[R] row-echelon form (was no subject)

[John Fox]

Dear Spencer,

The R matrix from the qr decomposition isn't quite in row-echelon form,
because the leading entry in each row is not 1, and the other entries in a
column with a leading entry aren't all 0. Some more examples:

> A  # nonsingular
     [,1] [,2] [,3]
[1,]    2   -2    0
[2,]    1   -1    1
[3,]    4    4   -4
> rowEchelonForm(A)
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1
> qr.R(qr(A))
          [,1]      [,2]       [,3]
[1,] -4.582576 -2.400397  3.2732684
[2,]  0.000000  3.903600 -2.3421602
[3,]  0.000000  0.000000  0.8944272


> B  # rank 2
     [,1] [,2] [,3] [,4]
[1,]   -2    0   -1    2
[2,]    4    0    1    0
[3,]    6    0    1    2
> rowEchelonForm(B)
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    1
[2,]    0    0    1   -4
[3,]    0    0    0    0
> qr.R(qr(B))
         [,1]      [,2] [,3]          [,4]
[1,] 7.483315 1.6035675    0  1.069045e+00
[2,] 0.000000 0.6546537    0 -2.618615e+00
[3,] 0.000000 0.0000000    0  6.336077e-16 


Regards,
 John

> -----Original Message-----
> From: Spencer Graves [mailto:spencer.graves at pdf.com] 
> Sent: Thursday, March 04, 2004 9:31 AM
> To: John Fox
> Cc: 'Aimin Yan'; r-help at stat.math.ethz.ch
> Subject: Re: [R] row-echelon form (was no subject)
> 
>       How about the following: 
> 
>  > A <- array(1:6, dim=c(3, 2))
>  > A.qr <- qr(A)
>  > qr.R(A.qr)
>           [,1]      [,2]
> [1,] -3.741657 -8.552360
> [2,]  0.000000  1.963961
>  >
>     I'm no expert, either, and I don't have time now to 
> research this further.  Perhaps someone else can further 
> enlighten us both. 
> spencer graves
> 
> John Fox wrote:
> 
> >Dear Spencer,
> >
> >I'd be surprised if the qr decomposition as computed in R 
> weren't a lot 
> >more stable numerically, but I'm no expert. As well, I don't 
> know how 
> >to get the row-echelon form from the qr decomposition -- though I 
> >suspect that you or someone else on the list is about to 
> enlighten me.
> >
> >Regards,
> > John
> >
> >  
> >
> >>-----Original Message-----
> >>From: Spencer Graves [mailto:spencer.graves at pdf.com]
> >>Sent: Wednesday, March 03, 2004 10:45 PM
> >>To: John Fox
> >>Cc: 'Aimin Yan'; r-help at stat.math.ethz.ch
> >>Subject: Re: [R] row-echelon form (was no subject)
> >>
> >>How does this compare with R of the qr decomposition?  
> spencer graves
> >>
> >>John Fox wrote:
> >>
> >>    
> >>
> >>>Dear Amin,
> >>>
> >>>I have a function (created just for demonstration, and reproduced
> >>>below) for finding the row-echelon form of a matrix. I'm
> >>>      
> >>>
> >>sure that many
> >>    
> >>
> >>>list members could produce something that's better 
> numerically, but 
> >>>this should be OK at least for toy problems.
> >>>
> >>>John
> >>>
> >>>--------- snip -------------
> >>>
> >>>rowEchelonForm <- function(X, tol=.Machine$double.eps){
> >>>   if ((!is.matrix(X)) || (!is.numeric(X))) stop("argument
> >>>      
> >>>
> >>must be a
> >>    
> >>
> >>>numeric matrix")
> >>>   Z <- X
> >>>   for (i in 1:min(dim(X))){
> >>>       if (i > 1) Z[i-1,] <- 0
> >>>       which <- which.max(abs(Z[,i]))  # find maximum pivot
> >>>      
> >>>
> >>in current
> >>    
> >>
> >>>column at or below current row
> >>>       pivot <- X[which, i]
> >>>       if (abs(pivot) <= tol) next     # check for 0 pivot
> >>>       if (which > i) X[c(i,which),] <- X[c(which,i),]  #
> >>>      
> >>>
> >>exchange rows
> >>    
> >>
> >>>       X[i,] <- X[i,]/pivot            # pivot
> >>>       row <- X[i,]                    
> >>>       X <- X - outer(X[,i], row)      # sweep
> >>>       X[i,] <- row                    # restore current row
> >>>       }
> >>>   n <- nrow(X)
> >>>   for (i in 1:n) if (max(abs(X[i,])) <= tol) X[c(i,n),] <-
> >>>      
> >>>
> >>X[c(n,i),]   #
> >>    
> >>
> >>>0 rows to bottom
> >>>   X
> >>>   }
> >>>       
> >>>
> >>> 
> >>>
> >>>      
> >>>
> >>>>-----Original Message-----
> >>>>From: r-help-bounces at stat.math.ethz.ch 
> >>>>[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Aimin Yan
> >>>>Sent: Wednesday, March 03, 2004 2:42 PM
> >>>>To: r-help at stat.math.ethz.ch
> >>>>Subject: [R] (no subject)
> >>>>
> >>>>how to produce a  Row Reduced Echelon Form for a matrix in R?
> >>>>Aimin Yan
> >>>>
> >>>>______________________________________________
> >>>>R-help at stat.math.ethz.ch mailing list 
> >>>>https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> >>>>PLEASE do read the posting guide! 
> >>>>http://www.R-project.org/posting-guide.html
> >>>>   
> >>>>
> >>>>        
> >>>>
> >>>______________________________________________
> >>>R-help at stat.math.ethz.ch mailing list 
> >>>https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> >>>PLEASE do read the posting guide! 
> >>>http://www.R-project.org/posting-guide.html
> >>> 
> >>>
> >>>      
> >>>
> >
> >  
> >
>