Lecture 04

So, in the last few lectures, we have talked about the course policy, and we have seen one example of numerical method, but we really never formally provided our definition or concepts of numerical methods or numerical analysis. So here in this lecture, we'll go in that direction. We started with an example in last lecture, but now we'll sort of make it abstract, make it formal. So we'll see that what are the fundamental processes behind numerical methods, its creation as well as implementation in practical problem. And also we'll talk about the advantages and disadvantages of numerical methods. So what do you know about the numerical methods from our previous example, we have seen that in our previous example, all we have done are arithmetic calculation. In fact, primarily we were doing counting and then one multiplication, one division. So the goal of numerical method is to reduce the problems of mathematics into arithmetic problem. And what do we mean by arithmetic problem? What do we mean that all problems of mathematics will reduce them to working with certain numbers mostly real numbers, but it could be also integer as well as complex numbers. And also the elementary arithmetic operations of numbers. And by these elementary operations, we mean addition, subtraction, etc. So this is the goal of numerical methods. Once we reduce the problem in arithmetic problem, what we do, this arithmetic problem that we have devised, we break it into a series of easy repetitive calculation. For example, in our last example, we were doing was counting. So that is a repetitive procedure. So all numerical methods are essentially repetitive procedure and we'll see more of them. Now methods like this, it has two step processes. What are these two steps? In step one, we first develop the idea or the process to solve the complex problem without any high expertise. So that is the goal always. We do not need the expertise of integration to solve integration. And just like that, we want to solve complex problems without having the expertise of that complex problem. Under this process, what we develop, this is something we call algorithm, we call pseudocode, we have different names. But essentially, we get the idea that how we'll do it. Once we get this step one right, we go to step two. And in step two, we implement this process in a machine capable of doing fast, accurate and simple calculations. In earlier days, in the initial days of numerical methods, these calculations were being done by human. But now we use computers to do it. In earlier days, computer was a designation that was given to human who will do all these calculations manually. But now we use a machine and machine could be fast and machine could be accurate and machine will not complain to do repetitive calculations. So in summary, the two step is first step is to develop the idea or develop the algorithm that how we are going to solve complex problem. And in step two, we'll implement our idea in a computer. So here in this course, we are about to learn both these steps. Since we use a software, and we solve problems using numerical methods, but we don't know how that is implemented in the computer. But here we are not going to do that. Here will be the user as well as will be the developer. Implementation in a computer that is the task of a developer. And using this is the task of an user will implement in computer and will use it. So at the end of this course, you will learn how to solve problem using numerical methods. And you will also learn how to write a computer program for numerical methods. So what are the advantages of numerical methods? Of course, we can solve problem where we do not have analytical solutions, which are plenty. Most practical problems, most problems of engineering applications cannot be solved analytically. So they do require numerical methods. And sometimes we do have analytical solutions of certain problems. But we want to solve them using numerical methods. It may happen that even if the analytical solution is available or possible, we ourselves cannot do the analytical solution or analytical solution is very long, tedious, and extremely complex that we avoid doing that. Rather, we take it for numerical methods to solve such problems. And the third advantage is that we can solve relatively complex problems with little knowledge of mathematics. And this is a great advantage, which we have already seen in our example, that we can have counting, we can have multiplication, and we can have division to evaluate and integral. So essentially, we didn't use any knowledge of calculus to evaluate integral. But with all these advantages, there are also certain disadvantages of numerical methods. For example, numerical methods only deals with numbers. So the answer you will get from numerical methods will come as numbers. But do we need really numbers for all problems? Sometimes it is not. Sometimes we need an analytical expression from a problem. In such cases, numerical methods can still be helpful. And we'll see later on in this course near the end that how it can be helpful. But it may not be optimal. In certain cases, it may be very clumsy to do use numerical methods when we do not want the answer to be in terms of numbers, but in terms of expression. And if we use numerical method there, we have to first get the answers in numbers, and then we have to convert the numbers to expression. That is possible, but that may be clumsy sometime. The second problem of numerical method that you have already seen that the result which will come that that will always be approximate. So some error will always be associated with the result. It will never be exact or almost never be exact. So if you are looking for exact answer, then numerical method most of the time probably will not work for you. And the third problem is that for any practical application of numerical methods, at least in nowadays, it will require considerable software and hardware investment. You need a reasonable quality computer, and maybe some compilers, at least, as well as a plotting software to use numerical methods. In early days of numerical methods, it used to be manual calculations, but now those days are gone. There needs to be an investment in hardware and software for using numerical methods. So just to repeat, we have already discussed it, but just to repeat that, then what do we do in this course? In this course, there are three parts. We have linear algebra, calculus, and then third is the data-driven learning. So in linear algebra and calculus, we'll talk about solutions of equations and eigenvalue problem and all this will solve using numerical methods. Similarly, in calculus, we'll see how differentiation, integration, we have already seen one method of integration. Similarly, we'll see other methods. We'll also see differential solution as well as solution of differential equations using numerical methods. And the third part, which is most modern aspects of numerical methods, where we'll talk about that how we can build theory from available data. During the course, you will see that the linear algebra and calculus, which are actually different books and different chapters in mathematics, in numerical methods, those distinctions are often not very sharp. Why is so? Because practical problems, they're often modeled in terms of differential equations, at least in engineering, most of the problems will come as differential equations. There are certain cases, for example, in radiative heat transfer, you may have integral equation, but in most cases, you will have differential equations. But these problems of differential equations as we'll go on in this course, you will see that they are solved using the techniques of linear algebra. And this is a beautiful aspects of numerical method is that you will see that an amalgamation of linear algebra and calculus here in this course. Another point is that in this course, as the title suggests, we'll talk more about the methods that how to solve certain things, how to evaluate an integral or how to solve a differential equation. But we'll not talk so much about application. Yes, we'll sometimes solve real application problems, but not all the times. Sometimes we'll just solve an abstract mathematical problem. Since all of you have engineering or science background, I think that connecting the actual mathematical problem with an application problem will not be very difficult for you. And examples of such connections will give you as the course progresses. So now in this lecture, we have discussed the fundamental processes behind numerical methods, the scope of numerical methods, what we can do, what we cannot do, and the advantages and disadvantages of numerical methods. For your exercise, you will list few problems where you want to solve those problems using numerical methods and why you take numerical methods in solving that problem. You can discuss them with example. Similarly, you find some problem where you should not use numerical methods. You discuss with example. That's all for this lecture. Thank you, and I'll see you in next class.