[MUSIC] Hi, and welcome to lesson six, Loops. This week, we'll be talking about a powerful programming concept, looping. A loop is a new control construct that makes it possible to repeat a block of code a number of times. So far, every statement and every function has been executed exactly once, or if some condition was false, not at all. While this is useful, it doesn't exploit the real power of a computer which is the ability to execute millions or billions of instructions per second. With what we've learned so far, to get our computer to execute a billion statements, we'd have to type a billion statements, and we'd be typing for many centuries just to get a program to run for one second. Fortunately, we can avoid all those centuries of work by using loops. In fact, we've already been using loops in MATLAB without even knowing it. To carry out the command n equals 1 colon 5 for example, MATLAB uses a loop internally to create the vector 1, 2, 3, 4, 5. And to carry out the command total equal sum of n, it uses a loop to add all those five numbers. These are implicit loops because we didn't use any explicit loop control constructs. Let's compute the sum of 1 through 5 without using the built-in sum function. We'll use a loop. And we need to design an algorithm that includes the loop. An algorithm is just a computer science term for a step-by-step procedure that solves a problem. And this figure, shows a very simple algorithm that will solve our problem. It uses a variable called total to store our sum as we compute it. And it starts by initializing total to zero. The large light blue box represents the loop. It specifies that the statement add in the total be executed repeatedly as n takes on the values 1 to 5, one by one. So the add n to total statement will be executed five times. After the fifth time, the loop ends and the program continues to the next statement after the loop. That statement prints out the value of the variable total, which is a result we wanted. Note the curved line with the arrows on it. It shows that after add n to total is ex, executed, control circles back and does it again, and it shows why this control construct is called a loop. That little circle looks like a loop. Let's look at the execution in detail. Warning, this part is really boring. We start by setting total to 0. This step is called initialization and it happens outside the loop. Now we enter the loop. Since the loop must repeat in, for n equal to each item in the list, 1, 2, 3, 4, and 5, the first thing that happens when the loop is entered is that n is set to the first item, one. Then n is added to the total. Since total is 0, n is 1. Total equals 0 plus 1 which equals 1. That completes the first repetition of the loop. A repetition is called an iteration, and now it's time for the second iteration. This time, n is set to the second value on the list, which happens to be 2. Again, n is added to total. Now, total equals 1, and n equals 2, so, total equals 1 plus 2 which equals 3. That completes the second iteration of the loop. Bored yet? Me too. Don't say I didn't warn you. But we need to go through this for at least one loop. Okay, for the third iteration, n is set to 3. Again, n is added to total. Total equals 3, n equals 3, so total equals 6. That completes the third iteration of the loop. For the fourth iteration n is set to 4. And as in every iteration n is added to the total. Now total equals 6, n equals 4, so total 6 plus 4 equals 10. And that's the end of the fourth iteration. For the fifth iteration n is set to 5, again n is added to the total. Now total equals 10, n equals 5 so total becomes 15. That's the end of the fifth iteration and we've hit the end of the list of values for n, so that loop ends. And finally, we get to the statement after the loop, which prints 15, and our algorithm is done. You may say this was a complicated way to add five integers, and you're right. Sure, typing 1 plus 2 plus 3 plus 4 plus 5 would have been simpler. Suppose we wanted to add the first 5,000 numbers. Using a loop, we can just change the five to 5,000 and it will add them all up in an eye-blink. Now we're ready to see how we can implement this algorithm in MATLAB. The most frequently used loop construct in MATLAB is called the for-loop. And that's what we need. Let's see how we can use it to add one through five. We start by initializing the variable total to 0 just like we did in our algorithm. Then comes our for-loop. It starts with the keyword for, and to the right of that is what looks like an ordinary assignment statement, but it's not. Its left, its left hand side is a variable called the loop index. In this case the loop index is n. On its right hand side is a vector. But here's where the difference from an ordinary assignment statement comes in. The vector will not be assigned to n. Instead, the meaning of this special assignment statement is that the index n will take on the elements of the vector one by one. At each iteration of the loop, n will get the next element of the vector, stopping with the last one. That's exactly what we want here. N will be equal to 1, then 2, then 3, then 4, then 5. At each iteration, n is added to the current value of total. So, just as for our algorithm, total keeps accumulating with sum, becoming equal to 1, then 3, then 6, then 10, and finally 15. Once the last element of the vector is assigned to the loop index, and used in the block of code inside the loop, the loop ends. And then fprintf prints the value the total, which is 15. Okay, let's see this example inside MATLAB. Here's our summing algorithm inside a function called sumintup. But instead of adding the integers from 1 to 5 we add up from 1 to n where n is the input argument. If we run it like this. We get 15, as we've just seen. The good thing with a function, though, is that it's easy to run our loop with different numbers. Let's try 10. Summing from 1 to 10 gives 55. What about 100? That gives 5,050 and so on. Lets runs the function with five again. First, let's set a break point right here. On the second line, I just clicked the little dash here, and the red circle shows me that my break point is set. Then we can use the debugger to step through. Okay lets run it with five as the input. The green arrow here means that execution has been suspended right where we put the break point and that we're ready to execute line two here. At this point, we have not executed this line yet. So the variable total is not yet appeared in the workspace. But there's already a capital N there, and it's equal to five. Remember, that's our input argument. And it gets its value from the function collar down here. And it comes into existence when its assigned the input value, which in this case is five. And that happens before line two is executed. Let's execute line two by clicking the step button. To get that button to show up, click the editor tab. That tab contains all the debugging tools. Here's the step button right here. Before I click it, you may remember from our introduction to the debugger that you can also take one step by clicking the function key F7 on a Macbook or F10 if you're using Windows. Okay, as I click step, watch in the workspace to see the variable total appear with the value we're assigning to it. Which is zero. There. Now we're at the beginning of the for loop. Okay, I'm going to execute the first line of the for loop. Clicking step executes that line. There. The workspace shows that we now have the variable lowercase n here, and it's been assigned the value one. Which is the first element of the vector one colon n here. This is a key concept for the for loop, so it's worth repeating. A loop index is assigned the first element of the vector produced by one colon n, not the entire vector, at least on the first iteration. On subsequent iterations, it will receive subsequent elements of the vector. Now we're ready to execute the body of the for loop here. With the loop index set to one. The body comprises just line four and that line adds n to total. The workspace shows their values before we take the step. Zero and one. Okay. I'll click step now, and we see the total has changed to one. Clicking step again here at the end statement results in Matlab taking the next element of the original one to five vector and assigning it to little n. We'll see that happen right now, there. Note that we did not see the green arrow go back up to the for line at the top of the loop. That will never happen again through the entire iteration. That line is actually executed only once at the beginning of the loop. It calculates the vector values, in this case, one, two, three, four, five, and it sets the loop index to its first value, which is one in this case. Then, after the first iteration, the next index is actually set at the bottom of the loop, it’s the same as if it were set at the top but that’s why you won’t see control jump back to the for line. Anyway, now were ready to execute the next addition. Total is equal to one and n is equal to two so after executing this line the total will be equal to three. And three it is. As we continue clicking the button to step through the code, we can watch the workspace to see how n goes through all the values to five. And how total reaches 15. Now let's do that. Click, now n is equal to three. Click, total is equal to six. Click. N is equal to four. Click total is equal to ten. Click, n is equal to five. Click total is equal to 15. And we take one more step, and we're out of the loop. The next step, will print the result in the command window. Let's do that. Click, and we see the result show up in the command window down here. Take another step. You see this arrow pointing down, means we're about to leave the function. The function ends. The step button disappears and we're done. Let's look at this example one more time. The expression with the for keyword is called the control statement. This is where the index variable is defined and it's subsequent values are specified, and it doesn't need a semicolon at the end. The block of code between the control statement and the n keyword closing loop, is the body of the loop. It's the body that gets executed repeatedly using successive values of the loop index. Here the body's a single statement but in general there can be any number of statements in the body of a loop. Most of the time the vector to the right of the equal sign and the control statement will be formed with a colon operator as it is here. But it doesn't have to be. Here is another example. Instead of a colon operator, here we simply use another variable that already contains a vector. In this case, it's a vector of five random numbers. This is perfectly fine. The loop index will be assigned the elements of this vector one by one. The body of this loop is a single if, else statement. It prints out one of two messages depending on whether or not the current value of the loop index, which is one of those five random numbers, is greater than one-half. We ran this program and this was the result. Note that depending on when and how the pseudo-random number generator was initialized, you may get a different random number set when you try. And unlike Myth Busters, we do encourage you to try this at home. Consider this slightly variation, slight variation of the previous example. We eliminated the variable name list. Because you can call a rand function or any other function for that matter, right in the control statement itself. In fact, any valid expression can go to the right side of that equals sign. The values assigned to the loop index don't have to be integers, they don't have to be regularly spaced, or even assigned in increasing order. In fact, they don't even have to be scalars. The general rule is that the loop index will be assigned the columns of the array generated by the expression on the right. Of the equal sign. If that array is vector, then those columns will be scalars. As for the body of the for-loop, any control construct can be used. If statements, other loops, and so forth are just lists of statements. Let's see the second example in the editor. And let's use the debugger again to watch it work. Place a breakpoint right here on line 2 and call the function. [SOUND] Well, we've stopped at line 2. Nothing's really happened yet, there's nothing in the workspace. So, let's take a step. Now we see N is equal to 5 over here. That happened right there on line 2. Now, we've gotten a set of numbers into list by calling rand 1, 5, because N is equal to 5. And here's our list numbers. There. One, two, three, four, five numbers. Let's enter the body of the for-loop. x gets the first number. Body of the loop consists of a single if-else statement. See what happens, let's see a, ahead of time. x is greater than 0.5, right? So that means the if branch will be executed instead of the else branch. And we're in that branch. And the fprintf executes and says that it's large. Now we're down at the end statement. When we click on that, x we'll get the next value, 0.9058. So let's do that. Now, we see x has the next value. That's also greater than 0.5, so this statement will execute again. And there we have it. The next step, set x equal to the next element, 0.1270. x is not greater than 0.5, so we'll be in the else branch. And there we are. Now we're in the body of the else branch. We'll carry out this fprintf, which says that this number is small. And we'll keep going in that way. You can see x going progressively though the list. Now it's on the last one. That's also large. So, it just so happened that in this position in the pseudorandom sequence, we had four large numbers and one small one. Another step gets us out of the for-loop. And one more gets us out of the function. By the way, before we started, the pseudorandom number sequence was initialized to the state it has when MATLAB starts. So, if you call rng of 0 before you call rand_check, like this, then you'll get the same numbers. Otherwise, you'll get a different sequence. An interesting question arises when the loop index is assigned a value explicitly by a statement within the body of the loop. The question is this. What's the value of the index on the next iteration? The answer is that it's the next value in the list of values given in the control statement, up here. Assignments to the loop index inside the body of the loop are temporary, they last only during the iteration which they take place. There's no effect on the original list of values given on the if line at the top of the loop, or on the next value to be assigned in the loop index for the next iteration. Let's see an example that illustrates this point. A modified sum int up function that we've named loop_test. Inside the loop, we first display the current value of n, then we add 1 to n, and then we add n to total. Let's run it. Why don't we give it the value 5. So what do we learn? The function printed the originally intended values for n, as given by 1 to n here, a 1:5 expression. We changed the value of n from 1 to 2. But at the next iteration, we got the originally intended next value from here, which was 2. As we said, changing the value of the loop index is temporary. It applies only to the current iteration. There was an effect of adding 1 each time, though, you can see in the total down here, which is 20. We added an extra 1 to n each time through the loop, so the total should be 5 more than the sum of the first five integers, which is 15. And we did get 20. This rule is ironclad. At the beginning of the nth iteration for every for-loop, loop's control statement will assign the loop index the nth term in its list of values, regardless of any value that may have been assigned to the loop index within the body of the loop during the previous iteration. Finally, a word of caution. Don't change it. Don't change the loop index inside the loop if at all possible. It'll just cause confusion when you read your code later, or when someone else reads it. There's very rarely a need to do so, and even then you'll be better off using an additional variable than messing with the loop index. Let's talk a little bit about handling vectors with loops. Let's look at this example. Here, MATLAB handled the subtraction of the element 5 from the element 5 and got 0. It handled the subtraction of the element 4 minus 5, got minus 1. 8 minus 7 gave 1. 8 minus 8. It handled every one of these subtractions element by element. Well, it did that with a loop. Loops behind the scenes. We didn't write the loop, so we call it an implicit loop. MATLAB does that a lot of times. But we can do it too, out here, in front of the scenes. Let's do it. As soon as I hit Return, my work is done, and the results are the same. The for-loop does exactly the same thing that the array subtraction operation did. We visit every element using the for-loop index, which indexes into u, and indexes into v, and indexes into w. We get the same result. We had to use the length function here. To find out how far to carry the loop. In this case, there's really no reason at all to use an explicit loop instead of the implicit loop. Implicit looping runs faster in general, it's easier to program in general, and it attracts fewer bugs in general. You should use it wherever you can. But, explicit looping is much more versatile than implicit looping. In addition to being able to do everything that array and matrix operations can do, for loops, loops in general can do many things that array and matrix operations can't do. Since we often need to do these additional things, MATLAB includes explicit loops as part of its language. As an example where we must use an explicit loop instead of an implicit one, consider the Fibonacci series. It's defined as a series where the first two elements are one, and all successive elements are the sum of the previous two elements. We got a function here that creates the first n elements for Fibonacci series called fibo. Let's see how it works. First of all, we do some error checking to make sure that the user has passed in a scalar, a number that's greater than or equal to 1, and a number that's an integer. In other words, a positive integer. We've done that before. And down here, we set the first element to 1 and the second element to 1. You'll remember the definition of Fibonacci series is the first two elements are equal to one and the successive elements are each equal to the sum of the two previous elements. Well now we enter a for-loop that goes from 3 to n, n is the input. Why 3? Well we've already done 1 and 2. And as you can see on the left side of this assignment statement, we're doing f3 and then f4 and then f5 and so on. On the right side let's say we're at three. 3 minus 2 is 1, 3 minus 1 is 2. So this is element f1. This is element f2. So f1 plus f2 gives f3. Let's do one more. Double i, the index, becomes 4. So f of 4 is equal to 4 minus 2, that's 2. And f of 4 minus 1, that's 3. So f of 4 is equal to 2 plus f of 3. And so it goes. Here we had to use an explicit loop since there's no built in operator or function that could have done what we needed, summing up the preceding elements of the vector to compute the current one. So the Fibonacci series serves as a simple example of why we need explicit looping. The body of a loop can contain any valid MATLAB statements, and that means that a loop can contain another loop. In other words, we can nest for-loops. That's in fact what we need to do if we want to perform an operation on every element of a matrix instead of a vector. Let's do this. There we got some random numbers and a, all integers. And let's do this. Of course you know what's happened here, we've used the array multiplication operation, so every element in a has been multiplied by the corresponding element in a. MATLAB did this with implicit looping. Just as with vectors, however, we can write equivalent code to do the same thing with loops. I've got an example right here in a file called mul.m. Let's see it. There. This is a script, by the way. You'll remember us talking about scripts at the end of lesson three. A script is a file that contains just a list of commands instead of a function. You can run it by giving its name as a command in the Command window, and you can also run it by clicking this Run button up here, with the great big green arrow. Let's click the arrow. There it shows that mul is carried out in the command window as if we'd typed it in there ourselves. Well, let's see what value it gave to p. Good, same as the array operation did with its implicit looping. We didn't make an error in our script up here, that's good. But let's look at the commands in the script. First we use the size function up here to find out the dimensions of a, and we put those dimensions in row and col. So row has the number of rows and a, col has the number of columns. Then we enter a for-loop here, whose index r would suggest row goes from one to row, so it scans through all the rows of a. Well you can have, as we mentioned, any MATLAB command, in the body of the for-loop, and so we have, a for-loop in the body of this for-loop. This inner for-loop is said to be nested in the outer for-loop. This for-loop's index c, which suggests columns, scans all the legitimate column indices in A. The actual calculation is done in the body of this inner loop, and that body consists of just this one statement. It's a multiplication, and multiplies element r, c of a by element r, c of a. When we exit this for-loop, we've done all the columns for a given row. We repeat by changing the row index by one and carrying out this loop again, for all the columns, of that new row index. And we repeat again, for a new value of r, and we carry out this inner loop for all these values of c for that given value of r, and so on. When we exit the outer loop, we're done with the script. Note that we've carried out the outer loop row times, and for each one of those we carried out the inner loop call times. So this body here, the single statement was executed the number of times it's equals to row times call. And that's the same number of elements that there are in the matrix A. So that's all there is to it. Or I should say, there's a lot to it. I say that because this script is a lot more complicated. In a dot star a. If we want to follow the order in which the elements are multiplied, we can insert a line right after the multiplication, like this, that prints things out. It'll print the row and the column in parentheses, using the percent D format with a space between them, and then it'll skip to a new line. Let's try it. And I'm going to need more room than this I think I'll sacrifice my workspace. So I'll close that and then move the command window over here beside the editor window. There. So let's try it again. This time, I'll type mul in the command window. There you can clearly follow what happened. Remember the first element is the row. So we did row 1 column 1, row 1 column 2, 1, 3, 1, 4. We're done with row 1. Now we move to row 2 and do columns 1, 2, 3, 4. Then we move to row 3 and do columns 1, 2, 3, 4. When we're doing these columns, we're staying inside this inner for-loop. When we're done with the inner loop we go to a new row in the outer loop and then repeat the inner loop. And we do that over and over and over. While we're at it, we could also insert another fprintf statement right here. Like this. It'll print working on row, space, and then it will print the row number r, using the %d format, dot, dot, dot, go to a new line and then print all this stuff out. Let's try that. There. Working on row 1. Well 1, 1, 1, 1, that’s row 1, 2, 2, 2, 2, 3, 3, 3, 3, and so on. This makes it nice and clear. This is called row major order, it means that you visit every element of the first row, then every element of the second row, etc. You can't do this sort of thing to look inside an implicit loop that MATLAB provides. But as a matter of fact, for implicit looping MATLAB uses column major order. Let's consider another example for nested for-loops. I've got an example right over here in a file called asterisks, there it is. Let's run it and see what we get. Well what we get is a pretty little half of a tree depicted in asterisks. In the first line, see that we get one asterisk. The next line two, then three, then four, then so on until we get to seven asterisks on the last line. Well over here in the code, there's nested for-loops here. And in the inside loop, we see that it prints one asterisk each time through the loop. Its index goes from one to mm, and as mm gets bigger, it prints longer and longer lines of asterisks. You can see the outer loop changes mm from 1 to N which happens to be set equal to 7, which is why we get seven lines with seven asterisks on the last line. And you'll notice that after each of the inner loops is finished, this fprintf takes us back to the beginning of the next line. The trick here was that we used the value of the outer loop index, to control behavior of the inner nested loop which gave us these longer and longer rows of asterisks. This again is something that we cannot do with implicit loops. [MUSIC] >> [APPLAUSE]
