R help - Dec 2014 - Difference between eigs() and eigen()

Dear R users and contributors,

I recently observed a difference between the outputs of the classic
eigen() function, and the Arnoldi variant eigs() that extracts only
the few first eigenpairs. Here is some sample code illustrating the
problem:

library(rARPACK)
library(speccalt)
set.seed(1)

# compute kernel matrix from rows of synth5
# then its Laplacian
kern <- local.rbfdot(synth5)
diag(kern) <- 0
deg <- sapply(1:(dim(synth5)[1]), function(i) {
return(sum(kern[i,]))
})
L <- diag(1/sqrt(deg)) %*% kern %*% diag(1/sqrt(deg))

eig1 <- eigs(L, 6)
eig2 <- eigen(L, symmetric=TRUE)

eig1$values then reads:
1.0000000 1.0000000 0.9993805 0.9992561 0.9985084 0.9975311

whereas eig2$values reads:
1.0000000 1.0000000 1.0000000 1.0000000 0.9993805 0.9992561
which is the correct result (eigenvalue 1 has multiplicity 4 in that example).

I guess there is an issue between Arnoldi methods and eigenvalues with
multiplicities greater than 1 (indeed at the end of the series the
unique eigenvals look identical), but as the problem is not documented
in the package PDF, I'm quite unclear if this is
implementation-specific or Arnoldi-general... The issue is quite
important in my case, as the associated eigenvectors then differ quite
significantly, and this impacts negatively my further operations.

I guess the next step is to dig into the mathematical literature, but
before this I wondered if someone already encountered this issue?

Any help would be appreciated,
Pierrick

eigs() is from a contributed package. No idea what it is about, but my 
guess is these are actually numerical differences coming from different 
algorithms used to calculate the eigenvalues.
For details, please ask the author of the corresponding contributed package.

Best,
Uwe Ligges


On 29.12.2014 19:02, Pierrick Bruneau wrote:
> Dear R users and contributors,
>
> I recently observed a difference between the outputs of the classic
> eigen() function, and the Arnoldi variant eigs() that extracts only
> the few first eigenpairs. Here is some sample code illustrating the
> problem:
>
> library(rARPACK)
> library(speccalt)
> set.seed(1)
>
> # compute kernel matrix from rows of synth5
> # then its Laplacian
> kern <- local.rbfdot(synth5)
> diag(kern) <- 0
> deg <- sapply(1:(dim(synth5)[1]), function(i) {
> return(sum(kern[i,]))
> })
> L <- diag(1/sqrt(deg)) %*% kern %*% diag(1/sqrt(deg))
>
> eig1 <- eigs(L, 6)
> eig2 <- eigen(L, symmetric=TRUE)
>
> eig1$values then reads:
> 1.0000000 1.0000000 0.9993805 0.9992561 0.9985084 0.9975311
>
> whereas eig2$values reads:
> 1.0000000 1.0000000 1.0000000 1.0000000 0.9993805 0.9992561
> which is the correct result (eigenvalue 1 has multiplicity 4 in that
example).
>
> I guess there is an issue between Arnoldi methods and eigenvalues with
> multiplicities greater than 1 (indeed at the end of the series the
> unique eigenvals look identical), but as the problem is not documented
> in the package PDF, I'm quite unclear if this is
> implementation-specific or Arnoldi-general... The issue is quite
> important in my case, as the associated eigenvectors then differ quite
> significantly, and this impacts negatively my further operations.
>
> I guess the next step is to dig into the mathematical literature, but
> before this I wondered if someone already encountered this issue?
>
> Any help would be appreciated,
> Pierrick
>
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