CHAPTER 05.03: NEWTON DIVIDED DIFFERENCE METHOD: General Order: Newton's Divided Difference Polynomial: Example: Part 1 of 2
Okay, in this segment, we're going to talk about Newton's divided difference polynomial, and we are talking about general order, and we'll take an example of that. So somebody, let's suppose, is giving you velocity versus time data. Velocity is given in meters per second, time is given in seconds, and somebody's giving you six data points, 0, 0, 10, 227.04, at 15 it is 362.78, at 20 it is 517.35, at 22.5 it’s 602.97, and at 30 it is 901.67. So those are the six data points which we are given, and what we are asked to do is to find out the value of the velocity at 16. We're asked to find the value of velocity at 16, and somebody's asking us to use a third-order . . . use a third-order Newton's divided difference polynomial. So when somebody's telling us that, hey, you have to use a third-order Newton's divided differences polynomial to calculate the value of velocity at 16, the first thing which we have to do is that we have to select the data points. That's why in all the derivations I have said chosen certain data points, find the Newton's divided difference polynomial, because many times you'll be given a lot of data points, but you'll be choosing the points properly to meet your requirements. If you have a third-order polynomial, you want to have four data points. So now, since you are trying to find the value of the velocity at 16, you've got to take 15 and 20. That's a given, because those are the closest two data points on x . . . on t which bracket the value of 16, so 15 and 20 have to be taken. Then I'm going to take 10, because the distance between 10 and 16 is only 6, and then I'm going to take 22.5, because the distance here is what, 6.5. I'm not going to take 0, because the distance is 16, I'm not going to take 30, because the distance is 14. So it looks like that these four data points, which I have here, are the ones which I need to select for finding the value of the velocity at 16. So let's go ahead and write down what t0, t1, t2, and t3 are, and what the corresponding velocities are so that we can have a good way of showing what . . . how we need to calculate these . . . the value of the velocity at 16. So what we have is t0 is 10, and the velocity at this particular data point is 227.04, then t1 is 15, and the velocity at t1 is 362.78, t2 is 20, and the velocity at this particular data point, or this particular time, is 517.35, of course, the units are meters per second for velocity, and the units for time are seconds, and at 22.5 seconds that the velocity is . . . corresponding velocity is 602.97. So as we know that the third-order polynomial is going to look like this, the third-order polynomial of the velocity, so third-order polynomial, is going to look like this, it's going to look like b0, plus b1 times t minus t0, plus b2 times t minus t0, times t minus t1, plus b3 times t minus t0, t minus t1, times t minus t2. So I have this third-order polynomial, where I also know that b0 is nothing but the first divided difference at the point t0 . . . or zeroth divided difference, it's the zeroth divided difference based on the value of the velocity at t0. Then b1 is the first divided difference based on the values of the velocity at t1 and t0. b2 is the second divided difference based on the values at t2, t1, and t0. And b3 is the value of the velocity at these four data points, at those . . . the velocity based on those four data points and the times. So what we need to do is we need to calculate these, and if we're going to calculate these, then we are done in terms of finding the third-order polynomial, and once we have the third-order polynomial, we can just substitute the value of time equal to 16 in the equation here, and we're done. So let's go ahead and see that how we're going to calculate these four constants, and I'm going to calculate these four constants based on the tree of divided differences which I showed you, because that's a better way of showing that how this whole thing is progressing. So let's go ahead and look at the tree here, so we have, I'm going to show time here and velocity here. So it's 10, 15 . . . 15, 20, and 22.5. So those are the four numbers for the time, then I have 227.04, 362.78, 517.35, and 602.97. I'm writing a little bit close to each other, I may need to be able to show you the tree of these divided differences. |