CHAPTER 07.02: TRAPEZOIDAL RULE: Multiple Segment Rule: Part 2 of 2
So let's go ahead and look at those numbers there. So for example, if I'm going to draw a table here, and I'm going to say, hey, I was going to calculate this integral, which is the same one, 0.1 to 1.3, 5 x e to the power -2 x, dx, by using the trapezoidal rule. So if n is the number of segments which I am using, what is the value which I am getting . . . what is the approximate value which I am getting, which is from the trapezoidal rule, or the multiple segment trapezoidal rule, and then I'm going to show you what the true error is, and also what the relative true error is in this case. So if I had 1, 2, 3, and 4, let's suppose, let's go ahead and see what kind of approximate values I'm getting. When I had one segment, I got 0.53530, when I had two segments, I got 0.78550, and when I had three segments, which I just calculated, I got 0.84385, and if I was going to do this problem with four segments, I'll get 0.86535. So I would like you to do this as your homework, that, we did calculate it for three segments in the example, we did calculate for one segment in the previous video segment, and I would like you to do this as homework, to see that for four segments, do you get this number here. Now, if you're going to look at the true errors which we got in each case, here we got 0.35857, here we got 0.10837, here we got 0.05002, and here we got 0.02852. And if you just look at those numbers, let me look at . . . let me write down, fill this last column also, this is about 40 percent, 40.1 percent, this is 12.1 percent, this is 5.60 percent, and this is 3.19 percent, so this is in percentages. So just, let's look at the relative true errors, because those are manageable numbers to look at. So you're finding out that when you used only one segment, you got 40 percent error, when you used two segments, you got 12 percent error, when you used three segments, you got 5.6 percent error, and when you got four, you had 3.19 as your relative true error. So you're finding out that as you keep on increasing the number of segments, your true error is also decreasing. So that's the end of this segment. |