I have used Mathematica for many years and have just started using R for programming.

In both programs we can define recursive functions. In Mathematica there is a way to save values of functions. I am not sure if this is the default setting for R. Take the Fibonacci numbers for example. In Mathematica

```
fibonacci[0]=1;
fibonacci[1]=1;
fibonacci[n_]:=fibonacci[n-1]+fibonacci[n-2];
```

Let’s say to find fibonacci[10]. It needs to find all fibonacci[i] from i=2, …, 9 first.

After that, if we want to find fibonacci[11], then the program needs to go through the whole process to find all fibonacci[i] from i=2, …, 10 again. It does not store the values it has obtained. A modification in Mathematica is

```
fibonacci[0]=1;
fibonacci[1]=1;
fibonacci[n_]:=fibonacci[n]=fibonacci[n-1]+fibonacci[n-2];
```

In this way, once we have computed the value fibonacci[10], it is stored, and we do not need to compute it again to find fibonacci[11]. This can save a lot of time to find, say fibonacci[10^9].

The Fibonacci function in R can be defined similarly:

```
fibonacci = function(n) {
if (n==0 | n==1) { n }
else {fibonacci[n-1]+fibonacci[n-2]}}
```

Does R store the value fibonacci[10] after we compute it? Does R compute fibonacci[10] again when we want to find fibonacci[11] next? Similar questions have been asked for other programmings.

Edit: as John has suggested, I have computed fibonacci[30] (which is 832040) and then fibonacci[31] (which is 1346269). It took longer to get fibonacci[31]. So it appears that the R function defined above does not store the values. How to change the program so that it can store the intermediate values of a recursive function?

### >Solution :

R does not do this by default, and similarly as you see Mathematica doesn’t either. You can implement memoise yourself or use the `memoise`

package.

```
fib <- function(i){
if(i == 0 || i == 1)
i
else
fib(i - 1) + fib(i - 2)
}
system.time(fib(30))
# user system elapsed
# 0.92 0.00 1.84
library(memoise)
fib <- memoise(fib) # <== Memoise magic
system.time(fib(100))
# user system elapsed
# 0.02 0.00 0.03
```