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

From: Stephane DRAY <dray_at_biomserv.univ-lyon1.fr>
Date: Thu 29 Jul 2004 - 00:57:20 EST

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)

# 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.

```++++++++++++++++++++++++++++++++++++

```

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"

Stéphane DRAY

Département des Sciences Biologiques
Université de Montréal, C.P. 6128, succursale centre-ville Montréal, Québec H3C 3J7, Canada

Tel : (514) 343-6111 poste 1233 Fax : (514) 343-2293 E-mail : stephane.dray@umontreal.ca

```Web                                          http://www.steph280.freesurf.fr/

______________________________________________
```
R-help@stat.math.ethz.ch mailing list