Albers' example Matrix with "Difficult" Eigen Factorization

An example of a sparse matrix for which eigen() seemed to be difficult, an unscaled version of this has been posted to the web, accompanying an E-mail to R-help (https://stat.ethz.ch/mailman/listinfo/r-help), by Casper J Albers, Open University, UK.

data(CAex)

Format

This is a \(72 \times 72\) symmetric matrix with 216 non-zero entries in five bands, stored as sparse matrix of class dgCMatrix.

Details

Historical note (2006-03-30): In earlier versions of R, eigen(CAex) fell into an infinite loop whereas eigen(CAex, EISPACK=TRUE) had been okay.

Examples

data(CAex, package = "Matrix")
str(CAex) # of class "dgCMatrix"
#> Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
#>   ..@ i       : int [1:216] 0 24 48 1 25 49 2 26 50 3 ...
#>   ..@ p       : int [1:73] 0 3 6 9 12 15 18 21 24 27 ...
#>   ..@ Dim     : int [1:2] 72 72
#>   ..@ Dimnames:List of 2
#>   .. ..$ : NULL
#>   .. ..$ : NULL
#>   ..@ x       : num [1:216] 0.999998 -0.001312 -0.000527 0.999999 -0.000903 ...
#>   ..@ factors : list()

image(CAex)# -> it's a simple band matrix with 5 bands

## and the eigen values are basically 1 (42 times) and 0 (30 x):
zapsmall(ev <- eigen(CAex, only.values=TRUE)$values)
#>  [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [39] 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## i.e., the matrix is symmetric, hence
sCA <- as(CAex, "symmetricMatrix")
## and
stopifnot(class(sCA) == "dsCMatrix",
          as(sCA, "matrix") == as(CAex, "matrix"))