CHAPTER 01.02: MEASURING ERRORS : Relative Approximate Error 

In this segment we're going to talk about relative approximate errors. So let's see what the motivation behind this is, that why do we need to talk about relative approximate errors?

So let's suppose if you have f of x equal to 7 e to the power 0.5 x. Somebody's asking you to calculate f prime of 2, and saying, hey, use delta x equal to 0.3, and use this formula to calculate your derivative of the function. This is an approximate formula, because in this case your delta x is not approaching 0, so you're choosing a finite value of delta x to be 0.3  Now somebody says, hey, I'm going to give you another function, which is similar to what you had previously. The only difference is that now the coefficient of this function is, the constant in this function which is multiplied here, is not 7, but 7 times 10 to the power -6. But I still want you to calculate the value of the function, the derivative of the function at 2, and use the same value of delta x, which is 0.3, and use the same formula for the derivative of the function.  So the question arises that, what numbers am I going to get for f prime of 2? So I'm going to write these down here, so here I have the function, which is f of x equal to 7 e to the power 0.5 x, here I have the function as 7 times 10 the power -6 e to the power 0.5 x, so same function here as here, except for there's a 10 to the power -6 being multiplied.  And here I'm going to show you what f prime of 2 is, and, using delta x equal to 0.3, and here I'm going to show you what f prime of 2 is using delta x equal to 0.15, and then let's go ahead and write those down. So if I take 0.3, what values do I get here?  I get 10.265. And what do I get for 0.15?  I get  9.8799.  Now if it is 10 the power -6 difference here, what's going to happen is that the f prime of 2 which you're going will be same number as here, except for it's multiplied by 10 to the power -6.  The number which you're going to get here is 9.8799 times 10 to the power -6. So what you are seeing here is that the approximate error which you are going to get between these two numbers, because this is with delta x equal to 0.3, and this is with delta x equal to 0.15, is going to some number which is going to be of the order of 1 or so, but when you calculate your approximate errors between these two numbers, it's going to be the order of 10 to the power -6. So similar formulas, you are asking to calculate the value of the function at the same point, you are using the same values of delta x, but the values which you are getting, and the approximate errors which you are getting for the two cases differ by 10 to the power -6. It might give you a false impression, in this case that, hey, my approximate error is small, and in this case that the approximate error is large.  So that's why we need to define the concept of the relative approximate error.  So let's go ahead and do that. So how is relative approximate error defined?  It is denoted by epsilon a, where epsilon stands for relative error, and a stands for approximate, is simply defined as approximate error divided by current approximation, or present approximation. So if we look at the first case, let's suppose, where f of x is equal to 7 e to the power 0.5 x, we're asked to calculate f prime of 2, and we are using delta x equal to 0.3, and then we're using delta x equal to 0.15. What we get as the approximate error is what we have at 0.15, the value, which is 9.8799 minus 10.265, so this is the approximate error which we are getting, this is the value of the derivative of the function by using delta x equal to 0.15, this is the value of the derivative of the function we're getting by using delta x equal to 0.3, and this approximate error is turning out to be -0.38474. So the relative error will be the approximate error, which is 0.038474, divided by the current approximation, which is 9.8799. And this number here, for the relative approximate error, is turning out to be -0.0389. And some people will write it as a percentage, so it will be -3.89 percent.  Some people also are interested in the absolute relative approximate error, so that will be just the positive of that, or the modulus value of that, absolute value of that, which will be 0.0389, or 3.89 percent.  So this is the relative approximate error which you are getting for this case. 

Now if we had the other example which I showed you earlier, that if I choose f of x equal to 7 times 10 to the power -6 e to the power 0.5 x, and somebody's asking me to calculate f prime of 2 using again the value of delta x equal to 0.3 and delta x equal to 0.15.  In this case what I'm going to get is the approximate error will be whatever value I get for the derivative of the function at 2  by using this value of delta x, and then this value of delta x will be the previous approximation.  I'll get 9.8799 times 10 the power -6 minus 10.265 times 10 to the power -6, and the value will turn out to be -0.38474 times 10 the power -6.  So your approximate error is a very small quantity, if you just look at it.  But your relative error will turn out to be the approximate error divided by the current approximation, which is 9.8799 times 10 to the power -6, which you get as -0.038942, or your -3.8942 percent.  And what you are finding out is that the relative error which you are getting for this case here, by having f of x equal to 7 times 10 to the power -6 e to the power 0.5 x is -3.89 percent. And if you look at the previous example, which is right here, in this, the function, the only difference in the function is 7 e to the power 0.5 x.  And again the relative error which you are getting here is -3.89 percent, exactly the same quantity.  So relative errors are a better judge, or better way of you judging how bad or good your error is. So that's why we have to talk about relative errors.  And that's the end of this segment.