In this segment we'll talk about approximate errors.
So, in a numerical method, the error can come from truncation error or from, or/and
from round off error. Truncation error is where you’re approximating a procedure –for example
you're trying to find the value of a series sum of a series and you're only taking a finite number of
terms. That is a type of truncation error. And round off error is coming from approximating
numbers. So, in any numerical method which you're going to use to be able to solve a
problem what you'll have to do is you'll have to identify where the error is coming from.
You’ll have to then quantify that error, and then of course minimize that error as per your needs.
So in this segment we are going to basically concentrate on the quantifying of the error
and the error which we are going to talk about here is approximate error.
And in a numerical method you won't be privy to the true value because that's why you're using
numerical method in the first place, so in an approximate error mean will mean that hey maybe
you're using some kind of iterative procedure what you have some kind of uh approximation, and
you had a previous approximation and what you're trying to do is, you’re trying to, based on that,
you might be able to define what the important approximate error is so let's go and define
approximate data and then we'll go through an example which would be a better way of
figuring out what this all about. So, the approximate error
which is donated by E sub a is equal to present approximation minus the previous approximation.
That’s what we call it. So, e sub a is the is the symbol by which it is donated and
denoted and then we have defined as present approximation previous approximation. So,
you got to have two approximations to be able to find out what the approximate error is.
And it will be clear once we do an example so let's go ahead and do that.
So, let's go and take an example of the approximate error so here we are given
that we are defining the derivative of a function by this approximate formula. And then we want to
find out the derivative of this function here at x equal to 2 but what we're doing is we're
using two different step sizes of h = 0.3 and h = 0.15. And this is what is going to give us the
current and previous approximation. So, we're going to say that hey this one will give us the
previous one because a larger step size, and this one is going to give us the present approximation.
So that's how we are going to be able to find the approximate error. So, let's go and see how do we
go about doing that. So, we are given that f’(x) is approximately given as this formula right here,
and what we're going to do is we're going to start with our
previous approximation which will be obtained by using a step size of 0.3,
x is of course 2 so that gives us f prime of 2 equal to this quantity right here.
So, what this implies is that we have to calculate the value of the function at these two different
points, so this number would be 7 * e^(0.5*2.3) – 7*e^(0.5*2) divided by 0.3 right here. And this
number will here turn out to be 10.263. So, this number here is my previous approximation
of f prime 2 because this is the largest step size.
Now let's go and redo this calculation exactly the same calculation but with a step size of 0.15.
So again, we have f prime of x which is approximately equal to this quantity
right. And f prime of 2 so x is 2 and h is 0.15, based on that, will be this quantity right here.
So, what this implies is that we are calculating the value of the function at two different points
and 2.15 and 2 so it would be as follows.
And this value here will turn out to be 9.8800. So, this is our current approximation.
So we have a previous approximation a current approximate, so we should be able
to calculate whatever absolute relative approximate sorry our
approximate. So, let's see what our approximate error is now:
Approximate error e sub a will be equal to current approximation minus the
previous approximation. And the current approximation is what we obtained with the
h=0.15, which is 9.8800 and what did we obtain with the previous approximation—10.263.
And this number here turns out to be equal to -0.38300.
So that is the approximate error. Now, if somebody asked you that hey does this
approximate error looks small or large that might be something which
just by looking at the number itself, may not give you a good glimpse. So, for example if we
the same example which we just did, we had 7 * e^(0.5*x). So, let's suppose somebody
changed the only one, one thing in the whole problem and said hey, your function is this
they just instead of 7 they put 7 times 10 to the power -6, do you know what's going to happen
to the absolute sorry the approximate error? The approximate error is going 0.38300 times
10 to the power -6. So just by doing a slightly different problem you may
think that hey this number is so small, so the approximate error is small. So
that's why in order to get a good glimpse of how much really the approximate error is,
what we do is we talk about something called the relative approximate error. So, the relative
approximate error which is denoted by epsilon a,
and that is defined as hey what is the approximate error divided by the current approximation.
That's it. So, what that will give you is a relative sense of
how much error do you really have. Let's go and take the same example and say hey let's
go and calculate the relative approximate for the same example as we had before
and see that hey is it any different, what do we gather from there.
So, this is the example which we had taken previously uh the only difference is that we have
we have we're calculating the relative approximate error. The function is still the same,
what are we calculating the derivative, what step sizes we're using is exactly the same.
So, let's go and see how we can calculate the relative approximate error. So, from the previous
example we have that the approximate error, that's e sub a, is equal to the present approximation,
which we had calculated in the previous uh example, which was these quantities
so this this is a present approximation previous approximation, and we get 0.38300.
So now the question is that hey what is the absolute what is the relative approximate
error? What is the relative approximate error? So, the relative approximate error
will be equal to the approximate error which we're getting divided by the current approximation.
So, we just found out that the approximate error is this,
and this was done also in the previous example, the current approximation is this, so we get an
approximate relative approximate error of -0.038765. And so now we have a good sense of
what the approximate error relatively is, and many times you may be asked to
write down epsilon a as a percentage, so it'll be just multiplying this number here by 100.
So, you will get 3.8765%. Many times, they may ask you to calculate the absolute relative
approximate error, then in that case it is just the magnitude of the numbers.
So, we have either this or this if you're writing in terms of percentages and that's the that's how
we calculate the relative approximate error. Later on, you will see that these approximate relative
approximate errors which we have calculated like here and here these are bearing or how
many significant digits will be at least correct in our answer. So, it's all uh related to each
other so don't uh look at this as a as an isolated case. And you are going to be talking about
true errors relative, true errors approximate errors, and relative approximate errors throughout
the course. So, grabbing the concept is extremely important. And that is the end of this segment.