outerって使えない場面があるよね？
剰余系での加減乗除での実際
> n=5 # prime number
> nn=1:(n-1)
> tasu <- function(x,y) (x+y)%%n
> hiku <- function(x,y) (x-y)%%n # row - col
> kake <- function(x,y) (x*y)%%n
> g=function(x) nn[which(x==1)]
> .M=outer(nn,nn,kake)
> G=apply(.M,2,g)
> gyaku <- function(x) nn[which(G==(x%%n))]
> waru <- function(x,y) (x*gyaku(y))%%n # row / col
> waru(3,2)
[1] 4
> xx=yy=c(0,nn)
> names(xx)=paste0('x',c(0,nn))
> names(yy)=paste0('y',c(0,nn))
> outer(xx,yy,tasu) # x + y
y0 y1 y2 y3 y4
x0 0 1 2 3 4
x1 1 2 3 4 0
x2 2 3 4 0 1
x3 3 4 0 1 2
x4 4 0 1 2 3
> outer(xx,yy,hiku) # x - y
y0 y1 y2 y3 y4
x0 0 4 3 2 1
x1 1 0 4 3 2
x2 2 1 0 4 3
x3 3 2 1 0 4
x4 4 3 2 1 0
> outer(xx,yy,kake) # x * y
y0 y1 y2 y3 y4
x0 0 0 0 0 0
x1 0 1 2 3 4
x2 0 2 4 1 3
x3 0 3 1 4 2
x4 0 4 3 2 1
> X=Y=nn
> outer(X,Y,waru) # WRONG!!
[,1] [,2] [,3] [,4]
[1,] 1 1 1 1
[2,] NA NA NA NA
[3,] NA NA NA NA
[4,] NA NA NA NA
> a=expand.grid(X,Y)
> b=matrix(mapply(waru,a[,1],a[,2]),length(X))
> rownames(b)=paste0('x',nn)
> colnames(b)=paste0('y',nn)
> b # x / y
y1 y2 y3 y4
x1 1 3 2 4
x2 2 1 4 3
x3 3 4 1 2
x4 4 2 3 1