# RE: Fwd: Re: [R] eigenvalues of a circulant matrix

From: Ted Harding <Ted.Harding_at_nessie.mcc.ac.uk>
Date: Tue 03 May 2005 - 21:07:50 EST

On 03-May-05 Globe Trotter wrote:

```> Looks like the files did not go through again. In any case,
> here is the kinv:
> please cut and paste and save to a file:
>
> [data snipped]
>
> --- Globe Trotter <itsme_410@yahoo.com> wrote:
```

>> Date: Mon, 2 May 2005 19:51:24 -0700 (PDT)
>> From: Globe Trotter <itsme_410@yahoo.com>
>> Subject: Re: [R] eigenvalues of a circulant matrix
>> To: r-help@stat.math.ethz.ch
>>
>> OK, here we go:
>>
>> I am submitting two attachments. The first is the datafile
>> called kinv used to create my circulant matrix, using the
>> following commands:
>>
>> x<-scan("kinv")
>> y<-x[c(109:1,0:108)]
>> X=toeplitz(y)
>> eigen(X)
>> write(X,ncol=216,file="test.dat")

length(x)
##[1] 216

Question 1:

Is this correct? Or has there been a problem with your posting of the data?

If it is correct, given that you seem to only use x[1:109], was there some point in giving the rest?

Question 2:

y<-x[c(109:1,0:108)]

I now get

(as expected). The "0" in "0:108" seems to have been ignored (again as expected), so this is equivalent to

y<-x[c(109:1,1:108)]

Is this as intended? If so, why use "0:108" instead of "1:108"? Check:

y[1] ##[1] 19.4495
x[109] ##[1] 19.4495

y[109] ##[1] -0.00116801
x[1] ##[1] -0.00116801

y[110] ##[1] -0.00116801
x[1] ##[1] -0.00116801

y[217] ##[1] -6.28085
x[108] ##[1] -6.28085

Can you confirm that this is as intended?

Comment 3:

You next command X=toeplitz(y): No apparent problems, it gives a symmetric result:

which(X != t(X)) ## numeric(0)

with 217 rows and columns:

dim(X) ##[1] 217 217

and looks circulant:

X[(1:5),(1:5)]

[,1] [,2] [,3] [,4] [,5]

```  [1,] 19.449500 -6.280850 -0.486405 -0.826079 -0.167792
[2,] -6.280850 19.449500 -6.280850 -0.486405 -0.826079
[3,] -0.486405 -6.280850 19.449500 -6.280850 -0.486405
[4,] -0.826079 -0.486405 -6.280850 19.449500 -6.280850
[5,] -0.167792 -0.826079 -0.486405 -6.280850 19.449500

```

Question 4:

Your next command, "eigen(X)", would simply output the results to screen and does not assign to anything.

Your next command "write(X,ncol=216,file="test.dat")" as it stands will write the toeplitz matrix X, constructed by your command "X<-toeplitz(y)" to file, but with 216 columns instead of 217. However, the result consists simply of numbers, and there is nothing like "NA" or "NaN" in the file which I get.

Nor are there any NAs or NaNs in X itself, of course.

But, when you yourself did "write(X,ncol=216,file="test.dat")", perhaps the "X" in this command was different from the "X" which is the toeplitz matrix. So, was it the result of an assignment from "eigen(X)" and, if so, which component or components?

So I have tried Z<-eigen(X). First of all, I get no problems with NAs or NaNs:
```  which(is.na(Z\$values))    ##numeric(0)
which(is.nan(Z\$values))   ##numeric(0)
which(is.na(Z\$vectors))   ##numeric(0)
which(is.nan(Z\$vectors))  ##numeric(0)

```

Next, trying various options for wirting to file:

write(Z,ncol=216,file="test.dat")

simply does not work (not a writable structure), while

write(Z\$values,ncol=216,file="test.dat")

produces simply a set of numbers, no NAs of NaNs, and likewise

write(Z\$vectors,ncol=216,file="test.dat")

(the only occurrences of non-numeric characters are "e", as in "e-05").

>> reports the following columns full of NaN's: 18, 58, 194, 200.
>> (Note that eigen(X,symmetric=T) makes no difference and I get
>> the same as above).

Therefore I find myself completely unable to reproduce your problem. However, for the various reasons stated in detail above, I am not at all sure that what you wrote as the statement of what you did in fact corresponds to what you really did!

I even wonder whether

Question 6:

Was the file "test.dat" the result of your "write" command? Or was it left over from a previous activity, the "write" from this session having failed to execute for some reason? (In which case the NaNs would have nothing to do with the results of "eigen(X)").

>> The second attachment contains only the eigenvectors
>> obtained on calling a LAPACK routine directly (from C).
>> The eigenvalues are essentially the same as that obtained
>> using R. Here, I use the LAPACK-recommended double
>> precision routine dspevd() routine for symmetric matrices
>> in packed storage format.
>> Note the absence of the NaN's....I would be happy to send
>> my C programs to whoever is interested.

Well, I didn't get any NaNs in R either -- quite consistent with your C program!

Please clarify according to the questions above.

Best wishes,
Ted.

E-Mail: (Ted Harding) <Ted.Harding@nessie.mcc.ac.uk> Fax-to-email: +44 (0)870 094 0861
```Date: 03-May-05                                       Time: 12:06:57
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