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[9 marks] The following nonlinear system of equations has solution at 1 = (1,1)7 .Ii-4=0 Ti + rfI2Assutning that x(0} (0.4,0.5)T and *() (0.95,1.17)7 use thc follow...

Question

[9 marks] The following nonlinear system of equations has solution at 1 = (1,1)7 .Ii-4=0 Ti + rfI2Assutning that x(0} (0.4,0.5)T and *() (0.95,1.17)7 use thc following Broyden $ update fortula to compute x(?) (rouncexl to two decimal digits). (yl) _ 4,-16E # 9()T3() How accurate is the solution?

[9 marks] The following nonlinear system of equations has solution at 1 = (1,1)7 . Ii-4=0 Ti + rfI2 Assutning that x(0} (0.4,0.5)T and *() (0.95,1.17)7 use thc following Broyden $ update fortula to compute x(?) (rouncexl to two decimal digits). (yl) _ 4,-16E # 9()T3() How accurate is the solution?



Answers

You graphically approximate the solution of the equation $$\frac{x}{x-1}-\frac{99}{100}=0$$ to be $x=-99.1 .$ Substituting this value for $x$ produces $$\frac{-99.1}{-99.1-1}-\frac{99}{100}=0.00000999=9.99 \times 10^{-6} \approx 0$$ Is -99.1 a good approximation of the solution? Write a short paragraph explaining why or why not.

By substituting X equal toe minus 99.1. It gives the value. It skips minus 99.1. Upon my 99 point one minus, find this 99 by 100 it is equal toe nine, my 99 extra power. My message. This is an article on it can be more smaller than that. So minus 99.1 is not very good. But it could solution approximation that Brooks makes. We can big, even much smaller, while attempting We can much the smaller of a load in there. It's small letter over you and then that.

We want to use in America method to approximate the first tree y values to disperse all the differential equation with this initial conditions instead of point to now how I learned from you looking about Suppose I have the curve. So this is my wife in I have this point X values x n one values y n Now I'm making use off the tension to this point I'm making use of that to find my next point Which is this Rick Color one The X value a koi X n plus one and the y value. It's why I m plus one Now we all know that the region off the engine at this point here is equals toe y m prime So greedy and formula is why m plus one minus whiten over. Now this part here I can call it Delta X. So over Delta X, it's equals the wine prime rearranging you get your Oiler formula Wish y m plus one request the y n. That's why in prime he x now, using any CS, we can find the exact one function. So this puts all the different station as you can see here are killing in. And this is the white function here, so I'm just going to write it down here. The why function is why it was toe minus e to the power off, minus x squared over two, minus e plus one. I'm just going to stop in the initial value here. Where Access one. Why is zero? So when you somethin, you get this rearranging and taking on both sides, you realize that he is minus huh? In putting back into this part here, I will have. Why equals toe and taking common basa minus e. I will have one minus because of this one here. One minus is the power off one minus X squared over two. Okay, let's build the table. I have end accent. I have lighten. I have one and crying, which is X m comes one minus y in. Okay. And then I have the Oiler method to find my next point. Which is why in plus one, it's equals toe. Why any That's one in prime time. 0.2 and then I need to find my exact Is that why I m plus one is actually one minus eat about one minus X and last one square with two. And then I will have to find my error. So my error in this column Yes. So X'd role is in the initial condition, as you is one. Why zero is zero provides your time is excellent, which is one comes one minus the y zero is zero. My wife time zero is one. Now my wife one. He's why zero, which is zero plus y zero prime we just found is one times you're going to hear. So my why one is 0.2 now my is it will be one minus eat about one minus x one square. Now my ex one will be taking this zero plus the step of 00.2. So will be 1.2 square over to, and that will be 0.19748 So the error is taking the exam, minus the estimated. So that means taking from this column Minuses column. My error is gonna be minus 0.252 Okay, the Knicks X one. Excellent. Just this one. Plus the steps to is 1.2. Why one is already created is you want to So why? One time is exchanges 1.2 one minus the world. One is 0.2 and 0.96 So why to will be 11 which is you're goingto plus y one prime, which is your 10.96 comes 0.2 here. So I get your 0.392 exit will be one minus e to the power one minus x two square. Now x two will be taking 1.2 plus Your 0.2 will be 1.4 square over to and that really is your point three one 22 So the error would be minus your 0.0 one through eight. Okay, next one. The next role, the two now next door is 1.2003 point two billion. 24. My story will be over here that we found with 0 2019. So wake in fright is next door which is one point mhm and one minus like toe, which is yes, the white collar crime is your point. It like me magazine I love white tree will be white to which is your country 92 plus Why do prime 2.8512 Time 0.2 And so why? Tree is 0.562 and that will be one minus. Eat about one minus extremely will be taking 1.4 plus 0.2. It will be 1.6 square over two, and this is 0.54159 Difference is minus 0.206 You can see that the error increase as n increases.

We want to use the American method to approximate the first three y values to this first order differential equation. Given this initial condition and a step of point to now, let's get the rice. Are white. Crime is easier taking out. Do why I x plus one. How are limited. Came about. Suppose I have the graph. This is my wife. Yeah, I have a point here. Israelis extend Why values why? And now I'm actually just using the engine to find my next point. The danger x n So the greedy in off this engine is the same as the white prime at this point here. Now, my next point can let it be here, which is X m plus one. Now, this difference is what we call the X, which is the step. So the coordinate will be S n +11 and plus one. Now we know that the greedy in formula is why m plus one minus y in over X and plus women sex And and that's just the X is equals to what am prime. So this is the just employing this ingredient formula here. So now, rearranging you get y m plus one equals two. Why? And that's why I am prime. Yes. Now this is our oil a method. So let's put in a table and this is X n. This is why n then we'll compute y m prime which is to why in comes X and plus one and using on limited will find the next point which is why in plus one which is why I m plus y m prime now the the extra is your 0.200. Suppose your 0.2 years now we need to find the exact right Wyden Last one and then computer error Now for the is it y m plus one. Just use any CS you can see here. I've already seeing the first order differential equation. The white function is given here. So let me just write down the white function here. So the is the actual one. The white function here is why it was e to the power off. So let me keep the inner. This is not the formula will be two x y. That's cool. Why? So the actual one will be this. Why equals to eat about X square plus two x plus C so it about X square plus two x to a C. Now we start this initial condition x zero vies tree taking long on both sides We have C is equals to long tree So our is it one m plus one would be tree because this well, why it goes toe eat about x square plus two x plus laundry We can split this long tree from the rest we will have Why and plasma Sequels to tree I eat a bar laundries tree thumbs Either box square plus two x Now this will be m plus one plus one and the era is taking the is that minus the approximated? So let's start with 00 over here and why zero is tree So why is your crime is two times y zero industry? An X zero will be zero last one. So we have 1.6 using all the methods why one will be y zero in this tree plus white prime zero that six times 0.2 and we will have 4.2 now as it will be tree eat about now we want x one squared x one will be taking a zero plus the step off 0.2. So a 0.2. So eat up our 0.2 square plus two times 0.2 and we will get 4.6512 Now the error. Let me just erase this pot. The error is taking this minus this we will get 0.45812 He makes one. We have x one x one is just taking this transplant. Ouzo with 0.2. Why one already found here, which is 4.2. Why one playing will be to off. Why one which is 4.2 times x one, which is your 10.2 plus one. We will get 10 point. So are limited wife to will be. Why one which is 4.2 plus why one prime And there is 10.8 times 0.2. So we have 12 is 6.216 Now the is there is e tree to the power off X toe square. No X two is gonna be 20.2 plus another 0.2 will be 0.4 square comes to come 0.4 and you will get 7.8351 So the ERA is 1.6191 Okay, the last one X Who is your point for taking these? Adding 2.2 million. Now why two is this one? 6.216 now likely. Fine. Will be to off white two will be this Times x two is your 20.4? Yes, one. So what? The point is something 0.0.4 or 48. So why tree is why to which is 6.216 plus Why do prime which is this'll computer something point for For it comes you're going to. And so my white tree is 9.69 seven nama is that is three either fall off ex tree that's tree square now extra. It will be taking 0.4 last point. You will get your 0.0.6 square last two times 3.6 and that will be 14.2765 Now the error will be 4.5795 You can see that as and increases the error greatly increases


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