CHAPTER 07.04: ROMBERG INTEGRATION: Multiple Segment Trapezoidal Rule: Derivation
In this segment, we're going to talk about the multiple segment trapezoidal rule, and we're going to talk about the error which is associated with the multiple segment trapezoidal rule. Now, if you look at a single segment trapezoidal rule, so let's suppose if I am trying to integrate this function from a to b, where this is a, and where this is b, then I know that the trapezoidal rule is basically going to give me the area under this curve here, which is a straight line here. So there's some amount of true error which has been created because of the approximation of the curve being a straight line from point a to point b, and this particular true error, without proof I'm giving it to you, is given by -1 divided by 12, b minus a, whole cubed, times f double-prime of alpha. Now, what does this mean? It is that you are basically integrating your function from a to be, f of x dx, and this quantity here is the amount of true error which is being created. Now, b is, of course, the upper limit of integration, a is the lower limit of integration, and alpha, the limitation on alpha is that it is somewhere between a and b. So we don't know alpha beforehand, what alpha is, because if we knew what alpha is, then we could just simply calculate the true error, add it to the approximate value, and simply you get the true value, but that's not the case, alpha here, in this case, is some point between a and b, which is not predetermined or known, but it is also known that it is between a and b. So this also tells you that if the second derivative is occurring in this formula for the true error, is that if you have a constant line or a straight line, the amount of true error which you're going to get will be equal to 0. So in a multiple segment trapezoidal rule, let's go ahead and see that what would be the error in that case. So if I have the same kind of function, let's suppose, and I am integrating from a to b, but in the multiple segment trapezoidal rule, what I am doing is I am breaking this interval from a to b into n equal segments, so I'm saying h is equal to b minus a, divided by n, and then this point becomes a plus h, this becomes a plus 2 h, and this last point here is a plus n minus 1 times h. So that's the way the multiple segment trapezoidal rule works, that you break it up from a to b into n equal segments, and then you are finding the individual trapezoids the areas of the individual trapezoids between these segments, so from a to a plus h, this will be one trapezoid, then the second, then third, then fourth, and so on and so forth. So what's going to happen is that each of these particular trapezoids is going to have some true error associated with it. So if you look at the true error associated with the first trapezoid, let's suppose we call it 1 here, it'll be minus a plus h, because that's the upper limit, minus a, which is the lower limit, raised to the power 3, of course, which is the b minus a, whole cubed, of course divided by 12, that's from the formula, multiplied by f double-prime of alpha1, where simply we know that alpha1 has to be between the lower limit and the upper limit, so it has to be some point between a and a plus h. So this expression which we wrote down here is basically from our knowledge of what is the true error for a single segment trapezoidal rule, and since what we are doing here, from a to a plus h, we are in fact applying the single segment trapezoidal rule in the multiple segment trapezoidal rule formula. So this E1 here turns out to be equal to minus h cubed, divided by 12, times f double-prime of alpha1, where alpha1 is between a and a plus h. So we can do the same thing for segment number 2, for example, so if I wanted to calculate what the error is in the second segment, it will be still minus h cubed, divided by 12, because the . . . because the upper limit is a plus 2 h, and the lower limit is a plus h, so the difference between the two limits is still h, just like it is h in the first segment. So the front coefficient here is simply minus h cubed by 12, however the second derivative will be for the function evaluated at some different point, alpha2, and that alpha2, of course, will be between a plus h and a plus 2 h. So you can very well see that how the trend is going. So if I wanted to calculate the true error for the ith segments, so I'm saying for the ith segment, which can be any segment going from 1 to n, it'll be minus h cubed, divided by 12, f double-prime of alpha-i, and where it can be going from a plus i minus 1 h to a plus i times h. So that will be the limitations of where alpha-i can be situated somewhere between this number and this number here. So this is the general formula for the true error in a particular segment. So what that means is that if I add all these segments, if I add all these errors for all the segments, that'll give me the true error for the whole . . . whole multiple segment trapezoidal rule. So the total true error will be equal to the summation of Eis, all the Eis, from i is equal to 1 to n. So that turns out to be, I'm going to substitute, again, the formula for Ei, it was minus h cubed, divided by 12, f double-prime of alpha-i, that's what I'm going to get. And what I'm going to do is I'm going to take minus h cubed, divided by 12 outside, I'm going to say summation, i is equal to 1 to n, f double-prime of alpha-i, so I'm going to do that, and then I also know that h is nothing but b minus a, divided by n, so it'll be b minus a, cubed, divided by 12 n cubed, summation, i is equal to 1 to n, f double-prime of alpha-i. And you will see in a little bit why I substituted h equal to b minus a, divided by n. It is why? Because I want to put something like this, b minus a, whole cubed, divided by 12 n squared, and then summation, i is equal to 1 to n, f double-prime alpha-i, divided by n. So I took one of the ns from the n cubed, put it right here. Now, what you can see here is that you have this particular quantity here, which is some approximation of the average value of the second derivative of the function, because you are calculating the second derivative of the function at n different points, which we don't know what they are, but we know what the location of . . . location of ranges for each of the alpha-is, and if I take some . . . all of these second derivatives which I have calculated at these n different points, and I divide it by n, it will give me some approximate value of the second derivative of the function, not an exact . . . exact value, but an approximate value of the second derivative of the function between point a and point b. So what we are left with now is this particular quantity, which is b minus a, whole cubed, which is a constant quantity for a particular integral, because a and b are fixed, 12 is fixed, negative is fixed, so the only thing which you can vary is n squared. So what this tells me is that the true error is approximately proportional to 1 by n squared. Now what does this mean when we say that the true error is approximately proportional to 1 divided by n squared? It means that if I double the number of segments in my multiple segment trapezoidal rule, my true error is going to get quartered, and so on and so forth. So that's how the true error works in the multiple segment trapezoidal rule. And that's the end of this segment. |