## Vectorize!

Here is an email sent by one of my students a few days ago:

Do you know how to integrate a function with an  “if”?
For instance:
>X=rnorm(100)
>Femp=function(x){
+   return(sum(X<x))
+}
>integrate(Femp,0,1)\$value

does not work.

My reply was that the fundamental reason it does not work is that integrate (or curve for instance) computes the function in several points of a grid by calling the function on a vector x and the comparison X<x does not make sense when both X and x are vectors. If one rewrites the code as

```X=rnorm(100)
Femp=function(x){
return((apply(as.matrix(outer(X,x,"-")<0),2,sum))}
```

the function can be integrated

```> integrate(Femp,-2,2)
1.877735 with absolute error < 5.6e-05
```

However, this kind of syntactic gymnastic can be avoided by a simple call to Vectorize.

```X=rnorm(100)
Femp=function(x){
return(sum(X<x))}
> integrate(Vectorize(Femp),-2,2)
1.877735 with absolute error < 5.6e-05
```

### 2 Responses to “Vectorize!”

1. I used a lot apply, but I did not know that Vectorize function… It looks great ! thanks….

• Obviously, it is time-consuming:

```> system.time(for (t in 1:10^3) integrate(f,-1,1))
user  system elapsed
0.104   0.000   0.101
> system.time(for (t in 1:10^3) integrate(Vectorize(f),-1,1))
user  system elapsed
0.416   0.004   0.418
```

although a preliminary definition removes part of the time:

```> vf<-Vectorize(f,SIMPLIFY=TRUE)
> #does not work with FALSE
> system.time(for (t in 1:10^3) integrate(vf,-1,1))
user  system elapsed
0.356   0.000   0.360
```

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