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We have seen that we can accomplish sequential repetition by having a function continue to call copies of itself, progressively simplifying the problem at hand, until all the needed repetition is done (recursion). In principle, this mechanism is sufficient to solve any programming problem, and moreover, it is very effective to use when the data structures themselves are hierarchical (we won't really treat hierarchical data structures in this semester).
But R does not always perform recursion efficiently. And for many algorithms, it can be more work to express them in terms of recursive function calls. In this lecture we will learn the classical alternative to recursion: the loop. In R, there are two main looping constructions: the counted loop and the conditional loop.
The counted loop is essentially another way to implement the map concept. Basically we start with a collection of objects, and we have a series of commands we wish to do for every member of the collection. In a counted loop, the system first sets a loop variable to the first member of the collection, and then does the desired commands. Then, the loop variable is assigned to the second member of the collection, and the desired commands are done again. This is repeated until the loop variable has been assigned the value of the last member of the collection and the commands carried out.
Imagine that we wish for instance to square the numbers 3, 9, 4, and 7. Of course, the best way to do this in R is c(3,9,4,7)^2, using the vectorized operator ^ to square the whole list. But I'm going to show you how you do this with the counted loop. To start with, we might just square them separately:
> 3^2 [1] 9 > 9^2 [1] 81 > 4^2 [1] 16 > 7^2 [1] 49 > 
> ii < 3 > ii^2 [1] 9 > ii < 9 > ii^2 [1] 81 > ii < 4 > ii^2 [1] 16 > ii < 7 > ii^2 [1] 49 > 
We could be fancier and ask the computer to print a message to the user each time:
> ii < 3 > cat("The square of ",ii," is ",ii^2,".\n") The square of 3 is 9. > ii < 9 > cat("The square of ",ii," is ",ii^2,".\n") The square of 9 is 81. > ii < 4 > cat("The square of ",ii," is ",ii^2,".\n") The square of 4 is 16. > ii < 7 > cat("The square of ",ii," is ",ii^2,".\n") The square of 7 is 49. > 
Let's see what the counted loop looks like. It is called for in R. In the example below, we have the for command itself, the loop variable, the sequence of values to loop through, and the commands to do for each value.
The square of 9 is 81 . The square of 4 is 16 . The square of 7 is 49 . > 
In many cases, you use the structure to perform an action a number of times:
I'm countingI'm at 2 ! I'm countingI'm at 3 ! I'm countingI'm at 4 ! I'm countingI'm at 5 ! I'm countingI'm at 6 ! I'm countingI'm at 7 ! I'm countingI'm at 8 ! I'm countingI'm at 9 ! I'm countingI'm at 10 ! > 
Let's do the factorial example using a counted loop. Remember we can always do it using prod; for example, the factorial of 8 can be computed by prod(1:8). So let's do it using a counted loop:
> ans < 1;
[1] 40320 
Let's use a counted loop to add up the elements of a list. For definiteness, let's take a look at data from the California recall election or 2003. According to the State of California, as of Oct. 13, 2003, with 100% of precincts reporting, the number of votes received by the top ten candidates are as follows:
> topten < c(3850982,2504640,1053968,218852,44201,22979,15875,13015,11257,10316); > tot < 0
[1] 7746085 
Here is another way to do the same thing. This time, we step through each element of the list, one at a time:
> topten < c(3850982,2504640,1053968,218852,44201,22979,15875,13015,11257,10316); > tot < 0
[1] 7746085 
We can do any map operation using the counted loop. So for instance, let's square the numbers from 1 to 10, in reverse:
> final < 10 > vals < final:1 > ans < rep(0,length(vals))
[1] 100 81 64 49 36 25 16 9 4 1 
Let's do another example. You've heard it said that the probability of heads on a fair coin toss is 0.5. What does this mean? It means that the relative frequency of heads should approach 0.5 if you do enough tosses. So here's what I want to do. Let's try to toss a coin (a simulated, computer coin, that is) a certain number of times, and keep track of the results. Let's try to toss a coin 10, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000, and 100000 times, and keep track of the number of heads. So we've got 11 simulations to do, and we'll have 11 different results; we'll need a vector of length 11 to hold the results.
> ntrials < c(10,100,200,500,1000,2000,5000,10000,20000,50000,100000) > nheads < rep(NA,length(ntrials))

For another example, let's look at one last sequential repetition. Let's look at a socalled birth process. In the science fiction movie Island of Terror, medical research accidentally creates some silicate monsters that are doubling every 6 hours. Imagine that we start with one of them, and follow the number of them every doubling time, for the first 100 doubling times.
> ntimes < 100 > nn < rep(1,length(ntrials))

As an exercise, see if you can convert the needle reuse computations from the previous lecture to use a for loop instead of recursion.
In the next lecture, we will learn the other main loop construct in R: the conditional loop while.