 We continue with our discussion on the analysis of dynamical systems. In the previous few lectures, we have been discussing the analysis or the time response of linear systems. Linear systems are amenable for what is known as Eigen value analysis, in which we can understand the properties of the response just by looking at the properties of the A matrix. Now, in today's lecture, we shall introduce you to a more general technique which is applied even to non-linear for the analysis of even non-linear systems that is the technique of numerical integration. So, we will talk about numerically integrating dynamical equations in order to obtain the response. Now, it should be understood that numerical integration is not limited to the analysis of linear systems. However, in today's lecture one or two examples that we will consider, we will consider numerical integration of linear systems. The reason why we do that is since we know the what the response is, it would be easier to understand the properties of the numerical integration algorithms which are used in doing numerical integration that is also called simulation of a system. Now, before we go on to numerical integration, we have been studying some very interesting topics in linear systems and how we can actually interpret the response of a linear system. So, we will just have a quick review of what we have been doing over the past few lectures. We started with the analysis of dynamical systems, what is an equilibrium, what are equilibria of a system, what are states, the issue of small and large disturbance stability. We also considered to illustrate this particular concept of small and large disturbance stability. The example of a single machine connected to a voltage source, then we moved on to linear time invariant systems, their analysis and we saw that we could analyze a system, linear time invariant system response using eigenvalues and eigenvectors of the A matrix of the system. That is if I got a system x dot is equal to A x, the properties of A really determine the stability of the system. The response of course is dependent not only on the eigenvalues and eigenvectors, but also depends on the initial conditions of the system. Whereas, the eigenvalues really determine the nature of the time response of individual patterns in the system, the eigenvectors, the right eigenvectors really give you some idea of the observability of certain patterns in certain states. So, that is basically a physical kind of interpretation of eigenvalues and eigenvectors and we had done a fairly detailed analysis of the time response of linear time invariant systems and we can actually write down the time response. Of course, there was a kind of a caution which I had asked you to bear in mind. That is if your A matrix has non distinct eigenvalues, it is possible that you will not be able to get n linearly independent eigenvectors, n being of course, the size of the A matrix. So, if you have got a A matrix which has got size n cross n and it does not have n distinct eigenvalues, this possibility that you may not have n distinct or linearly independent eigenvectors. In that case you cannot diagonalize the matrix, you will get terms like t e raise to lambda t and so on under such situations. But the basic point which I wanted to you know you to understand that the behavior of a linear system can be quite properly understood. There is no mystery as far as the functions which appear in the response. The time functions that is e raise to lambda t, sin omega t is t e raise to lambda t is all are well known and well understood functions. So, the response of the system can be quite easily obtained and understood. In addition, we can also understand how the system will behave like if you have got a system x dot is equal to A x plus B u that is u is an input, then the input the first response for this input also can be written down. So, that is one major advantage of linear time analysis of linear time invariant systems. Remember of course, that linear your linear time invariant systems, dynamical systems arise in our day to day life and in power system dynamics in two ways. The system may be inherently linear or you may be analyzing a basically originally non-linear system by linearizing it around an equilibrium point to understand its small disturbance behavior. So, linear systems arise in these two possible contexts. An important point which we understood in the previous lecture was that some systems not all systems, some systems have very widely varying Eigen values or in other words, the patterns which are seen in the final response. Remember the response is superposition of patterns. The patterns which are seen in the response, the time response is made out of patterns, you will find that there is a wide variation in the speed of response of the various patterns. For example, you may find that there is a one component of the response which varies very fast and one component which varies very very slowly. So, these modes so to speak are you know you can say they are fast and slow modes and in case you have got fast and slow modes, you can actually make approximations in the model which you are using. One of the important things was that if you can identify the states which are in some way associated with the fast modes and the states which are associated with the slow modes could actually make modeling simplifications and in the basic point is that you could reduce the order of the system, you could reduce the number of differential equations in the system. This will be very important because in power systems you will find that there are transients which you know kind of are very fast like lightning transients and network transients and some very slow transients associated with the mechanical systems like the governor, boiler and so on which are very slow. So, when you are studying a particular system, you need not model every you know you may make modeling simplifications depending on the thing you are interested in. In case you are for example, interested in fast transients, you will pay more attention for example, in modeling a transmission line. You may even think of modeling a transmission line by its partial differential equation model whereas, if you are understanding slow transients or perhaps like for example, loss of synchronism is a relatively slow phenomena compared to lightning transients and oscillations, electromechanical oscillations which are seen in the grid. In that case, you may wish to even represent the network by lumped you know by lumped dynamical system and sometimes even neglect the dynamical equations themselves of the network. You may treat the network to be in quasi sinusoidal steady state. Of course, in some jumping steps, you have to recall this when we come to modeling of transmission lines and other components. The basic point is that depending on what phenomena is of interest whether it is fast or slow, you can make modeling simplifications. To make things a bit more precise, if you have got a system x dot is equal to A x or you have got a system x dot is equal to G of x, where x actually is a set of vectors. I mean this is actually the shorthand for large number of equations. So, you have got in fact, x 1 dot is equal to G 1 x 1 x 2 x n x 2 dot is equal to G 2 x 1 x 2 x n and so on or in this particular case x 1 dot is equal to this is a linear system A 1 1 x 1 plus A 1 2 x 2 and so on and x 2 dot is equal to A 2 1 x 1 plus A 2 2 x 2 and so on. So, you can have a system this is a linear system this is a non-linear system. If you can identify if it is possible to identify states say you know there are certain states say x 3 onwards that is x 3 x 4 x n which you can identify as associated with the fast transients. In that case you would be at least in this non-linear both in the non-linear and linear systems case you can make this particular approximation this is of course, unchanged this x 3 dot we just set equal to 0 and replace it by an algebraic equation G x 1 is equal to 0. This is a differential algebraic model of the system and if you take this particular model of the system you if of course, one important point is you know a priori that these states x 3 onwards are in some way associated with the fast transients what do you mean by association is of course, something we define in the last part of the last lecture I will just recall what it was in some time. So, what you can do is if you are interested only in the slow transients then you can replace the differential equations corresponding to the fast state variables or the state variables corresponding to the fast modes or the fast transients of the system and you can replace them by algebraic equations. So, you convert a set of differential equations into a set of differential algebraic equations of course, in the non-linear case it may not be easy to you know you could use these algebraic equations to eliminate or rather write x 3 to x n in terms of x 1 and x 2 this is possible in linear systems quite easily in non-linear systems it may not be possible to eliminate in that sense. So, in linear systems we can get rid of the fast variables in some what I mean by get rid is just write them in terms of the slow variables and the resulting differential equation then can be written simply as in case of linear systems you can have differential equations only in x 1 dot and x 2 dot that is what we did in the example in the previous class. In fact, they were you know when we analyze the slow and fast transients. So, if you are interested in the slow response you can kind of be blind to the fast response. So, this is what we discussed in the previous class conversely if you are interested in the slow response or rather you are interested in the fast transients for a very short duration of time. So, you are interested only in viewing the fast transients for a short duration of time you can assume the states associated with the slow transients are just frozen at their predisturbance values. So, these are the kind of modeling simplifications we can make it turns out that these kind of systems it of course, this presumes that the system can be broken up into fast and slow subsystems this may not always be possible, but in case you can do it these modeling simplifications are possible we shall see that once you make the modeling simplifications you may be able to analyze systems in a better fashion. For example, when we understand numerical integration by removing the fast you know the transient the state variables corresponding to fast transients by eliminating them or writing them in terms of the slow variables you can actually start using some simpler numerical integration methods. So, of course, this is something we will discuss later when we I introduce you to numerical integration methods somewhat later in this particular lecture. Of course, the key point which I mentioned last time was to associate certain states with certain patterns fast transients you know how do you do that in linear systems I mentioned that if you take the right eigenvectors and the left eigenvectors corresponding to a system for example, in the previous class we studied this particular system and what we did was to find out the association of fast transients and slow transients we evaluated the right and left eigenvectors corresponding to various modes we found out the right eigenvector matrix P inverse matrix is also has the left eigenvectors as its rows this is called Q. Now, if I take this P and P inverse and I do this operation please refer to the previous lecture if I do this operation then the matrix which results gives us the participation of a certain state in a certain eigenvalue or certain mode in this case. So, this is basically one way suppose the participation of a particular state in a particular eigenvalue lambda 3 is 1 then we say this state is completely associated with this eigenvalue lambda 3 of course, I L 1 and V C sorry I L 1 and V C both participate in the mode corresponding to this eigenvalue. So, this is what we did last time. So, this was of course, what I mentioned sometime ago you could you can use get an association of states and modes using the participation factor matrix note that in a coupled system sometimes the participation is distributed amongst all states. So, sometimes it may not be possible to really make this you know partitioning of systems into fast and slow transients you may find that every state in some way or the other in some significant sense participates in all the modes. So, that may of course, preclude using this fast and slow transients to make modeling simplifications. Of course, the aim of this course is to study power system dynamics and control and although the initial part of this particular course I have been concentrating on general analysis of systems. We shall soon in may be a couple or three lectures go into domain specific modeling issues, but it is to be noted that you should have some background in the analysis of dynamical system. So, I recommend that you can read as far as linear time invariant systems are concerned the books by Ogata or some equivalent book you will find many and of course, I would also recommend that you read some books on basic eigenvalues and eigenvectors. For example, matrices for scientists and engineers which I have recommended here, but of course, there are many other equivalent books which are just as good. There are other many, many other topics relating to linear systems and as we go along we shall also talk about transfer function representation and so on, but that will be when we understand excitation and prime mover controllers. So, for the time being now let us move on to the next part of our course that is numerical integration. Before I really start the topic of numerical integration, let us look at the course overview again so that we do not really get lost in what we are doing. Remember that the first part of this course was a basic introduction to analysis of dynamical systems where we have understood the Eigen analysis of linear time invariant systems. Now, we shall go on to study numerical integration techniques. What is still to come is modeling of synchronous machines, modeling of excitation, prime mover systems, modeling of transmission lines, loads in other components and of course, what is the main important thing in this course is to understand the stability of interconnected power systems. So, we shall use the models which we have developed and analysis tools which we have developed in part one of the course and go ahead and understand the stability of interconnected power systems, power system stability analysis tools and methods to improve power system stability. So, this is basically what we shall do in this particular course. Now, let us move on to the other topic of numerical integration. Now, if you are studying a system x dot, continuous time system x dot is equal to f x of t, you may one of the ways to analyze the system is to numerically integrate the system given the initial conditions. Now, the simplest way to do that of course, is what is known as Euler's method or equivalently forward Euler's method. So, I shall use of course, Euler's method and forward Euler's method interchangeably they mean the same thing x dot can be approximated as x k k minus 1 minus x k upon h in this fashion. This is known as Euler's method of numerical integration. Now, what do we really mean by numerical integration? What I mean is suppose I know suppose this is time t is equal to 0, I know the initial value of x that is x of 0 in some sense, I know the initial value of x I evaluate. So, k is equal to 0 or you can take the index is 1 does not matter k is equal to 0 you evaluate this function at this point using this particular equation you can since you know what the function is you can evaluate it at this point k is equal to 0 and get what k plus 1 is that is k plus 1 is of course, 1 in this case. So, from x 0 this is x of 0 you can get x of 1. So, you can say evaluate this. So, x k plus 1 is equal to x k plus h times x k t k of course, what is this h? This is the value of the variable or the h is essentially the time duration after which x k plus 1 is evaluated. So, x 1 means actually x of 1 implies x of h x of 2 x 2 is nothing, but x of 2 h. So, I am evaluating this x at discrete points. So, this is x of 2 this is x 1 this is x 0. So, of course, in between I do not know the values these values are not in fact, given by the numerical integration method I have just interpolated in between. So, what I am going to get is a discrete set of points now by interpolating between these points if I can roughly get the actual continuous time response of this I would say my objective has been fulfilled and I have got the response of the system. Remember of course, that the basic numerical algorithm will only yield the values of the variable at discrete points of time. Now, one important point you should note is that this is an approximation Euler's method is an approximation you will not of course, get the exact value of x by doing numerical integration in all cases there only some systems which will give you correct answers if you numerically integrating using Euler's method. So, what I will do is let us take a system this is let us take a linear system the reason why I am taking a linear system is I know the response of the linear system in terms of well known functions there is no need except in very complicated linear system with lot of switching and lot of you know intermediate disturbances there is no need to numerically integrate a linear system because you can actually write down its time response. So, but the reason why I am studying numerical integration using a linear system is that I know the response of the linear system. So, I will be able to tell how the numerical integration method behaves you know as compared to the correct response because I know the correct response. So, if you take for example, a system x dot is equal to A x. So, this x dot is equal to A x this dot is coming by mistake. So, if I try to what is the solution of this x of t is equal to e raise to a t x of 0 and of course, if I the correct response if I sample it at discrete points you will get x of k is equal to e raise to a k h into x of 0. So, this is the correct correct sampled response, but if I use a numerical integration method like Euler method this is a linear system. So, what I will get is this is what I will get. So, x of k in this case will be 1 plus a h raise to k x of 0 that is x of 0 is this. So, the correct response is and this numerically integrated response is this that is if I use forward or forward Euler method or Euler method. So, these two of course, are not the same the point is this is a good approximation of this the answer of course, depends on the value of h if h is very small then one may expect that you are going to come close to the solution your accuracy will be if h is extremely small now we will just take a simple example to understand this point, but before we do that let us look at a qualitative issue when is this correct solution stable when is the system stable the system is stable if a is less than 0 if a is a real number it should be less than 0 then the system is stable. However, here this particular system when is it stable you note that if the modulus of 1 plus a h 1 plus a h of course, is a constant if you choose your h if it is a fixed h in that case this particular 1 plus a h is always a constant in your numerical integration raise to k. So, the point is if mod of 1 plus a h is less than 1 then Euler method will say that the system is stable. Now, what is the important thing is that the actual system is stable when a is less than 0, but Euler method says that it is stable only if this is true these two conditions in fact are not synonymous they are slightly different. In fact, if h greater than 2 by a then even if a is negative this may be greater than 1. So, the stability of Euler method will depend on its time step it does not not only depend on a, but it also depends on the time step. So, it is obvious that Euler method can give a qualitatively wrong picture of the stability of the system if you do not choose your h appropriately. So, what we shall do is numerically integrate a system x dot is equal to a x using Euler method. So, for this purpose I will use psi lab as I was doing in the previous lectures. Now, this is the basic window of psi lab in which I will run a program now instead of typing out all the commands one by one I have done it already in a separate file which I shall show you now. So, we shall study numerical integration using Euler method what we will do is this is the variable x we start with a clean slate we clear this variable. Let us say that a is equal to 5 a is equal to minus 5 is it a stable system or an unstable system it is a stable system because x dot is equal to a x if a is a negative is a negative real number you will find that the system is actually stable. Now, let us try to integrate the system x dot is equal to a x using Euler method for that I should define what my time step is let us say right now I choose the time step h is equal to 0.1. Let us simulate the system for say I am sorry let us simulate the system for 10 seconds actually need not simulated for 10 seconds let us do one thing let us simulate it for 5 seconds. So, the number of steps to simulate from 0 to 5 would be given by this formula. So, you have got t final is 5 h is 0.1. So, the number of steps is you will get by rounding of t final by h. So, that is of course, an integer. So, that is why we have to round it off suppose the initial condition remember we are solving the dynamics of a system for a initial condition which is not the equilibrium for x dot is equal to a x the equilibrium is x is equal to 0. So, we are giving an initial condition x 1 is equal to 1 incidentally I am not written x 0 here because Sylab does not rather Sylab does not normally permit you to give index which is 0. So, that is why it is x 1. So, you have shifted all the indices by 1 as I mentioned sometime back we shall take a is equal to minus 5. So, what is the response of the system it is e raise to minus 5 t into x of 0. So, you know we expect that about in a second or so the system should settle down it is a stable system the time constant of the system is 1 upon 5 that is 0.2 seconds 4 times this time constant is roughly the time it will take to settle down to its steady value which is 0 that is the equilibrium value this is a stable system. So, if I want to numerically integrate I will use this particular command for i is equal to 2 to the number of steps i is the index instead of using the variable k I am using the variable i here where x of i is equal to 1 plus h into x of i minus 1 and then I plot time these are the time steps individual time steps. So, I plot the time steps and the corresponding x values. So, of course, if I execute this this is what I get as a response. So, if you look at this response I will slightly expand it you see the response is fairly well captured this is remember the time step I have chosen is x is h is equal to 0.1 point yeah was it 0.1 yeah it is 0.1. So, for this particular system h is equal to 0.1 seems to give you this response the system settles down in around 1 second. Of course, this seems to be a continuous graph that is because that is because Sylab is doing the interpolation between this individual points. So, that is not a very big surprise suppose I change the time step from 0.1 to 0.2. So, first what we will do is redo this with 0.1 and then redo this with 0.2. We see that our response is slightly changed you see here it is gone slightly different there is a different response here. On the other hand if I change it to 0.5 things start looking very very different the response is completely different this is the response and of course, it is not a correct response it is in fact going unstable the discrete time system is going unstable. So, what we see is that if you have got Euler method and I choose a time step which is not compatible. In fact, if you look at my condition which I had written down you can focus on the what I am writing 1 plus a h less than 1 implies system is stable as per Euler's method the discretized system is stable. So, in this particular case 1 minus 5 h less than 1 implies the discrete time system is stable remember the original system x dot is equal to minus 5 x is stable the discrete time system which is obtained by using Euler method gives you this condition is not the same as this condition. In fact, if I choose h which is greater than 2 times a in fact I should I should write as modulus of 2 times of a then I should basically get this condition will not be satisfied and will get a totally a wrong response. So, actually this happens when h is greater than 2 by minus 5 this is nothing but 0.4. So, if h is greater than 0.4 1 minus 5 h is greater than 1 and Euler method gives you a completely wrong information about stability. So, obviously whenever you are using a numerical integration method we need to be careful about the stability of the numerical method whether it gives a good qualitative understanding of how the system behaves. So, we shall now consider other methods obviously Euler method under certain circumstances does not give you a good response. In fact, we shall see later that when dealing with stiff systems Euler method is particularly unsuitable especially if one is interested only in the slow response. So, this is something which will of course understand in a few moments from now or perhaps in the next lecture. Now, this takes us to other methods of numerical integration whenever we talk of numerical integration there several terms which we will come across one is what is known as the order of the method. So, one of the things which you will come across is the order of the method second whether it is what is known as a explicit or an implicit method. Third thing which you will come across is whether it is a single step or a multi-step method. Now, there are two issues which we or two errors which result whenever we use a numerical integration method one of the error is because of the fact because of the fact that the numerical integration is in fact the discretization is almost always an approximation like Euler method is an approximation of the original continuous time system. So, that error which is introduced because of the approximation is called truncation error. The second thing which causes an error is because you will be of course doing this numerical integration on a computer and a computer will introduce another error which is called as round off error because any number can be represented only to a finite precision in a computer. But of course round off errors in normally you will not have a problem of round off error because these days you can specify a very high precision in computers. So, you have to normally bother about the approximation which is made that is are you going to use the Euler method or some other method. So, let me first introduce you to some other methods then we will try to understand what is order what is explicit or implicit method and what is a single step or a multi-step method. First thing suppose you have got a system x dot is equal to A x, x k plus 1 is equal to or rather x minus x k upon h is equal to A x k this is Euler or forward Euler. Backward Euler on the other hand is x k plus 1 minus x k upon h is equal to A x k plus 1. So, this is an approximate so you are using the value of the function A x at the point x k plus 1. Now, this is you may see you are using x k and you are using x k plus 1. What is the significance of this? In fact, if you have got a non-linear system backward Euler would look like this is the function g which you have discussed sometime earlier. So, for a non-linear system backward Euler will look like this. So, backward Euler requires x k plus 1 to evaluate g x k plus 1 and then you have to solve this. Now, what do you notice what the problem with this is as compared to this? See in this case if even if it was a non-linear system you would have got x k plus 1 minus x k by h is equal to g of x k. So, typically you will know what x k is. So, you can evaluate g of x k you know what x k is here too then you can get x k plus 1. In contrast look at this problem here you have got x k plus 1 you know what x k is, but you do not know what x k plus 1 is x k plus 1 is to be obtained from this. In fact, what you get is an algebraic you know x k. So, you have got an algebraic equation in x k plus 1 you would need to solve this implicit equation in order to get x k plus 1. In a linear system this would imply that of course, you will have to do a division, but here you will find that you will have to in fact, use a numerical method if g is a non-linear function you will have to use a numerical method for solving algebraic equations in order to get the solution of this. So, backward oilers involves a bit of complexity forward oiler or oiler method is easy to evaluate this x k plus 1 is obtained explicitly from x k x k plus 1 here is obtained implicitly from x k. So, in fact, without going to the any formal definition I hope you are getting what we mean by explicit and implicit method. If a method requires you to know x k plus 1 in order to compute this function g in that case we would call this method an implicit method. In fact, would you call this an explicit method or an implicit method suppose you have got a system x dot is equal to g of x this is what we are considering is this explicit or is this implicit this is another approximation this is another way of doing a numerical integration it is called trapezoidal rule. Trapezoidal rule is actually an implicit method because to get x k plus 1 from x k will require to solve this non-linear algebraic equation in case it is a non-linear system and that itself is an implicit algebraic equations then implicit algebraic equation which will probably require numerical method. So, in fact, in every time step if it is a non-linear system you may have to iterate in order to get x k plus 1 remember. However, then the numerical method in order to get x k plus 1 from x k is a method for solving numerical algebraic or rather non-linear algebraic equations like Newton-Raphson or Gauss-Seidel method. So, trapezoidal rule backward Euler are known as implicit methods. Now, why should you use trapezoidal rule or backward Euler if it is slightly more difficult to solve as compared to Euler method. One of the problems is the properties of Euler method. Euler method unfortunately for stiff systems is not a very good idea we will do an example in the next class a few examples in the next class to show this. Whereas, backward Euler and trapezoidal rule in fact are quite suited for numerical integration of stiff systems. Now, whenever I say suitable what do I mean a numerical integration should be reasonably fast it should not require you to use a very small time step otherwise to simulate or numerically integrate over interval which is large it will take you a very long time. So, numerical method to the extent possible should be able to use time steps which are not too small compared to the interval which one wants to simulate. Now, in a stiff system as you may imagine since you have got fast and slow transients there will be an issue about choosing your time steps. This is of course, something we will discuss later in the maybe in the next lecture or so. So, the second thing the thing which we discussed of course, was implicit or explicit methods. Now, let us talk about order of a method whenever we talk about an order of a method what we really mean is suppose the response of x of t suppose the actual correct response of x of t can be written down as a polynomial. For example, if x of t is can be it suppose is alpha 0 plus alpha 1 of t suppose this is a time response. In fact, all the time responses we have been talking of involve the exponential function, but suppose this is the time response then x dot of t is nothing but alpha 1 of x sorry alpha 1 that is all. So, x of x dot of t is nothing but alpha 1. So, what this means is of course, this particular system if I numerically integrate with back with Euler method or backward Euler method I will get the correct solution without any truncation error. So, backward Euler and forward Euler will give you the exact solution no a truncation errors in case and numerically integrate a system of this kind specifically of this kind. So, in fact, backward Euler and forward Euler or simply Euler method they are what are known as first order method methods. Of course, if you have got a system x dot is equal to a of x you know not alpha 1 alpha 1 here is a constant. If x dot is equal to a into x in that case your responses remember e raise to a t x of 0 this is the x of t. In such a situation of course, you know the expansion of e raise to a t is an infinite series in t. So, obviously the responses going to have many more terms. So, whenever you use Euler or backward Euler to simulate this system you are bound to get some truncation error. So, remember first order method means if your responses can be written down simply as a first order polynomial of rather the first order polynomial in t in that case backward Euler and forward Euler which are first order methods will give you correct response. You can show that if your system response can be written as then method like trapezoidal rule will give you the correct no truncation error. So, this kind of system is known as a second order system. So, trapezoidal rule will not give any truncation error for a system which has got a response of this kind. Again trapezoidal rule will always give some truncation error for a system which has got this response because this is an infinite series in t. So, it is not just a second order polynomial in t. So, the trapezoidal rule is this is something you can show I will not you know prove it here you can just try to do it at home trapezoidal rule is in fact a second order method. There are other methods for example, rangekutta method. So, if you have got x dot is equal to f of x f of x t sometimes t may appear explicitly in this function then rangekutta method uses first it evaluates the function k 1 which is nothing but then it evaluates k 2 which is nothing but then it evaluates the function k 3 rather it evaluates k 3 where k 3 is nothing but and then k 4 nothing but h k 3. So, it evaluates this k 1 k 2 k 3 k 4 this is rangekutta fourth order method fourth order. So, rangekutta fourth order method eventually calculates x k as or rather x k plus 1 as x k plus h by 6 into k 1 plus 2 k 2 plus 2 k 3 plus k 4. So, this is basically what rangekutta method does it is basically evaluating k 1 k 2 k 3 k 4 which are in fact intermediate points intermediate values of this function between the interval t k and t k plus 1 and then using this particular formula. So, this is rangekutta fourth order method first question is it an explicit or an implicit method the answer it is an explicit method because it requires the evaluation of x k plus 1 but it does not require x k plus 1 itself. So, it you can evaluate all these functions explicitly you do not require x k plus 1 in these evaluation. So, this is an explicit method and something which of course, I will not prove here but this is what is a fourth order method. So, if your response can be written down as a fourth roughly as a fourth order polynomial in t then you will have no truncation error. So, this is in fact rangekutta method. So, we have of course, discuss now order of the method whether it is an explicit or an implicit method we have really discuss this without going into any great amount of rigor as in the I can refer you to a few books at the end of the lecture you can go through them for more detailed analysis of this. There is another point which we missed out that is multi step and single step methods. Now, I will not write down the various multi step and single step methods but all the three methods of four methods which I have told you Euler, Trapezoidal, rangekutta and backward Euler all require simply x k to get x k plus 1 you do not require anything more. You could in principle use for example, x k minus 1 also in your calculation in to get k plus 1. So, when you are trying to get k plus 1 your discretization uses values which are two time steps before or three or you know even a time step ahead in time in that case of course, it will like t k plus 2 or t k plus 3 in that case it becomes an implicit method. So, such methods which use not only the previous time step but time steps other than the previous time step values of x at time steps other than the previous time step they in fact are known as multi step methods. So, we are going to order of the method whether it is explicit or implicit and whether it is single step or multi step method. In most of what we are going to do in this course we shall restrict ourselves to single step methods. Explicit methods of course, are easy to evaluate because they do not require to solve an algebraic equation especially this is a problem when you are talking of non-linear systems because the algebraic equations is non-linear and you will have to use Gauss Seidel or Newton-Raphson at every time step to evaluate iteratively what x k plus 1 is given x k. So, that creates a bit of a you know kind of a problem when one tries to numerically integrate a non-linear system using an implicit method. So, you should have really some nice tangible benefits when you are trying to use an implicit method. These in fact, implicit methods are known to be slightly better when like backward Euler or trapezoidal rule are usually better suited when you have got stiff systems and you want to study the slow response. So, this is something of course, we shall do in the next lecture. Before we stop this lecture, let me just give you a few references. For example, for numerical integration methods there are lots of books, but some of the classic books are one by gear ordinary differential equations by Prentice Hall. I am sure it will be there in a well stocked library Englewood-Cliff's 1971 and there is another one that is L Lapidus and J H Seinfeld numerical solution of ODE's. ODE of course, is ordinary differential equations academic press New York 1971. These are the what I would say is the classical books in this field numerical integration method. In some cases, they may be fairly rigorous, more rigorous than we will ever be in this particular course. Therefore, you can also look at other books which are somewhat simpler which may be available. I am sure there are many other books which are available in your library, but these are the classical books in numerical integration methods. In the next lecture, we shall understand the numerical integration of stiff systems in more detail and also understand some of the stability properties or the you know properties, general properties of the numerical methods in particular Euler method, backward Euler method and trapezoidal rule.