# arpack_defaults

##### ARPACK eigenvector calculation

Interface to the ARPACK library for calculating eigenvectors of sparse matrices

##### Usage

`arpack_defaults`arpack(
func,
extra = NULL,
sym = FALSE,
options = arpack_defaults,
env = parent.frame(),
complex = !sym
)

##### 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 input matrix is never given explicitly.) The second argument is

`extra`

.- 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 you're sure that the results will be real, then supply

`FALSE`

here.

##### 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`eigen_centrality`

,`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.

This parameter is sometimes overwritten by the various functions, e.g.

`page_rank`

always sets ‘`LM`

’.- nev
Numeric scalar. The number of eigenvalues to be computed.

- tol
Numeric scalar. Stopping criterion: the relative accuracy of the Ritz value is considered acceptable if its error is less than

`tol`

times its estimated value. If this is set to zero then machine precision is used.- ncv
Number of Lanczos vectors to be generated.

- ldv
Numberic scalar. It should be set to zero in the current implementation.

- ishift
Either zero or one. If zero then the shifts are provided by the user via reverse communication. If one then exact shifts with respect to the reduced tridiagonal matrix \(T\). Please always set this to one.

- maxiter
Maximum number of Arnoldi update iterations allowed.

- nb
Blocksize to be used in the recurrence. Please always leave this on the default value, one.

- mode
The type of the eigenproblem to be solved. Possible values if the input matrix is symmetric:

- 1
\(Ax=\lambda x\), \(A\) is symmetric.

- 2
\(Ax=\lambda Mx\), \(A\) is symmetric, \(M\) is symmetric positive definite.

- 3
\(Kx=\lambda Mx\), \(K\) is symmetric, \(M\) is symmetric positive semi-definite.

- 4
\(Kx=\lambda KGx\), \(K\) is symmetric positive semi-definite, \(KG\) is symmetric indefinite.

- 5
\(Ax=\lambda Mx\), \(A\) is symmetric, \(M\) is symmetric positive semi-definite. (Cayley transformed mode.)

`mode==1`

was tested and other values might not work properly.Possible values if the input matrix is not symmetric:

- 1
\(Ax=\lambda x\).

- 2
\(Ax=\lambda Mx\), \(M\) is symmetric positive definite.

- 3
\(Ax=\lambda Mx\), \(M\) is symmetric semi-definite.

- 4
\(Ax=\lambda Mx\), \(M\) is symmetric semi-definite.

`mode==1`

was tested and other values might not work properly.- start
Not used currently. Later it be used to set a starting vector.

- sigma
Not used currently.

- sigmai
Not use currently.

- info
- Error flag of ARPACK. Possible values:
- 0
Normal exit.

- 1
Maximum number of iterations taken.

- 3
No shifts could be applied during a cycle of the Implicitly restarted Arnoldi iteration. One possibility is to increase the size of

`ncv`

relative to`nev`

.

- iter
- Number of Arnoldi iterations taken.
- nconv
- Number of “converged” Ritz values. This represents the number of Ritz values that satisfy the convergence critetion.
- numop
- Total number of matrix-vector multiplications.
- numopb
- Not used currently.
- numreo
- Total number of steps of re-orthogonalization.

On output the following additional fields are added:

ARPACK can return more error conditions than these, but they are converted to regular igraph errors.

Please see the ARPACK documentation for additional details.

##### Value

A named list with the following members:

Numeric vector, the desired eigenvalues.

Numeric matrix, the desired
eigenvectors as columns. If `complex=TRUE`

(the default for
non-symmetric problems), then the matrix is complex.

A named
list with the supplied `options`

and some information about the
performed calculation, including an ARPACK exit code. See the details above.

##### Format

An object of class `list`

of length 14.

##### 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).

##### See Also

`eigen_centrality`

, `page_rank`

,
`hub_score`

, `cluster_leading_eigen`

are some of the
functions in igraph which use ARPACK. The ARPACK homepage is at
http://www.caam.rice.edu/software/ARPACK/.

##### Examples

```
# NOT RUN {
# 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[1] <- (length(x)-1)*x[1] - sum(x[-1])
for (i in 2:length(x)) {
y[i] <- x[i] - x[1]
}
y
}
arpack(f, options=list(n=10, nev=1, ncv=3), sym=TRUE)
# double check
eigen(laplacian_matrix(make_star(10, mode="undirected")))
## First three eigenvalues of the adjacency matrix of a graph
## We need the 'Matrix' package for this
if (require(Matrix)) {
set.seed(42)
g <- sample_gnp(1000, 5/1000)
M <- as_adj(g, sparse=TRUE)
f2 <- function(x, extra=NULL) { cat("."); as.vector(M %*% x) }
baev <- arpack(f2, sym=TRUE, options=list(n=vcount(g), nev=3, ncv=8,
which="LM", maxiter=2000))
}
# }
```

*Documentation reproduced from package igraph, version 1.2.5, License: GPL (>= 2)*