[R] a bug with LAPACK ? non orthogonal vectors obtained with eigen of a symmetric matrix

[Roger D. Peng]

This is interesting.  I can reproduce your results but cannot come up 
with an explanation.  However, using svd(LINPACK = FALSE) seems to 
work every time.  Might you consider trying that instead?

-roger

Stephane DRAY wrote:
> Hello,
> I have send send this message one week ago but I have receive no answer. 
> Perhaps, some of RListers were in holidays and do not read my message. I 
> try again..
> My problem is that I obtained non orthonormal eigenvectors with some 
> matrices with LAPACK while EISPACK seems to provide "good" results. Is 
> there some restrictions with the use of LAPACK ? Is it a bug ? I did not 
> find the answer. Here is my experiment:
> 
>  I have obtained strange results using eigen on a symmetric matrix:
> 
> # this function perform a double centering of a matrix 
> (xij-rowmean(i)-colmean(j)+meantot)
> dbcenter=function(mat){
> rmean=apply(mat,1,mean)
> cmean=apply(mat,2,mean)
> newmat=sweep(mat,1,rmean,"-")
> newmat=sweep(newmat,2,cmean,"-")
> newmat=newmat+mean(mat)
> newmat}
> 
> # i use spdep package to create a spatial contiguity matrix
> library(spdep)
> x=dbcenter(nb2mat(cell2nb(3,3),style="B"))
> 
> #compute eigenvalues of a 9 by 9 matrix
> res=eigen(x)
> 
> # some eigenvalues are equal to 0
> eq0 <- apply(as.matrix(res$values),1,function(x) identical(all.equal(x, 
> 0), TRUE))
> 
> # I remove the corresponding eigenvectors
> res0=res$vec[,-which(eq0)]
> 
> # then I compute the Froebenius norm with the identity matrix
> sum((diag(1,ncol(res0))-crossprod(res0))^2)
> 
> [1] 1.515139e-30
> 
> # The results are correct,
> # then I do it again with a biggest matrix(100 by 100)
> 
> x=dbcenter(nb2mat(cell2nb(10,10),style="B"))
> res=eigen(x)
> eq0 <- apply(as.matrix(res$values),1,function(x) identical(all.equal(x, 
> 0), TRUE))
> res0=res$vec[,-which(eq0)]
> sum((diag(1,ncol(res0))-crossprod(res0))^2)
> 
> [1] 3.986387
> 
> 
> I have try the same with res=eigen(x,EISPACK=T):
> 
>  x=dbcenter(nb2mat(cell2nb(10,10),style="B"))
> res=eigen(x,EISPACK=T)
> eq0 <- apply(as.matrix(res$values),1,function(x) identical(all.equal(x, 
> 0), TRUE))
> res0=res$vec[,-which(eq0)]
> sum((diag(1,ncol(res0))-crossprod(res0))^2)
> [1] 1.315542e-27
> 
> 
> So I wonder I there is a bug in the LAPACK algorithm or if there are 
> some things that I have not understood. Note that my matrix has some 
> pairs of equal eigenvalues.
> 
> Thanks in advance.
> ++++++++++++++++++++++++++++++++++++
> 
> I have continue my experiments in changing the size of my matrix :
> (3^2 by 3^2, 4^2 by 4^2... 20^2 by 20^2)
> 
> EISPACK is always correct but LINPACK provide very strange results:
> 
> 
>  > for(i in 3:20){
> + x=dbcenter(nb2mat(cell2nb(i,i),style="B"))
> + res=eigen(x,EIS=T)
> + eq0 <- apply(as.matrix(res$values),1,function(x) 
> identical(all.equal(x, 0), TRUE))
> + res0=res$vec[,-which(eq0)]
> + print(sum((diag(1,ncol(res0))-crossprod(res0))^2))
> + }
> [1] 7.939371e-30
> [1] 2.268788e-29
> [1] 9.237286e-29
> [1] 1.806393e-28
> [1] 3.24619e-28
> [1] 5.239195e-28
> [1] 9.78079e-28
> [1] 1.315542e-27
> [1] 1.838600e-27
> [1] 3.114150e-27
> [1] 5.499297e-27
> [1] 5.471782e-27
> [1] 1.075098e-26
> [1] 1.534822e-26
> [1] 1.771326e-26
> [1] 2.342404e-26
> [1] 3.462522e-26
> [1] 4.310143e-26
>  > for(i in 3:20){
> + x=dbcenter(nb2mat(cell2nb(i,i),style="B"))
> + res=eigen(x)
> + eq0 <- apply(as.matrix(res$values),1,function(x) 
> identical(all.equal(x, 0), TRUE))
> + res0=res$vec[,-which(eq0)]
> + print(sum((diag(1,ncol(res0))-crossprod(res0))^2))
> + }
> [1] 1.515139e-30
> [1] 1.054286e-27
> [1] 9.553017e-29
> [1] 2.263455e-28
> [1] 5.641993e-27
> [1] 4.442088e-26
> [1] 3.996714
> [1] 3.986387
> [1] 3.996545
> [1] 7.396718
> [1] NaN
> [1] 7.980621
> [1] 7.996769
> [1] 3.984399
> [1] NaN
> [1] NaN
> [1] NaN
> [1] NaN
> 
> Note that I have do the same with random number and never find this kind 
> of problems
> 
> 
>  > R.Version()
> $platform
> [1] "i386-pc-mingw32"
> 
> $arch
> [1] "i386"
> 
> $os
> [1] "mingw32"
> 
> $system
> [1] "i386, mingw32"
> 
> $status
> [1] ""
> 
> $major
> [1] "1"
> 
> $minor
> [1] "9.1"
> 
> $year
> [1] "2004"
> 
> $month
> [1] "06"
> 
> $day
> [1] "21"
> 
> $language
> [1] "R"
> 
> Stéphane DRAY
> -------------------------------------------------------------------------------------------------- 
> 
> Département des Sciences Biologiques
> Université de Montréal, C.P. 6128, succursale centre-ville
> Montréal, Québec H3C 3J7, Canada
> 
> Tel : (514) 343-6111 poste 1233         Fax : (514) 343-2293
> E-mail : stephane.dray at umontreal.ca
> -------------------------------------------------------------------------------------------------- 
> 
> Web                                          
> http://www.steph280.freesurf.fr/
> 
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