R中的函数式编程：带vandermonde矩阵的图示

Question

I would like to get a feel of functional programming in R. To that effect, I would like to write the vandermonde matrix computation, as it can involve a few constructs.

In imperative style that would be :

vandermonde.direct <- function (alpha, n) 
{
  if (!is.vector(alpha))  stop("argument alpha is not a vector")
  if (!is.numeric(alpha)) stop("argument n is not a numeric vector")
  m <- length(alpha)
  V <- matrix(0, nrow = m, ncol = n)
  V[, 1] <- rep(1, m)
  j <- 2
  while (j <= n) {
    V[, j] <- alpha^(j - 1)
    j <- j + 1
  }
  return(V)
}

How would you write that elegantly in R in functional style ?

The following does not work :

x10 <- runif(10)
n <- 3
Reduce(cbind, aaply(seq_len(n-1),1, function (i) { function (x) {x**i}}), matrix(1,length(x10),1))

As it tells me Error: Results must have one or more dimensions. for list of function which go from i in seq_len(3-1) to the function x -> x**i.

Answer 1

It does not seem very natural to use Reduce for this task. The error message is caused by aaply , which tries to return an array: you can use alply instead; you also need to call your functions, somewhere.

Here are a few idiomatic alternatives:

outer( x10, 0:n, `^` )
t(sapply( x10, function(u) u^(0:n) ))
sapply( 0:3, function(k) x10^k )

Answer 2

Here it is with Reduce :

m <- as.data.frame(Reduce(f=function(left, right) left * x10, 
                          x=1:(n-1), init=rep(1,length(x10)), accumulate=TRUE))
names(m) <- 1:n - 1

Answer 3

Here's another option, that uses the environment features of R:

vdm <- function(a)
{
    function(i, j) a[i]^(j-1)
}

This will work for arbitrary n (the number of columns).

To create the "Vandermonde functional" for a given a , use this:

v <- vdm(a=c(10,100))

To build a matrix all at once, use this:

> outer(1:3, 1:4, v)
     [,1] [,2]  [,3]  [,4]
[1,]    1   10   100 1e+03
[2,]    1  100 10000 1e+06
[3,]    1   NA    NA    NA

Note that index a[3] is out of bounds, thus returning NA (except for the first column, which is 1 ).