# [R] taylor expansions with real vectors

bilelsan sanbilel at yahoo.fr
Wed Jul 18 15:47:05 CEST 2012

```Dear list,

I have a big deal concerning the development of a Taylor expansion.

require(Matrix)
e1 <- as.vector(1:5)
e2 <- as.vector(6:10)

in order to obtain all the combinations between these two vectors following
a Taylor expansion (or more simply through a Maclaurin series) for real
numbers.
We have f(x) = f(0) + f'(0)(x-0) + f''(0)(x-0)^2/2! + … + f^(k)(0)(x-0)^k/k!
with
f(x) = e1 + e2 for Taylor expansion (r = 1)
+ 1/2!*e1^2 + 1/2!*e2^2 + 1/2!*e1*e2 for Taylor expansion (r = 2)
excluding e2*e1
+ 1/3!*e1^3 + 1/3!*e1^2*e2 + 1/3!*e2^2*e1 + 1/3!*e2^3 for Taylor
expansion (r = 3) excluding e2*e1^2 and e1*e2^2
...
I already write the number of possible combinations as :
x <- as.vector(0)
for (r in 1:r){x[r] <- 2*(sum(choose(2*q+r-1,r))-sum(choose(q+r-1,r)))}# q:
number of lag of e1 and e2; r: order of taylor expansion
nstar   <- sum(x) # N* number of total combinations

How to write f(x) in a general framework?
Quid of this framework when e1 and e2 are completed with their lags if q >
1?