Many of us are faced with the confusing terms in different Sciences. But is very few people who are not afraid of strange words, but rather up and forced to enter more deeply into the subject of study. Today we will talk about such things as interpolation. This method of plotting the known points, allowing, with a minimal amount of information about the function to predict its behavior in specific areas of the curve.
Before proceeding to the substance of the definition and tell me more about it, go a little further in history.
Interpolation has been known since ancient times. However, in its development, this phenomenon owes to some of the most outstanding mathematicians of the past: Newton, Leibniz and Gregory. They developed this concept using more advanced mathematical methods available at that time. Prior to this interpolation, of course, applied and used in calculations, but did so quite imprecise ways, requiring large amounts of data to build the model, more or less close to reality.
Today we can even choose which method of interpolation is more suitable. Everything is translated into computer language, which with great accuracy can predict the behavior of a function in a particular area bounded by known points.
Interpolation is a fairly narrow concept, so that its history is not so rich in facts. In the next section we will understand what interpolation actually is and how it differs from its opposite – extrapolation.
What is interpolation?
As we have said, is the common name of ways to graph the points. In school this is mainly done with the help of tabulation and identifying points on the graph and approximate drawing lines connecting them. The last action is done for reasons of similarity of the studied functions other graphs which are known to us.
However, there are other, more complex and accurate ways to perform the task of graphing the points. So, interpolation is in fact a “prediction” of behavior functions in the specific area bounded by known points.
There is a similar concept that is associated with the same area – extrapolation. It is also a prediction of the function, but outside the known points on the graph. In this way the prediction is done based on the behavior of the function at a known interval, and then apply this feature for an unknown gap. This method is very convenient for practical application and actively used in the economy to predict the UPS and downs in the market and to predict the demographic situation in the country.
But we moved away from the main topic. In the next section, we will understand what kind of interpolation happens and with which formulas it is possible to perform this operation.
Types of interpolation
The simplest type is the interpolation method is nearest neighbor. Using this method we get a very rough diagram consisting of rectangles. If you ever seen an explanation of the geometric meaning of the integral on the chart, you will understand what a graphic is about.
In addition, there are other methods of interpolation. The most famous and popular associated with the polynomials. They are more accurate and allow us to predict the behavior of a function at a fairly meager set of values. The first method of interpolation, which we consider will be linear interpolation by polynomials. This is the simplest method in this category, and they certainly every one of you enjoyed at school. Its essence lies in building a direct between known points. As you know, two points of the plane is a single line whose equation is can be found based on the coordinates of the data points. By building these lines, we get the polyline graph that somehow, but reflects the approximate values of the functions and in General coincides with reality. And implemented linear interpolation.
Complicated types of interpolation
There are more interesting, but more complex interpolation method. It was invented by the French mathematician Joseph Louis Lagrange. That is why the interpolation calculation by this method named after him: interpolation of Lagrange. The focus here is this: if the method outlined in the previous paragraph, uses to calculate only the linear function, the decomposition method Lagrangian involves the use of polynomials of higher degrees. But not so easy to find the formulae of interpolation for the different functions. And the more points is known, the more accurate is the formula of interpolation. But there are plenty of other methods.
There is a more perfect and close to reality the calculation method. The interpolation formula used in it is a set of polynomials, each of which depends on the phase function. This method is called spline-function. In addition, there are also ways to spend such a thing as interpolation of functions of two variables. There are only two methods. Among them, bilinear or double interpolation. This method makes it easy to plot points in three-dimensional space. Other methods to address will not. In General, interpolation is a universal calling for all of these methods of plotting, but the variety of ways to implement this action forces us to divide them into groups depending on the kind of function that is subject to this action. That is, the interpolation, an example of which we have considered above relates to direct methods. There is also a reverse interpolation, which is characterized in that allows to not calculate the direct and inverse function (i.e. x of y). To consider the latest options we will not, as it is quite difficult and requires good mathematical knowledge base.
Let's move on to perhaps one of the most important sections. From it we learn how and where we are discussing the set of methods applied in life.
Mathematics, as we know, the Queen of Sciences. So even if you don't see the point in those or other operations, this does not mean that they are useless. For example, it seems that interpolation is a useless thing, with which only graphics can build, which are now less in demand. However, any calculations in engineering, physics and many other Sciences (e.g., biology), it is important to present a fairly complete picture of the phenomenon, while having a certain set of values. The values scattered throughout the schedule, do not always give a clear picture about the behavior of a function at a specific site, the values of its derivatives and points of intersection with the axes. And this is very important for many areas of our life.
And how is it useful in life?
A similar question can be very difficult to answer. But the answer is simple: nothing. It is that knowledge that you do not need. But if you understand this material and the methods by which these activities take place, you potreniruetes its logic, which in life is very useful. The main thing – not knowledge itself but the skills that one acquires in the process of learning. No wonder there is a saying: “live and learn”.
You can understand how important it was (and still does not lose its importance), this area of mathematics by looking at the variety of other concepts related to this. We talked about extrapolation, but there's still and approximation. Maybe you've heard that word. In any case, what it means, we also looked at in this article. Approximation, and interpolation, are two concepts associated with graphing. But unlike the former from the latter in that it is an approximation of the plotting based on similar well-known graphs. These two concepts are very similar to each other, and more interesting to study each of them.
Mathematics is not a difficult science, as it seems at first glance. It is rather interesting. And in this article we tried to prove it to you. We reviewed the concepts associated with the graph, I learned what a double interpolation, and dismantled in the examples where it is applied.