Iteration. Rice University Technical Report TR95-13, Department of Computational and Applied Mathematics. B.N. Parlett & Y. Saad, Complex Shift and Invert Strategies for Real Matrices. Linear Algebra and its Applications, vol 88/89, pp 575-595, (1987).

evcent, page.rank, hub.score, leading.eigenvector.community are some of the functions in igraph which use ARPACK. The ARPACK homepage is at http://www.caam.rice.edu/software/ARPACK/.

Aliases
• arpack
• arpack-options
• igraph.arpack.default
• arpack.unpack.complex
Examples
# Identity matrix
f <- function(x, extra=NULL) x
arpack(f, options=list(n=10, nev=2, ncv=4), sym=TRUE)

# Graph laplacian of a star graph (undirected), n>=2
# Note that this is a linear operation
f <- function(x, extra=NULL) {
y <- x
y <- (length(x)-1)*x - sum(x[-1])
for (i in 2:length(x)) {
y[i] <- x[i] - x
}
y
}

arpack(f, options=list(n=10, nev=1, ncv=3), sym=TRUE)

# double check
eigen(graph.laplacian(graph.star(10, mode="undirected")))
Documentation reproduced from package igraph, version 0.5.5-3, License: GPL (>= 2)

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