Inverse of the central difference operator

Question

The central difference is a method to approximate numerically the derivative of a function sampled at discrete intervals. In R, one would do:

n<-100
y<-cumsum(rnorm(n))
y_p[1]<-diff(y[1:2])
for(i in 2:(n-1)) y_p[i]<-diff(y[c(i-1,i+1)])/2
y_p[n]<-diff(y[c(n-1,n)])

My question is, what is the comonly accepted inverse to this operator? (preferably in pseudo code form to avoid ambuiguity)

Best,

Answer 1

Up to a constant difference (equal to y[1]) you can reconstruct this with

z <- rep(NA,n)
z[1] <- 0 
z[2] <- z[1] + y_p[1]
for (i in 2:(n-1)) { z[i+1] <- z[i-1] + 2*y_p[i] }

though you might prefer for (i in 3:length(y_p)) { z[i] <- z[i-2] + 2*y_p[i-1] } as the last line.

You can see the constant difference with

plot(y-z)