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.

Value

A named list with the following members:

values

Numeric vector, the desired eigenvalues.

vectors

Numeric matrix, the desired eigenvectors as columns. If complex=TRUE (the default for non-symmetric problems), then the matrix is complex.

options

A named list with the supplied options and some information about the performed calculation, including an ARPACK exit code. See the details above.

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.

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’, standard 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.)

Please note that only 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.

Please note that only 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.

On output the following additional fields are added:

Please see the ARPACK documentation for additional details.

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 that use ARPACK.

Author

Rich Lehoucq, Kristi Maschhoff, Danny Sorensen, Chao Yang for ARPACK, Gabor Csardi csardi.gabor@gmail.com for the R interface.

Examples

# Identity matrix
f <- function(x, extra = NULL) x
arpack(f, options = list(n = 10, nev = 2, ncv = 4), sym = TRUE)
#> $values
#> [1] 1 1
#> 
#> $vectors
#>             [,1]         [,2]
#>  [1,] -0.3726998  0.112735486
#>  [2,]  0.1195827  0.041517695
#>  [3,]  0.4062664 -0.421475667
#>  [4,] -0.4452969 -0.709645619
#>  [5,]  0.3098338  0.177672414
#>  [6,]  0.1653131 -0.125415897
#>  [7,] -0.1946057  0.482549613
#>  [8,]  0.4619532  0.006154024
#>  [9,] -0.2346540  0.121470373
#> [10,]  0.2319333 -0.096874464
#> 
#> $options
#> $options$bmat
#> [1] "I"
#> 
#> $options$n
#> [1] 10
#> 
#> $options$which
#> [1] "XX"
#> 
#> $options$nev
#> [1] 2
#> 
#> $options$tol
#> [1] 0
#> 
#> $options$ncv
#> [1] 4
#> 
#> $options$ldv
#> [1] 0
#> 
#> $options$ishift
#> [1] 1
#> 
#> $options$maxiter
#> [1] 3000
#> 
#> $options$nb
#> [1] 1
#> 
#> $options$mode
#> [1] 1
#> 
#> $options$start
#> [1] 0
#> 
#> $options$sigma
#> [1] 0
#> 
#> $options$sigmai
#> [1] 0
#> 
#> $options$info
#> [1] 0
#> 
#> $options$iter
#> [1] 1
#> 
#> $options$nconv
#> [1] 2
#> 
#> $options$numop
#> [1] 4
#> 
#> $options$numopb
#> [1] 0
#> 
#> $options$numreo
#> [1] 4
#> 
#> 

# 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)
#> $values
#> [1] 10
#> 
#> $vectors
#>  [1] -0.9486833  0.1054093  0.1054093  0.1054093  0.1054093  0.1054093
#>  [7]  0.1054093  0.1054093  0.1054093  0.1054093
#> 
#> $options
#> $options$bmat
#> [1] "I"
#> 
#> $options$n
#> [1] 10
#> 
#> $options$which
#> [1] "XX"
#> 
#> $options$nev
#> [1] 1
#> 
#> $options$tol
#> [1] 0
#> 
#> $options$ncv
#> [1] 3
#> 
#> $options$ldv
#> [1] 0
#> 
#> $options$ishift
#> [1] 1
#> 
#> $options$maxiter
#> [1] 3000
#> 
#> $options$nb
#> [1] 1
#> 
#> $options$mode
#> [1] 1
#> 
#> $options$start
#> [1] 0
#> 
#> $options$sigma
#> [1] 0
#> 
#> $options$sigmai
#> [1] 0
#> 
#> $options$info
#> [1] 0
#> 
#> $options$iter
#> [1] 1
#> 
#> $options$nconv
#> [1] 1
#> 
#> $options$numop
#> [1] 3
#> 
#> $options$numopb
#> [1] 0
#> 
#> $options$numreo
#> [1] 3
#> 
#> 

# double check
eigen(laplacian_matrix(make_star(10, mode = "undirected")))
#> eigen() decomposition
#> $values
#>  [1] 1.000000e+01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#>  [6] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 3.552714e-15
#> 
#> $vectors
#>             [,1]          [,2]         [,3]          [,4]          [,5]
#>  [1,]  0.9486833  0.000000e+00  0.000000000  0.000000e+00  0.000000e+00
#>  [2,] -0.1054093  9.251859e-18  0.000000000 -9.251859e-18  1.850372e-17
#>  [3,] -0.1054093 -9.316303e-02  0.079217213 -5.526928e-02 -5.860537e-02
#>  [4,] -0.1054093  9.295069e-01  0.020878444 -5.681485e-02  2.420873e-02
#>  [5,] -0.1054093 -1.772059e-01 -0.072634120 -1.521102e-01  1.990548e-02
#>  [6,] -0.1054093 -1.772059e-01  0.014765896  3.109043e-01  1.456789e-01
#>  [7,] -0.1054093 -1.246177e-01  0.008447773  6.043444e-02  8.117572e-01
#>  [8,] -0.1054093 -6.376015e-02 -0.659513765  4.360577e-01 -3.697494e-01
#>  [9,] -0.1054093 -1.772059e-01 -0.124196522 -7.876367e-01 -2.016864e-01
#> [10,] -0.1054093 -1.163484e-01  0.733035080  2.444346e-01 -3.715091e-01
#>                [,6]        [,7]       [,8]          [,9]      [,10]
#>  [1,]  0.000000e+00  0.00000000  0.0000000  0.000000e+00 -0.3162278
#>  [2,]  9.251859e-18  0.00000000  0.9428090  6.476301e-17 -0.3162278
#>  [3,]  3.877514e-02  0.07938499 -0.1178511  9.196470e-01 -0.3162278
#>  [4,]  2.633343e-02  0.05841931 -0.1178511 -5.158318e-02 -0.3162278
#>  [5,]  8.820081e-01 -0.01683834 -0.1178511 -1.912244e-01 -0.3162278
#>  [6,] -1.516578e-01  0.81604496 -0.1178511 -1.912244e-01 -0.3162278
#>  [7,] -1.927091e-01 -0.39623153 -0.1178511 -5.158318e-02 -0.3162278
#>  [8,] -1.420878e-01 -0.29370999 -0.1178511 -5.158318e-02 -0.3162278
#>  [9,] -3.596255e-01  0.03503021 -0.1178511 -1.912244e-01 -0.3162278
#> [10,] -1.010365e-01 -0.28209962 -0.1178511 -1.912244e-01 -0.3162278
#> 

## 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_adjacency_matrix(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
  ))
}
#> Loading required package: Matrix
#> .....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................