 Hello. So, it is very nice to interact with all of you. So, yeah, if you have any questions about the course or the exam and various things, you can definitely ask. So, I will try to try my best to clear this questions to the best possible. Thank you. Hello, sir. This is Satish sir. Satish from one theory college. Sir, at assignment number 6, problem number 2, you give the initial guess of that one is 1 comma 1, sir, and the question. But at the solutions, you give pi 0 is equal to 0 comma 0. Okay. Oh, in the solutions, is it? Yeah, yes sir. Okay. Yeah, so that, okay. So, basically the initial guess has to be given as 1 comma 1. In this particular problem, using LSU non-LIN, whether you give the initial guess as 0 comma 0 or 1 comma 1, you will get the same solution. So, therefore, for the final solution, it would not matter whether we gave 1 comma 1 or 0 comma 0 for this particular problem itself. In some other problems, it would matter and that is why we specify what initial guesses have to be given. But in this particular problem, the final solution does not depend on the initial guess. Okay, did that answer your question? Did that answer your question? Yeah, yeah. Thank you. Thank you. Okay. So, exams, so I think all of you might have seen the overall exam pattern. So, those are the type of questions that will be there. So, the first, yes. Okay. So, the first one type of question is going to be multiple choice question. And multiple choice question, there's only one correct answer. Then there is match the columns. Match the columns is the application on the left column has to be matched with, let's say, one of the applications is given over there and the correct answer over there, let's say, is the command sum. So, you just have to pick whatever box that sum corresponds to. So, that's the match the column questions. Then we have a short answer question where the final answer would be numerical answers. So, let's say you have a vector one, two, three, four, five, and you want to find some one, two, three, four, five. The value of the answer to that is going to be 15. So, you just have to enter the answer 15 in the numerical columns and that's all you need to do. So, that's the short answer type question. The next question, the next set of questions would be on a short coding type questions. So, the short coding type question would be codes that you need to write. That would be one line codes. You don't, these are not lengthy codes. These are very short one line codes. Let's say you want to find factorial of a number. And factorial of a number can be found by using the Rode command, Rode command, Rode one, full in N. So, if you write that, that is what is expected. So, that's the short MATLAB code type question. So, those are single line codes. Then there are questions where I have a screenshot of MATLAB code. The screenshot will have up to 10 lines of code and there would be an error in one of the lines. And you need to write out which line that error is in or identify what's the type of error. And finally, there will be one coding question and that's to write a MATLAB code. Now, students who are writing exams do not have access to MATLAB. So, therefore, we are not looking at whether the code will run in MATLAB accurately or not. We are looking at one, the logic that you have used and second, whether you have a reasonable understanding of MATLAB. So, for example, if the command is prod or come prod, you should have a reasonable understanding of that command and that's sufficient. If there are minor syntax errors, you don't have to worry about it because this will be manually graded and we won't run it in MATLAB because you guys don't have access to MATLAB. So, these are the various type of questions that will be there in the final exam. And those have been indicated in the model question paper. So, yeah, so that's what I had to discuss from my side about the question paper for the final exam. Sir, I am not able to find the solution of Newton-Raphson method. So, can you explain about the solution? Sir, I have a question number 4 and question number 1. Sorry? Assignment 4, question number 1. Question number 1. Okay. So, have you looked at the solution that were posted? Yeah, sir, but not understanding it. Not able to solve using MATLAB. I know this by mathematical. Wait. So, assignment 4 question 1 is Gauss-Seidel. It's not Newton-Raphson. So, you have a question on Newton-Raphson or Gauss-Seidel? No, sir. Newton-Raphson, sir. Sorry, assignment 5, sir. Assignment 5. Okay. Just give me a few seconds I will bring up. So, the Newton-Raphson, I'll just write down on the sheet over here. I hope you can see that. Is the next new guesses, new guesses are given by xi plus 1 equal to xi minus f at xi divided by, okay. So, can you see this equation? Yes, sir. Yeah. So, this is the equation that you use in order to get new guesses of x using Newton-Raphson method, right? So, you will start with certain initial guess. So, let's say our initial guess was x0. Let me just look at this problem again. x0 equal to 20, let's say. So, that's what we started off with, okay. And then we need to go from 4i equal to equal to 1. So, let's say 50. Let's say we want to do 50 iterations. This is how we are going to generate the new guess. Okay. Now, keep in mind MATLAB arrays start from 1. MATLAB arrays do not start from 0. So, if we write x0 equal to 20, we are going to get an error. So, we need to write x1, sorry. Yeah, we need to write x1 equal to 20 as the first guy. And then in this particular loop, we are going to calculate using Newton-Raphson's, this particular guy, right? And we need to check error is absolute value of the difference between xi plus 1 minus xi. This is the definition of error. This is how we have defined the error. And we need to say that if the absolute value of error is less than certain tolerance 1e minus 4, then we will stop. We will give the command break n and then we will also end the loop. So, this is what Newton-Raphson's method is going to look like. So, now the question is what is this f and what is this f dash? This f is the function 40 multiplied by x to the power of 1.5 etc. that is given in the problem. And f dash is also given in the problem. So, instead of writing f of xi and f dash of xi, you will have to write the actual expressions down over here. So, once you do that, you will get the solution. So, the core part of the code is really this particular lines which have been put inside the for loop. If you do not want to use for, you can use while instead. And as we have shown in the first module, if you use while, then this particular command that is there will go inside the while loop. In this example, you did not have to do, you did not have to do this error checking. You only had to run for the loop for four iterations. That was the problem statement. So, this is how you will run that in the assignment example. So, these statements you will not have, you will just have this. So, that is the Newton-Raphson code that you will have. Okay? So, does that answer your question? Hello, I cannot hear you. Can you speak louder or closer to the mic please? Assignment 2? Assignment 2, question number 3, Maclaurin 3D. Okay. Yeah. So, you want me to show how the solution works? So, for assignment 2 problem 4, you want to see how the solution works, is it? Okay. So, let me look at the problem. Okay. So, okay. So, in this problem, what you need to do is to write the Maclaurin series for e to the power x. Now, the Maclaurin series for e to the power x is, e to the power x is 1 plus x plus x square by 2 factorial plus x cube by 3 factorial and so on. Okay. So, the first step is, again, in this particular example of solving this. So, the first step that we did was to get a vector VEC. Okay. Now, what does VEC involve? So, VEC should, the vector VEC should be 1. Yeah. Yeah. Okay. Yeah. So, now the VEC is basically going to be 1 x x square by 2 x cube by 3 factorial x to the power 4 by 4 factorial and so on, which is nothing but 1 x x square x cube x to the power 4 dot slash 1, 1 into 2, 1 into 2, 2 into 3, 1 into, sorry, 1, 1 into 2, 1 into 2 into 3, 1, 1 into 2 into 3 into 4 and so on. Okay. So, this is how VEC is going to be generated. So, this is how you can generate your VEC and store them in the vector VEC. So, how do we do that? We do that using the command VEC equal to x dot carrot 1 to n. Okay. x dot carrot 1 to n will generate this guy over here. Okay. And to generate this, you will have to use comprod 1 to n. Okay. And now we need to do element by element division. So, we have dot slash that will do element by element division of VEC. Okay. So, now this VEC, if you see, contains everything from this point onwards. It does not contain the first element. So, we need to add that first element in that vector by saying VEC equal to 1 comma VEC. Okay. So, this will generate our vector VEC. Now, the first assumption, the first guy is let's say F2 is or sorry F1 is going to be 1 plus x. F2 is 1 plus x plus x square by 2 factorial. F3 is one more term added so on and so forth. So, what does that what does that mean? The subsequent values of F are nothing but cum sum of VEC. So, this will give us the values of e to the power x approximation to the first term, second term, third term and so on and so forth. Okay. And now you can find a difference between subsequent terms. Find out what difference is less than let me see the tolerance less than 5 e minus 3 and that will give you the solution. So, this was how the solution of that particular problem was given in the solution manual. Okay. This is not the only way for you to be able to solve it. What we can do is we can solve it using a for or a while loop also. So, if you are doing this using a for loop or a while loop you can let's say we want to do for loop. So, for i equal to 1, 1, 2 let's say 15. Okay. Again, 15 is an arbitrary number that I am taking over here. We don't know whether it will converge in 15 or not. Okay. Now, our value of F or exp val is going to be nothing but you know the VEC that we have formed earlier that's going to be our exp val. So, we will do some of VEC 1 to i plus 1. That's going to be our exp val. Okay. Our error is going to be abs of exp val minus exp old. Okay. And we need to give the value exp old as exp val in order to store it. Okay. And we will end this. Now, if we run it only in this manner we will get an MATLAB will have a problem when it comes across this particular command and that's because exp old is not defined. So, we need to pre define our exp old equal to 1. Why we are doing exp old as 1 because the first guy is just going to be 1. Second guy is 1 plus x. Third guy is 1 plus x plus x square. So, we want to get the first guy. So, for the first guy exp old is 1 and then in the loop we will do all these three steps. Okay. So, this is the way that you can do this particular solution. This was the way that one of the students in the forums had asked how to do this this particular way of solution. This was something that I had shown you in one of the video lectures if you wanted to calculate e to the power x using first order term, second order term, third order term up to nth order terms. Okay. So, these are two possible ways to do this particular problem. So, whichever way you find comfortable you can use use this way. Sir Nassaman number 8. So, in the third problem, bring the damper swing equation. Okay. Yes, tell me. In the question number 3, pneumatic damper non-linear equation. Yes. There is a square x by dt square plus c into modulus of dx by dt and dx by dt. Correct. So, how can we solve that one? I mean I have to multiply the twice times or how can you solve that one? Yeah. So, actually this question good question because this was also asked in one of the forum question which we have replied to. I'll yeah. I'll just go to that. Okay. So, the first thing is as we had shown in the first lecture, we are going to write dx by dt as v. Okay. And then our dv by dt is going to be the right-hand side of the given equation. Okay. So, this is the standard procedure we had used in the lecture. Okay. Now, what does the right-hand side actually include? The right-hand side of that equation is going to include c dx by dt into dx by dt plus kx squared negative of this divided by m. This is going to be the right-hand side. Okay. Now, we don't have dx by dt, but instead we have replaced dx by dt with v. So, what we do will replace wherever dx by dt comes, we are going to replace it with v. Okay. So, this is going to be your equation. How to get this mod? You get this mod using ADS command in MATLAB. And once you do that, the problem is remaining part of the problem is same as how we have solved in the lectures. Okay. Thank you, sir. Sir, in the solution of ordinary differential equations module, we discussed RK2 method, RK3, RK4 methods, and Rosan methods, PS methods. But we feel difficult to feel, memorize the formulas. How to memorize the formulas? In this way, sir. Okay. So, first of all, you don't need to memorize any formula. Okay. So, if you, there is no easy way to memorize the formula. And I don't expect, we are not going to expect you to memorize the formula. So, with respect to the ordinary differential equations, what you need to know is this. Let's say we have the RK2 method. If we have RK2 method, the best RK2 method is going to give us the order of accuracy from global truncation error as X to the power 2, local truncation error is X to the power 3. Those are the things which are not to be memorized, but those are the things that are to be understood. And we have tried to understand them by solving several examples. One of the examples was in assignment seven. The other thing we have tried to understand is by comparison using Euler's method, RK2, Hewn's method, and RK4 classical method. So, those are the things that I expect you to understand. I am not going to test students on their memorization ability. So, you don't have to memorize whole formulas. What you need to do is basically, if you have done all the assignments, all the students have done assignments on their own, they wouldn't need to memorize those formulas. So, for example, ODE45, the syntax for ODE45 is ODE45 at t, y followed by function named t, y. That part, if you have done ODE45 in assignment eight, you would automatically, the students would automatically know that particular formula. So, those are the things which one would expect you to kind of quote unquote remember, but not the wholesale formula of the form, Hewn's method is K1 equal to f of t, y, K2 equal to f of t plus h comma y of t plus h, y of K1 plus h. So, those are the things I do not expect you to memorize. Was that clear answer or do you have a follow up question on that? I call sir. Okay. Sir, in MATLAB input function ODE45, we use two elements as output element, input element, but in other MATLAB input function, these really one element. How about now sir, one element as output element, two element as output element? Sorry, last part I can you repeat. So, in ODE45, we have two output arguments. Two input argument. Two input argument. Yes, correct. But in other MATLAB input function, they use only one as element as input arguments. Correct. In ODE45, we have two output arguments. Okay. So, have you completed the question? So, I am not sure if I understood the question. So, let me try to answer based on what I have understood and if I have not understood it correctly, then correct me. Okay. You can see the notes, right? So, in non-linear equation solving, for example, we had to solve equation. We had to solve equations of the type f of x equal to 0. Okay. Not sure. So, this was the equation that we wanted to solve for non-linear equation solving. Okay. The equation that we wanted to solve for linear equations was of the form A x bar equal to B where A is known and B is known, right? When it came to, for example, integration, we had to solve integral f of x dx. Okay. Everywhere, you see that there was, as we had, as we write down the equations on using pen and paper, we have only one dependent or independent variable in which we write. As against that, when we write ODE is, we write the ODE of the form dy by dt equal to f of t comma y. An example of that is, let's say dy, the example is, let's say dy by dt is minus 2 ty. So, dy by dt, the right-hand side of the function is a function of both t and y. y itself can be a vector, but the meaning of t is time, meaning of y is the variables that we want to solve for. If you look at the other examples, non-linear equation solving, linear equation solving or integration, your x bar, various things had the similar meaning. Those were the things that we wanted to solve for. Here x bar had same meaning, x bar were the things that we wanted to solve for. And the integration, x bar, x had the same meaning, which was x was the independent variable, which varied independently of the function. Okay. Whereas here, t is the independent variable and y is the dependent variable. And that's why we have this as in the form f of t comma y. Okay. So, this is the reasoning behind why ODE solving has equation or has equation of this sort, of this sort, whereas all the others had equations of this type. Okay. That's one part of the answer and that's from understanding how the system behaves. The other part of the answer is this is how MATLAB has chosen to code ODE solver and fsol. So, therefore, we have to use the structure that MATLAB has used. So, we don't have a choice in that case. Okay. So, I've tried to give the answer in two forms. One is we are doing it this way because there is a reasoning behind this. But really, the complete answer to that is also because that is how MATLAB has its internal commands lift and ask. Okay. So, I'm not sure if this was indeed your question. So, please let me know if I misunderstood your question. So, was that your question or I misunderstood the question? Did I answer your question? Okay. Sir, your answer is an understandable one, sir. The answer is an understandable one, sir. Okay. Okay. Fine. Sir, is there any difference between the open source Sylab with respect to the MATLAB commands? Okay. So, if you are looking at open source software, Sylab and MATLAB are two different software. So, it's like, I mean, it's a wrong example, but let's say MATLAB is like C++ and Sylab is like C. So, they are different. Some 80% of the commands might be similar, but they are two different software. There is another open source software called Nu Octave 260hp Octave. That is much more similar to MATLAB because Nu Octave says this is if there is any command, any MATLAB file that runs in MATLAB, it has to run in Octave also. So, if you want to use open source software that is very close to MATLAB, use Octave, but Sylab is a similar software to MATLAB, not the same. Have you required to submit MATLAB code from given estimate? I just unmuted your microphone. Please go ahead. Yes, sir. Yes, final exam. Have you required to submit MATLAB code from given assignment? So, there will be one problem of writing a MATLAB code and that problem will be from one of the assignments or one of the video lectures or very similar to assignment or video lectures. So, you will have to write the code. However, MATLAB will not be available while you are doing your final exam. So, what I am looking for is not accurate syntax, but your understanding of MATLAB and your understanding of that particular numerical technique. So, that is what I would be looking for. So, I have just an hour back, I have posted a response to one of the questions on the forum, the same question, how do we write MATLAB code and so on and so forth and that will also give you more detail. So, if there is for example, some error in the syntax, that is not going to be a problem, but you are not expected to write an algorithm over there, but you are expected to write as close as possible to something that looks like a MATLAB code. What is the negative mark estimate for the final exam? The negative marks are only for the multiple choice questions. So, I think there are maybe five or six something like that multiple choice questions. So, for that, if you look at the model question paper, you will see that only one of these questions have negative markings. So, if you get the answer correct, you will get three points. If you get it wrong, you will get minus one point. None of the other questions. So, for example, MSQ, multiple select questions that are there and match the columns and all that, they do not have any negative marking. Negative marking is only for multiple choice question, where there is only one correct answer. Sir, I have a PPT of final exam pattern. In this exam pattern, I am watching a sixth section as for the error. I am not able to understand this section. So, let me do this because I think it will be better if I am able to So, I do not have that particular PDF shown on your screen. But okay. So, what this spot the error question is going to be is I will write a code, a small code, which would be say a five or ten line code in MATLAB. Okay. And one of the lines in the code will have an error. So, you need to find out which line in the code has an error or what is the error that MATLAB will flag. So, if you have your PDF open in front of you, it's a two line code that I have given in the PDF. Let me just try to do this over here. Okay. Can you see the notes journal? Okay. Okay. So, the code that I have given is x equal to one to ten. That's line number one. And y equal to x carrot two. Okay. Now x equal to one to ten means that it's a vector. We have seen in multiple examples that if you do vector power two, you are going to get an error in MATLAB. Okay. After this session is over, you can try this particular code out in MATLAB. You'll see you'll get an error. The reason why you will get an error is because the vector is a one by ten vector and one by ten vector cannot multiply with another one by ten vector. So, it cannot multiply with itself. Okay. What is the correct answer? The correct answer would be you will do a dot carrot. Dot carrot is element by element power. Okay. So, the error because this dot wasn't there, it was like this. So, therefore, the error is in line number two. Okay. So, one type of question would be which line has an error? In this case, your answer will be two. The other type of question will be what would be the correct way of writing this particular line? If that is the question, your answer will be y equal to x dot carrot two or your answer could be power x comma two, y equal to x comma two. Okay. Any of these would be correct or if you come up with a different way of writing this, that's also actually perfectly fine. So, there are a couple of type of questions that we have in spot the error to find out the line number in which there is an error or if there is an error, what is the correct way of writing that particular line of code? Okay. Also, keep in mind that if the code is completely correct, then you put the answer as zero because there is no error in any of the line numbers and if there is no error in any of the line numbers, do not leave that answer blank, put the value equal to zero over there. This problem does not have negative marking. So, please, you know, try to answer something over there so that there's some chance of getting, getting points. Okay. Sir, I have one more question. Sir, have you required to matlab software in for final exam? No, we will not be able to provide matlab software in final exam. So, you do not have a way of checking in matlab whether the code is syntactically correct or not. So, you kind of have to write what is in, in, in books and all that. This is called pseudocode. Pseudocode means something that is almost like a code, but there might be a syntax error. Is there any, any command for, for what do you want to say to calculate specific times? Yes, it calculates at specific, okay. Yeah, that's a good question because I did not cover this. Okay, can you see the journal screen? Sir. Yeah. Okay. So, we have been using Tsol comma Ysol equal to etcetera, etcetera, where we give Tspant let's say equal to 1 comma 10 or sorry, 0 comma 10. Okay, just to give an example. If you give this, then ODE 45 calculates the values between 0 and 10 at all the points that it determines internally. Okay, however, if instead let's say you wanted the values at time 1, 5, 10, 25 and 50, you can give your Tspant as 0 comma 1 comma 5 comma 10 comma 25 comma 50. In that case, what MATLAB is going to do is, this is your initial time. Okay, and MATLAB will give output only at these five times, not at internally computed points, but only at these points. Okay, so in this particular case, if you give this as a command, your Tsol will be the same as Tspant and your Ysol will have six rows, sorry, yeah, will have six rows, first row corresponding to 0, second row corresponding to 1 and sixth row corresponding to time 50. Sir, I have another door, sir. Yes, please. Sir, will you please suggest any Tspant for something MATLAB versus an example, which is indicated by the math right here? I am, have they specified anything? No, because I am not much aware about what requirements that there are, so that's why I am asking you. Have they specified something in their website? Do you know? Okay, if they have not specified an excellent book for MATLAB numerical methods is a book by Fawcett, that's the book that we have specified as a possible textbook for this course, if students are interested in learning more about MATLAB for numerical computation. However, MATLAB associate certification is not just for numerical programming, it is in general general purpose MATLAB usage, including plotting, including analysis and so on, which the Fawcett's book does not cover. So if you want more information, please post it on the forum and someone might be able to answer to that. I do not know the answer of the top of my head right now. Sir, how to calculate the inverse for a non-square matrix? Are the matrix values, matrix limit values 0? Okay, so okay, two-cat answer. The first part is inverse of a non-square matrix is not defined. What is defined is what is known as a pseudo inverse and so we won't we won't get into that because those are you typically advanced concepts we usually don't do them in an engineering course for BTECs. So there is something called pseudo inverse and you can look it up in MATLAB help or online, but inverse is defined for a square matrix which has full rank. So if it's an n cross n matrix rank of which is equal to n, only then an inverse is defined otherwise an inverse is not defined. Okay, however, there are other concepts which is known as pseudo inverse and Jordan decomposition and singular value decomposition something that we do not cover in a typical BTEC course. Good afternoon sir. Sir, if I have a array of 100 random numbers and I want every fourth element starting from the 20th element till the end sir, can I do this sir? Can you do what? Sir, I am repeating the question sir. Okay, yes, so the question I heard are you typing something on the chat? Sir, I am repeating the question sir. Sure. If I have an array of 100 random numbers and I want every fourth element starting from the 20th element till the end sir, how will I do this sir? You tell me how will you do this? What's the starting element? I want to do this without making use of loop sir. Right, what's the starting element? The starting element will be sir 24th element sir. 20. Right, what's the step? What's the step that you are taking? Every fourth element you want. So what's the step that you are taking? I will assign the value of, I will take a particular variable, then I will assign the location of the 20th fourth variable, then I will increase its consequence by 4. Right, and how do you do that using colon notation? Using colon notation, if how will you do that? Okay, can you see the journal screen? Okay, so when you write 20 colon 4 colon 100, what this gives is 20, 24, 28, 32, so on and so forth. Right, this is what the colon notation in MATLAB means. So if you want to access elements of any matrix or any array from the 20th element till the 100th element in steps of 4, you can give a 20 colon 4 colon 100. Okay, this is one way of doing things. MATLAB actually makes it a little bit easier if you want to do it right till the end. If you want to do it right till the end, you can also give this particular command. So you can write a in brackets in circular brackets 20 colon 4 colon end or 20 colon 4 colon 100. Both of these will work. Thank you, sir. Sir, for a square matrix, it is very zero. How to calculate the inverse for the matrix, sir? If what is zero? Sir, for the square matrix, determinant value is zero. How will you calculate the inverse? If the determinant of a square matrix is zero, then the matrix will not have an inverse. Okay, so inverse is defined only for a full-ranked square matrix. Full-ranked square matrices means that its determinant is non-zero. Okay, so inverse is not defined. There is something called pseudo-inverse that is defined, which you can use PINV instead of INV. But PINV is not an inverse of a matrix. So inverse is not defined. Yeah, so if there are no further questions, I think we'll end this call. So wish you all the best for the final exam and for the final eighth assignment. Again, for the final exam, what we expect really is that you should have a pretty decent understanding of the things that we have covered in this particular course. So for revision of this course, the various PDF files that we have uploaded, those will be sufficient. Again, not the examples, but what the learning that has been through those PDF files, those are the things that we are expecting. So for example, in differentiation, your central difference formula, your forward difference formula, the fact that forward difference is h to the power one accurate, central difference is h to the power two accurate. The fact that there is a minimum, there is an optimum value of h, step size h for which the errors are minimum. Those are the things that we have covered in differentiation. When it came to integration, we covered trapezoidal rule, since it's one third rule, and then the two MATLAB commands, trapeZ and quad. So those are the things that we expect you to understand and know. Then in linear equations, the direct methods that we covered were the Gauss elimination, we can use Gauss elimination for solving equations, followed by Gauss-Seidel method, which is iterative method, and the EDMA algorithm. So those are the things from linear equations. Again, the idea of linear equation to solve Ax equal to b, the rank condition, what the condition number means, condition number as the ratio of eigenvalues. So those will be the things from linear equation. With respect to non-linear equation, the Newton-Raphson and bisection rule that we use, and the fixed point iteration method, those were the numerical methods that we use, and the MATLAB algorithms that we use were f0 and fsol. So those are the things from module 5. In module 6, linear regression, non-linear regression, how to do linear regression using the matrix method, how to do non-linear regression using LSU non-LIN, those are the things that we covered in module 6. Module 7 and 8, ODE45 and ODE15S, what we mean by stiff systems, when to use ODE45 and ODE15S, and how to use ODE45 and ODE15S. Finally, the Euler's method, RK2 and RK4 methods, not the equations of RK2, RK4 methods, but the accuracy, and RK2 uses two terms, RK4 uses four terms, and so on and so forth. So those were the things that in a nutshell, we have covered in this particular module. What I'm also going to do is list of all the assignment problems that we have solved. So you'll get an idea of the fact that we have solved some really interesting problems from electrical engineering, chemical engineering, mechanical engineering, civil engineering. So those I will post after the assignment 8 is over, so that you can get an idea of the type of problems that you have solved in the assignments. So yeah, so I hope this has been a very useful learning experience for all of you guys. So and thank you very much for attending this.