On 7/12/05, Robin Hankin <r.hankin@(protected):
> Hi
>
> I want to write a little function that takes a vector of arbitrary
> length "n" and returns a matrix of size n+1 by n+1.
>
> I can't easily describe it, but the following function that works for
> n=3 should convey what I'm trying to do:
>
>
> f <- function(x){
>  matrix(c(
>   1       ,  0    ,  0 , 0,
> x[1]       ,  1    ,  0 , 0,
> x[1]*x[2]   , x[2]   ,  1 , 0,
> x[1]*x[2]*x[3], x[2]*x[3], x[3], 1
> ),
> 4,4,      byrow=T)
> }
>
> f(c(10,7,2))
>    [,1] [,2] [,3] [,4]
> [1,]   1   0   0   0
> [2,]  10   1   0   0
> [3,]  70   7   1   0
> [4,] 140  14   2   1
> >
>
>
> As one goes down column "i", the entries get multiplied by successive
> elements of x, starting with x[i], after the first "1"
>
> As one goes along a row, one takes a product of the tail end of x,
> until the zeroes kick in.

I have not checked this generally but at least for
the 4x4 case its inverse is 0 except for 1s on the
diagonal and -x on the subdiagonal. We can use
diff on a diagonal matrix to give a matrix with
a diagonal and superdiagonal and then massage that
into the required form, invert and round --
leave off the rounding if the components of x
are not known to be integer.

round(solve(diag(4) - t(diff(diag(5))[,1:4])+diag(4) * c(0,x)))

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