Quantifying Errors: True Error

So, before we go and talk about true error what we need to figure out is that why do we need to talk about it. So, in numerical methods you're going to get two sources of errors which are going to come which are called truncation error and round off, error truncation errors where you are approximating a formula, let's suppose, or maybe taking only a few terms of a series uh because you're truncating a mathematical procedure. While the round of error is coming from approximating numbers. So those will be the source of the error so how do we so we have to quantify those errors right and so what we need to figure out about errors is that hey errors are not good and we need to figure out how much error we can live with so the first thing which we have to do is do we have to identify where the error is coming from, and that's something which you will see throughout the course in terms of whenever we discuss the numerical procedure and then second thing is that hey we need to quantify the errors. So, I want to identify theirs we've got to quantify the errors and this quantification of the errors comes in two forms one is called true error and the other one is called approximate error. Approximate error we'll discuss in a different segment, but in this segment, we're going to focus our attention on true errors. And of course, we understand that hey what we're going to do is that we will minimize the error as per our needs so that's how we're going to go about dealing with errors. Now many people might say that hey why do we need to how can we calculate true errors? Because we are using numerical methods in the first place because either the exact values are not possible to find, or exact values will take a long time for us to find. So, what is the benefit of true error? The benefit of true error here is that hey if you are developing a numerical method, let's suppose, and you want to test it out, you may have an exact value for some special cases and that's how you're going to test it out and see that hey whether your numerical method program or the algorithm which you have developed is working or not. So, let's go and concentrate on what the definition of the true error is and see, through an example, and you'll be able to figure out the other parts of the flowchart which is here in um as you go through the course. So, let's go and see how true error is defined. So true error is denoted by the symbol E sub t, uppercase e standing for error and this for t. And that is defined as a true value minus the approximate value. Many times, this true value here which you are seeing here is also being going to be called the exact value. That means exactly the same thing. So, whether the is a true value there or exact value there means the same thing so it's just the difference between what is the exact value and what you're getting as the approximate value from our numerical methods. So that's the definition of true error as simple as that and in the next slide we will look at an example to see how we can calculate a true error. So, here's the example which we'll be taking but we are being given an approximate definition of the derivative of the function as follows. And then we are taking this function with a step size of 0.3, or the value of h, and we're going to find the approximate value of the derivative of the function at 2, we're going to find this true value and henceforth find the value of the true error. So, our definition for the derivative of the function is given as f prime of x is approximately [f(x + h) – f(x) ]/h . Our value of x is we're trying to find the derivative of the function at 2, the step size ‘h’ is given to us as 0.3, so that gives us f prime of 2 is approximately this. So that gives us that hey we have to calculate the value of the function at 2.3 and 2, and the difference as we divide it by 0.3. So now all we have to do is to substitute the value of the function at 2.3 and 2. Since our function is 7e^0.5x, so in this case its 2.3, minus 7e^0.5x, but in this case it’s 2, divided by 0.3. And this value here gives us 10.263. So that's the approximate value of the function at 2. Now you know how to find the exact value what is f prime of 2, and we have to find the exact value. We start up with our knowledge of differential calculus so f(x) = 7e^(0.5x), and we already know that from differential calculus that the derivative of ce^(ax) for example, where c and a are constants, is nothing ca*e^(ax), so both c and e are constants in this. So that's what we get from differential calculus. So, we apply the same thing here, where d/dx (7e^(0.5x)), and they will give us 7*0.5*e^(0.5*x), and that gives us 3.5*e^(0.5x), and that is the exact value of the derivative of the function. So, f prime of x is 3.5*e^(0.5x), so hence we get f’(2) = 3.5e^(0.5*2), x is 2 and that value turns out to be equal to 9.5140. So that's the exact value up to 5 significant digits here and we in the earlier part we found out what the approximate value is. So here what we're going to do is so we're being asked to calculate the true error. The true error will be true value minus approximate value. And the true value found from part b, which was 9.5140. The approximate value found from the first part was 10.263. So, if we take the 2 and subtract, we get -0.749 as our true error. And that's the end of this example here. But one of the things which you're going to see here is that we're getting a true error of -0.749. How do we know whether this true error is a small quantity or a large quantity? So, for example, if our if f(x) was given as 7*e^(0.5x), that’s what was given in the example, let's suppose uh instead we have f(x) = 7*10^-6 * e^(0.5x). Let’s suppose the previous example the only thing which was changed was the function; rather than being given as 7*e^(0.5x), it is given as f(x) = 7*10^-6 * e^(0.5x). If we repeat the problem, the true error would be -0.749*10^-6. So suddenly, for the same problem just with a different function in terms of the constant being uh a slightly different function in terms of the constant being 7E-6 rather than 7, we get a true error which seems to be a very small quantity. So same problem but we find out that in one the number looks very small another one number looks reasonably large. So how do we distinguish between the two is what comes in the picture because we have to figure out whether the numerical method is working properly or not. So, we might be calculating two errors based for trying to figure that out. So, what we do have to define is something called the relative true error. Because now what we're going to do is we're going to do is normalize our true error by saying that hey it is the true error divided by the true value. So, this basically gives us a proportion of it, and the relative true is denoted by epsilon of t, so we had true error, which was denoted by e sub t, relative true error denoted by um epsilon of t. So, let's go and take an example, same example as we had before and see hey what do we get is the relative true error. So here is the example which we are taking for the relative true error, same as the first example but the only difference is that we are now asking for the relative true error as opposed to the true error in the case before. So, this is pretty straightforward in this case because we will take the numbers from the previous example. We got true error as -0.749, that’s from the previous example. We had the true value was also from the previous example was 9.5140. So, what is the true error, the relative true error? The relative true error will be the true error divided by the true value, and that gives us -0.749/9.5140. And this number here is -0.078726. That's what we get as the relative true error. Now one of the things which you have to think about is as follows. So, what we obtained so far is that the relative true error, which we got was -0.078726. And sometimes people will say that hey uh give us this number relative true error in terms of percentages, what you're going to do then? Then in that case all you're going to do is you're going to take the number and multiply by 100. That's all you're going to do. And you’re going to get 7.8726%. So relative true error can be written as a fraction, or as a percentage. And depending on how the question is asked if. If it's not asked, then you can give the answer in either terms. But if it is asked to be given as a percentage then you give it as a percentage. If it's not, if it is asked to be given as a fraction then you give it as a fraction. And that's how it will work. Now um let's uh also sometimes what they will do is they will ask you to calculate absolute relative true error. So, in that case what are you going to do? In that case you should be going to the absolute value of the relative true error. So, in this case is that that is just the positive, the magnitude of the number turns out to be this, or if it is in terms of percentages, you would just write it as this That's all there is to it Now going back to the example which we're taking we said hey, the function given in the example was 7e^(0.5x). Now instead if the function was given in the example was given as 7*10^-6 *e^(0.5x) for the same example, what's going to happen is the true errors which you will get for true errors you get for this problem and for this function for the same problem they'll be different they'll differ by 10 to the power minus 6 magnitude. However, so far as the relative true errors are concerned, they will be the same. So that's the power of each of these quantities which are defined whether you're defining true error or whether you're defining relative true error. so relative true error gives you the relative mechanism of seeing that hey how bad your error is. So that's the end of this segment.