CHAPTER 02.02: DIFFERENTIATION OF CONTINUOUS FUNCTIONS: Backward Divided Difference: Part 2 of 2
Let's go ahead and see that what kind of numbers I get, or if I choose delta x to be a smaller number, does the true error decrease, and how much does it decrease by? So I'm going to draw a table which I found by simply doing some calculations in an Excel spreadsheet. So if I write down my . . . this is the function, f of x equal to 2 e to the power 1.5 x, I'm calculating my derivative of the function at 3, so I got delta x here, f prime of 3 here, and my true error here. When delta x is 0.1, I just calculated my f prime of 3, from the example, which turns out to be 250.77, and the true error is 19.27. For 0.05, which is half of the delta x which I chose, I get 260.17, and the error is now 9.88, that's the true error. Now I choose delta x equal to 0.025, I get f prime of 3 to be 265.05, and what I get for the true error is 5.00. So what you are finding out is that as delta x is becoming smaller and smaller, I'm halving delta x, the approximate values of f prime of 3 are getting closer and closer to the exact value of 270.05 as I make delta x to be half of the previous one. The true error is also, what you are finding out, is decreasing as you keep on decreasing delta x. But one of the things which you are noticing here is that the true error as I halve delta x, I find out that the true error here is becoming also half of what it was previously, and when I go from here to here, the true error is, again, becoming approximately half of what it was before. So what I would like you to do is, these are the questions which I would like you to answer on this segment. One is, why does the true error get halved . . . I should say approximately halved, not exactly halved, because it's approximately halved, as delta x is halved? So I would like you to answer that question.
And the second question is, again, if true value is not known, so let's suppose if you didn't know what the . . . is not known, if the true value which you've found out to be 270.05, let's suppose it's not known, because we are using numerical methods basically to calculate approximate values, so the true value is not known, how would you use the concept of relative approximate error to find values correct up to some given significant digits? The question which I am asking is that, how would you use the concept of relative approximate error, which you can calculate as you are making delta x to be half of what it was before, to find values to be able to say that, hey, how many significant digits are correct? So let me go back to the table right here, so if you look at the table right here, these are the approximate value which you are getting for different values of delta x, right? So you should be able to calculate the relative approximate error here, the absolute relative approximate here, the absolute relative approximate error here, and based on what numbers you get, you should be able to say, hey, can I trust this 2 to be correct, can I trust this 6 to be correct, can I trust this 5 to be correct, can I trust this 0 to be correct, can I trust this 5 to be correct? In this case, what I am trying to basically say is that how many of these significant digits, starting from the left, can I trust in this solution, without having any knowledge of what the exact value is. So that's where the concept of the relative true . . . relative approximate error comes into the picture, to be able to figure out how many significant digits are at least correct in the answer. And that's the end of this segment.