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COSC 4364 Numerical Methods - Lecture12_Splines
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00:00 well they always have a copy to . Mhm. Happy thoughts. You
00:09 . $20. Oh yes, I it would be that one. Oh
00:21 , no. They do eat in asia. Yes, I'll come back
00:32 them. Stop them. So. hmm hmm hmm hmm hmm hmm hmm
00:39 . Hopefully everyone also remote uh schedule is after script that was well.
00:58 the second thing we have to do and this is our story customs,
01:05 presidents from one of the problems The implants and support what or like
01:16 similar. So I guess what? you for that. Given that the
01:22 was actually enough Africans that have been . I don't feel so bad enough
01:35 touted as anything class victor. Obviously is a serious issue or something that's
01:47 class. What about remember stuff. we'll see what response the preferences inquest
02:02 the gambles because yes, I do for a remote taxes. But I
02:07 that formal schedule it is Of So I guess I presume it's
02:24 yep. So, and as I , so that's what I will do
02:34 when next week I will be once week and I would hope together.
02:44 examples this week. So you have . Mm hmm That's close. Close
02:53 . Closed notes. Yes. no, no. Mm hmm.
03:08 is my reference. The former Either way obviously exposed books. There
03:20 different types of questions and his open are not so sure. Mhm.
03:31 question. But usually attractive design So it's not a question of memorizing
03:44 all. Ah but so that more now that reminds me very much they
03:56 some principle and that should be probably generation or for stuff solution four clients
04:07 you're not thinking about it. That's . I definitely remember pretty much.
04:22 about you? What formally shades can have formally sheets? Yes in the
04:31 except maybe some simple things that I to remember of his life status us
04:40 simple very common function that resolved I corporate on that almost every lecture.
05:01 yeah find them is to design So I tried to do with the
05:12 , the reasoning and telling us what understood from it that it's important part
05:16 to memorize got some defense. You there is something there's something and examples
05:32 kind of innovation calculations simple for us do it. So it's a conservative
05:46 to problems think about this. they're coming. Mhm. Yeah.
05:59 ? Right. We'll understand there's someone that that were attending the high performance
06:11 we were talking about having the student thinking that we're probably ending the hybrid
06:19 of the class that they couldn't quite . What? Oh I see.
06:24 for the lectures will continue to do . So for the rest of the
06:28 is just at the exams they It's easier for us to manage and
06:36 will do them in class but all lectures we will continue to offer that's
06:43 hungry. So it's okay other questions it's not. So I will and
07:05 to start start start with calculus clients I mentioned very briefly at the end
07:12 the spectrum so I can't. Okay various the reason because France as I
07:22 this, let's place the collection of owners instead of falling over. So
07:28 talk about the contestants but there are to construct what. So I think
07:41 difference is I said last time when sort of alluded to with you.
07:51 What same song that was very easy understand in that case was biographical and
07:59 meals from the integration. And we a way off constructing those conduct them
08:09 that integration method was. But the between what did you sing or Simpson
08:22 compliance is that there is conditions on movement. So when you move from
08:27 interval to the next, there was conditions. Yeah. Well we did
08:35 symptoms in addition to the state board in the functions that there was 10
08:42 but otherwise of course no condition or that should connect and the continuous
08:47 Yes. How smooth. Mr Right. So that's respond to be
08:54 difference about all those clients. And want to talk about what kind of
08:59 conditions you can refer and what the and how you can construct. So
09:12 they're going to be I guess one concept that this sounds. No,
09:16 not already knows being used for the and the nose were the ones or
09:26 independent variable um values public. We to do interpolation. Now there's these
09:37 that independent variable values where you want have smooth movements, condition. So
09:48 the nuts and the notes can be same one shoulder and several examples but
09:55 are not necessarily saying you can construct so that you have in the the
10:04 points kind of a different from wherever want. And I'll show you an
10:10 support because it's just so the two have different purposes. That's the thing
10:20 conceptually. That should be alright. . So we have this already in
10:29 of rule effectively, but now they're approximation of the piece um function.
10:40 want to talk about dr admission So I think that most of
10:45 I hope all of them systematically. is kind of used for they probably
10:53 smoothness conditions that in this case it's explicit on the sidelines with another bunch
10:59 examples. But one thing that is and this uh construction of this destroys
11:11 function that it is continuous and that the case also chapters or because the
11:18 is all go we have the function from the state fund segments constructed subject
11:24 function to them. Um I'll come to that, wow. So and
11:39 . So this is coming, that's what it would look like that.
11:44 now instead of having the single Maryland not because it's peaceful seniors. So
11:51 a different peaceful single function between their . And it's one of the so
11:58 the spine is that this whole collection as and when it comes to the
12:03 base line weapons, all these polynomial about the most first degree. So
12:11 one of the polynomial looks like this they can all of the different coming
12:16 . But in this case different. huh. This place slopes on the
12:22 times between the not which are also used their notes. And this particular
12:29 because there is also where the interpolation done. Mm hmm mm hmm mm
12:34 . But the knots miss than where corner no meals doing each other.
12:39 that's what you have broadcast competition. here's some of the requirements for thanks
12:49 countless is fine. And I think one of the first degree that that's
12:56 to be closer. That means that end points. Oh, They are
13:01 one the access of interest between A B. That the end points should
13:07 included for the polynomial. Uh Sorry. But it's starting at
13:15 Should, you know the first three normal. But it should also be
13:18 for a consumer. The last function divide that big. The enforcer bunch
13:25 interior points published from the US. conditions. Um, so, and
13:38 continuity sense. Now there are different um, on this interval and different
13:50 over this interval. So the what's bottom here. Despite this the harder
14:00 continuity and there's going to approach. on election point whichever. Sometimes I
14:08 such that to take the limited to to us from the left side and
14:13 coming to the value s from the side for larger X values the limit
14:21 process. It's the second that that's working community conditions. Um So your
14:32 go back and look at what they . And the chapters are also that
14:38 in fact what this stuff is all use this in defining the integral in
14:46 space for the function there in And B. This construction with
14:52 Is that formally? It's fine. not? So the condition was the
15:01 should be included and that's obvious the space firstly in the function that is
15:09 in the Iraqi Soldier. But I it will be. So that's kind
15:16 what else is it? You it is Each of these different polynomial
15:22 the first order polynomial. Just checking too. And it wasn't next.
15:31 was supposed to be continuous. Yes, correct. Nice. Here
15:41 . So in this game. the fact of the matter and basically
15:46 chapters or will kind of defined. , that's fine. The first two
15:52 then use the stuff for the That was it. So not
15:58 A great example the pictures are trying figure out with this separate polynomial.
16:04 they have There's three different inter rocks one. It's 1 - zero.
16:10 is the one and 1 15 and each one we have it polynomial
16:17 So now we're going to try to out if this is dang what?
16:25 no you have to have a closed too. Finance one is included.
16:33 the first polynomial A certain well defined -1 And the other is the last
16:42 in which is also one of the on certain posts in there.
16:50 That's also so I might contain sm for that zero is equal to
17:02 But then the as X approaches zero sms is from above. Is
17:12 Yes. Okay. So yes. they're not there are one or the
17:21 . Right? That's what to pay . They kind of going at
17:27 It's an internal 10 and one is internal one particular next time. So
17:35 a right so controlled. Um And first ah not here from the best
17:45 X. and the gun goes to . On the other hand this polynomial
17:51 becomes is continuous. Actually points out happens I have been putting on the
18:02 the fact that The other not for value Then it is in fact there
18:09 is continuous at the other one. it doesn't help you. It has
18:13 be continuous with all the uh space . So now that's the question,
18:24 they're constructed and they come down with most is perfect and simple example.
18:31 , well we talked about how Manami is there can use this formulation.
18:37 the question is, can we use for also or just specifying what common
18:53 ? So why not just use each of them on the wall? So
19:14 it kind of worked for the first is flying but that's kind of the
19:19 thing that it doesn't work. Mhm reason is that by doing the construction
19:28 in the bottom of the interval, doesn't capture the small press condition between
19:36 things for us. It needs something . And working currently smooth for the
19:43 degree is finding was only continuity but thinking about activities and the game continues
19:49 well. Mm hmm. That was that. Oh with the film.
19:59 , so this is just yes will more or less the same thing as
20:04 we have what they did for the all known except again think of
20:12 But then there's a perspective between two points. So then this one best
20:21 right there down. Let's go get think the board some cases. So
20:26 use this new formulation where you then it's one endpoint another. So
20:32 hmm. Mm hmm mm hmm. to the next 10 points. Some
20:37 their kids um for each polynomial then needs Us to specify two things I
20:47 to specify the the starting value from interval. Am Nr cases.
20:54 so these are two parameters are holding pints straight sides. It's always going
20:59 have to end parameters. two Parameters every interval. Mm hmm. This
21:06 never end into the wall to the . Mm hmm. Ha. Also
21:13 doing the interpolation here, We basically two conditions. One police, one
21:19 the two conditions as well. So are and that I was using um
21:29 the success some comments and I would to write to gold And in that
21:35 then yes to find a particular you need to figure out, switch
21:40 , exit the argument if you use formulation instead of spending that kind of
21:47 them a no meals and we kind already have in the world that's fixing
21:54 something else. And that's one. look at the accuracy and it looks
22:01 little bit different the structure. So , I want to talk to It
22:09 just using this while there is a which is sometimes for continuous and rice
22:18 related to the make a point. say that the derivative. It's a
22:30 function that at least the Derivative between main points. Yes. Equal to
22:41 . So, um, that's important one single. And I don't mind
22:47 this module of the continuous deserves especially look at function violence at three different
22:54 . You will the United States and between A and B. And then
23:02 the maximum difference between these two function . Ah that will be different.
23:09 you and me anywhere in the But the point that I want to
23:13 this is basically if you will be then it's just a nice function
23:20 well behaved then the maximum value Difference function values of those points eventually zero
23:29 As opposed to zero then well then has not helped in the function.
23:34 . Yes. Well you're the So that's something for the ceremony is
23:43 couple of sites that follows here. continuity and addressed themselves Specific and a
23:55 870 positive number then there is. so you have a function that takes
24:04 input arguments the X. And then . Value of Earth that the same
24:11 the other the way also we have learning this simplicity. So if this
24:18 an important that as an output through different input values ah and regardless of
24:28 Excellent. What about this book than some sense, if there are close
24:36 , then also the difference in the values are close enough and it's kind
24:42 stated backwards in some sense of you this difference and function values to be
24:50 than whatever epsilon depicts. And if uniformly continues, you can always find
24:55 extreme violence. There are also closed they're coming up on the functions that
25:01 held by the late the principal. . Um, so if you just
25:08 the small difference in the they just to figure out small, someone needs
25:15 be to satisfy variation for difference in function. Violence after it's small
25:21 And as soon as probably continues, can always find figure out yourself.
25:25 parallel victor parra between A and C. And 3/4 of the way
25:38 . Ah, So that's good. and that's what said, well,
25:46 it's continuous about behavior can use that . There can be no beef.
25:51 deliberative distraction. That's kind of what 40 cents. So it's just
26:00 Let's continue. Okay. That is just to complete itself from here.
26:11 terms of the river things, you think of it. There's basically three
26:15 squared And if that's where effects is than export is 16 and that
26:24 Yes, you are Than users and morning and that's But this notion of
26:34 continuity, it's done. And here's an example please discussed earlier Suggested about
26:46 interject one direction losing so that everyone or used tractors always or any one
26:56 these rules. It doesn't work because over X. The derivative of one
27:03 Rex. It's not bounded that they're to do m one sufficiently both.
27:12 , except for zero them. can't find this uniform or comfortable.
27:26 are other forms of integration. Integration interaction and that's this sport. So
27:39 this case to go back to the and they didn't put a normal approximations
27:44 picking up what the error and the nation was between the function F and
27:52 polynomial. And in that case he to take your serious expansion and the
27:59 term was that um less than the or the driver? The proper factor
28:09 terms of in front of the derivative the border that was not into.
28:21 . Some indicate a serious expansion. then we have the intermediate point between
28:27 endpoints on the interval. And then was the tail issues in terms of
28:34 is done highly related. They choose use this module is a continuity that
28:39 it's the function Now that is continuous has narratives. Then you can use
28:46 they had on the previous slides and error between the functional polynomial. The
28:55 and obvious in terms of these And we can look at each one
28:58 the intervals in this kind of this of knowledge, he's going to maximize
29:03 benefit. How much the function of and No, no, something
29:14 Mm hmm. So it looks the . But there's some intuition about thinking
29:20 the intervals and assuming that's a nice . That's not a lot of the
29:25 smaller the variable costs. Yes. , promotional consideration. And there's a
29:35 of expressions I so of the book overlooked at the as an exercise that
29:43 managed love to this one. That the first derivative. Oh the
29:53 And with a maximum of first limited meat. Of all of the h
30:01 Any any one of these. It's length intervals but it doesn't have to
30:09 but any interval of length and the term of the derivatives in that think
30:14 all. And then you have this all. Ah this one, it
30:22 a little bit better if it also a second relative that behaves well.
30:26 this one is actually Electoral eight And the version and it was about one
30:37 the length of the innovation. But have a sense the way of proving
30:40 this is actually true. So from . Mhm. Um So, so
31:03 this clients, because again, we um this unicorn continuity is dealing with
31:10 intervals. Um when we did the population before it was wonderful and normal
31:21 . Um So the whole interval is be as opposed these boys are smaller
31:28 of all. So in that case we wanted to talk traditional daughters stuff
31:35 increase the number of interpolation points Better approximations. But when it came
31:47 this, another function that wasn't right? Because we added points and
31:51 they were equal space, they started get more and more mature is so
31:55 deviation the maximum difference between the polynomial and the active german function that increasing
32:02 more points. So this notion all of dealing with this modules are continuity
32:14 looking at small intervals and test As said that one. And the point
32:21 it's better we expect to approximation you have low order polynomial on small
32:34 so you can have the same number points. But instead of trying to
32:38 one polynomial off to use a a small number of points that no
32:44 no meals and then you get But that's the reason why clients are
32:54 very widely used. So we get collection of the enormous and 70% of
33:02 over because you have a better approximation what function. Any questions on
33:15 Start the second. So I have great force. So that's simple.
33:29 the polynomial is on each one of symptoms. Maybe up to a degree
33:35 be constant, first or second degree not higher than second degree. Still
33:43 same role. That should be post And now for the second degree polynomial
33:52 have both the polynomial itself. For collection there should be continuous at the
33:59 while Camilla's join each other but also first thing of it. So you
34:10 at the first degree polynomial is the , The inner function. Obviously the
34:15 was not continue to consider the but everyone is also the first derivatives.
34:26 standards from the left side on the side of the other side. Um
34:33 one of them. Alright, so see that's an example. Let's try
34:40 figure out whether this one is what explains. So now the conditions was
34:48 to distribute them. I understand that what we are included. There is
34:58 first people in order to have is the second degree this from 2nd reading
35:05 but it's not voluntary so that's two less. So this has the potential
35:10 be a. That explains for that need to verify not only that it
35:19 um but also that the derivative is . Okay but this one was eating
35:31 institute so now we need to figure out one third. Ah The function
35:40 up for the polynomial. The correctional are continues so there's the right value
35:47 the zero not in the 11 not the same but also driven itself.
35:56 see and that's why I'm here. can start with the function itself accused
36:05 polynomial Are the values of three knots same for the left, left and
36:13 from the normal. I want to it zero and 0. So he's
36:26 on the same values that's right. off one of our respect one.
36:36 . Alright. So it's continuous. was good. So then you put
36:40 on the next side. So now a question about the driven um but
36:47 also have said this continues right Now can look at the derivative at
36:55 That should be if the driven team explain when it comes to x.
37:02 this is -3. That's what zero. The same values five
37:12 that one. And on the next is uh, one person. So
37:20 one turns into minus T. Rex the derivative two. That's one.
37:29 , this one um, the derivative mine too. Um, so the
37:42 . So now is also continuous. that was one that closed interval and
37:52 the functionally the collection of calling your form a continuous function and the derivative
38:01 this production of polynomial of the knots also continues. So I want to
38:09 case for now. It's a question the reconstruction, it's true that the
38:18 . Mm hmm. But I may a first degree in investment again.
38:25 then you're gonna stop formulation with each interpolation exported to the endpoints that the
38:31 of the line and we have enough to specify. Now we have a
38:41 bit more. Mm hmm. Are worried about somehow? We have don't
38:48 up for me going on right and that things a little bit more because
38:57 only have three. Something on the of the slide, right, has
39:06 values that needs to be said specifying there first a great, we only
39:12 D. N C. Effectively. they have three product military reforms.
39:19 . So now we just believe three foundations. Now the interpolation part.
39:30 best for ourselves that the yes, best for the endpoint conditions are right
39:37 them. Yeah, mm hmm. have a condition on the value of
39:48 cubes but the next and the right the interpretation but that's too. But
39:55 the Simpson rule was stuck in preparation in the mid point but we don't
40:03 that thing. You don't know the . Y you just have to.
40:13 . So this is something that I right, we have interpolation conditions against
40:19 this future and conditions right there for , given what we know what with
40:26 . So there's still to learn conditions it was for the first degree.
40:30 that now we're still missing and conditions we needed three. So right.
40:41 what they did. So look, , I don't know what to use
40:48 tradition on what that explains that not the function but the derivative also needed
40:53 be 13. So when we look and not the derivative of the following
41:01 the left and the on the right for any intervals that obviously and
41:11 not in the middle Double two and there is. Mhm. So that
41:23 with us. Now you can have things up three and - launched so
41:29 still missing. Well, well, see, wow. So that's what
41:37 said and the one is used for expands is one degree of freedom that
41:45 can use sponsors slightly different problems. one has to suggest someone can for
41:54 specify the derivative is one of the conservative in terms of this movement condition
42:01 only a the interior, not not . So you can arbitrarily choose
42:08 We want the for instance of zero the second derivative this area or something
42:13 that. But wanted to your freedom these interpolation conditions and the smoothness doesn't
42:27 this in terms of the conditions and . So the first thing is to
42:33 enough conditions that they can specify for parameters will be 2nd step, how
42:39 we use it to actually get So here we are actually pathetic
42:51 And those are probation conditions we talked and that is enough conditions. And
43:02 there were foundations on the river. and we'll find that sort of a
43:12 procedure is being used in the Yes. Two. Uh huh.
43:22 we're still close enough. I think is. Okay, so so the
43:33 um starting point ah that has been in the book for defining responds.
43:44 this darks. So the highest order that is supposed to be continued in
43:53 second of a difference. 1 2nd is flying only the first derivative is
44:02 to be continued. So if you a quadratic polynomial, take the derivative
44:10 we get the first degree polynomial. so so this is kind of now
44:17 of doing your phenomenal itself, it like distributive but it's kind of doing
44:23 straight kind of approximation between. Do you have any values at the
44:28 and 4 but they don't know So we're going to have a machinery
44:34 figure out how to get those But if they were known they can
44:38 a straight line approximation that started first and make sure it's continuous. The
44:45 derivative doesn't have to have an additional condition. It just has the big
44:52 . So, so this is essentially little bit just the water is written
44:59 different things But this is kind of new 10 or so formulation from a
45:04 line between ah Ci Ci plus one the now the derivative five past
45:14 that one. And so it's just straight fund in terms of the request
45:23 , when we looked up earlier neutral intended to write the constant first at
45:29 of the endpoints and second term was slope of the line fans moving along
45:36 lines of years before. So and of course they needed continuity of the
45:48 polynomial themselves. So the left and right polynomial should have the same
45:53 So now I think that's an equation the views and that's fine. No
46:01 integration for this than the and it's of victim in his performance. So
46:12 wines, it's the concept that it the relative it disappears. Um So
46:19 we can see the derivative of this . This is fresh. So now
46:25 in so basically in this one we figured out that this constructive subjects
46:33 It's your crime but then did not to serve them. This new function
46:39 also considered. Did that here? we'll figure out if that's my
46:47 So yeah. So quick plug in T. R. And this expression
46:55 this film scores And what is that function value or at the left hand
47:01 of that is what it was supposed be in an Interpol insert function.
47:09 And that's the content. What's that have done here. But um let's
47:17 at the function value of Q. . C. I. Plus
47:22 Yeah. We're also sorry not jumping . So let's look at. Uh
47:32 . Sorry. Uh He was first by construction must begin T.
47:39 And this expression is for the ways becomes er and plus one This becomes
47:44 one. And then you can see the story is private, straightforward to
47:52 expression. It was constructed. That shouldn't be surprised that now it looks
47:59 this if you believe in that if look at the construction of the next
48:05 ci last one just to face. mean I passed one everywhere there's also
48:11 that then Next polynomial tr plus one its left hand point. The ci
48:19 one which is also the value of left for no one at the
48:25 So you can see that the derivative Wisconsin. Okay, so now let's
48:35 what happens here. So now this mhm right. It is necessary and
48:47 shoulders. The he's palla normal as right does the same value of why
48:58 respond in this case as the next . So the ipad first interval has
49:04 left. If that's constructed in the way that everything here will just
49:09 That's one. So it's obvious that actress 1st polynomial has its left end
49:15 is that's one sort of thing is show that The left hand point the
49:22 pulling over for respect to your class also. Yes sir. So
49:34 so if they just insists that that true University of Drug Ntr plus one
49:42 this expression um hmm. Then you it so we can see our first
49:51 whatever it is. X. By arguments you left last month then it's
49:58 right hand side should evaluate to by custom. So if you do that
50:08 . Now and This equation plugging in . R. Plus one then you
50:14 tr plus one minus T. And we have a denominator disappears and
50:20 um denominator. And then this is the managed ci and then you just
50:27 terms. So we had it to equal to the high plus one.
50:32 you can say that that's on the hand side and we can move the
50:35 the left hand side. Um These parenthesis became tr plus 1, 1
50:41 so They spent around two and clean up. And then the question more
50:48 to um relationship in this case between derivative value of ci one of the
50:57 with the nervous but there are other and if this relationship holds then both
51:05 . Prime, thank you are continuous all. Now these were given interpolation
51:15 and these were the given not. were often used for incorporation in this
51:21 . So this is a known So basically You can just start at
51:27 equals zero and then all the other followers. And then you can go
51:33 and stick it into this question and you have the definition of pulling
51:39 Mhm. So I think that's for seventh year. So and this is
51:47 of disease for the derivative values. So if you start with the zero
51:56 all the rest of it follows as function of Z0 Now. And that's
52:02 the C0 was a derivative. The 10 points if you should agree it
52:11 on the previous slide but you can that to zero and then all of
52:14 other ones. So so this is aware of them getting the coefficients
52:26 All of us 2nd degree polynomial was all the do we know?
52:37 So that's where that was the That was. That's a good
52:44 Yes. So that's where we have conditions from the smoothness conditions inc gave
52:53 the two man and I never said conditions on the first day of
52:59 Did you have ammunition and -1. we were missing. And that's a
53:03 of freedom. So One can do against one type of super spies.
53:10 you use this criteria it could be one and their their policy. But
53:16 need to make one decision on your if the machine doesn't. And so
53:23 this this would be zero. And you said that 200 then what we
53:30 follow. So 74 27. So have a instruction in the polynomial secondary
53:49 . And this is something for not for that defines the second degree
53:56 each of the intervals. And then was needed was the seas and the
54:01 . You get out of this machinery is defined except for Say the starting
54:09 . That was an arbitrary choice. these are known from the circulation.
54:17 is again that summarizes and this is example of it. Just for the
54:25 is fine darkness. Yes. The thing is well, you know,
54:35 can see this that the derivative on left and the right is the same
54:39 in my school Burns Doctor Shopping. all the difference. Yeah. And
54:47 this finishing use the different obviously because that it is not serious and responsible
54:52 something. So let's illustrate again the rule as I said a few
55:07 Doesn't have the smokiness conditions between wow conservative intervals just to see if someone
55:14 this ambulance. But there's no small foundations don't english with the loss.
55:30 . Both into all. That's It's a little more time. But
55:34 one who continues yes, continue conservative . That's why the correctional quadratic polynomial
55:47 is underlying the Simpsons. Ooh, not define this part. Yeah,
56:00 more question. That's the point. the next example is one that is
56:12 trying to illustrate that one type of . They don't nodes, nodes and
56:22 . I'm not the same. So choose that smoothness conditions and other
56:32 Mm hmm. Right from the So let's go. So in this
56:47 . So in this case the interpolation looks like Tuesday nights and get points
56:58 these intervals I guess where we know landings. Thus the two imports have
57:05 in the first so once no, collection of projected polynomial bring him to
57:13 in this case for the first section third separate but that is falling on
57:20 for these three intervals but they're supposed approximately function at the left and right
57:29 . Thus the three hit points and the interpolation of the voice and then
57:38 smoother condition supplies wherever or no one's to each other so small. This
57:45 to be. So the condition that notations you stand that we have the
57:58 . Are they interpolation points. And tease here the knots, the mother
58:09 smokiness condition or something important conditions as smoothness conditions have taken points because there's
58:18 one polynomial so we can have some . But what? That was part
58:25 the thing that happens in and those were the same in the previous
58:31 Then we have to and an orbiter . That was only wonderful.
58:39 So uh let's see another. Uh . So so now we have this
58:52 a normal again, the first thing the derivative is um first the hateful
58:58 no help the second degree and yeah retrospective formulation for so I have a
59:18 and gracefully. It's still between the 10 points respectively. See And this
59:32 the interval that if you have an constant then is the difference between
59:40 no the and then define the solution that tells the point for them.
59:56 , it's a bit message to verify . In fact that's some of my
60:05 use the finance this particular way are . Well let's see from now again
60:17 can see what the values are first not. Um And point of the
60:29 law. So in that case we the 17 years. Another minus
60:38 So I hope so playing around with thing what yet um So minus the
60:49 and it's just an expression for talent the rest yet half of the financial
60:58 in um And the defendant. Mm . This Very, very point type
61:12 seven points. And we can also used for. Mm hmm. The
61:18 important ci plus one instead of this expression. You simplify it knowing what
61:26 is and then plug it in when won't talk to us. Now it
61:31 out. So it is a straight approximation. That's satisfying foundation. It's
61:38 dream that he would have been known marketing and evaluated. But I still
61:42 know the sea ice and this we need more conditions in order to the
61:49 . I want those. Um, I'm standing with you integration. So
61:58 we have accused and now we have cues. Then we can use the
62:05 condition double surprise. And that was of this was the constant that disappears
62:16 this depression zone. This is where can see. But that's a good
62:19 to see to make sure that this does what it's supposed to do.
62:25 which is the, it's just coming now the interpolation conditions and this
62:32 So this is the polynomial. And . Mhm. Inter promotion point index
62:40 one more than that index of the . So that this space we should
62:46 and the towel And that is my one in because it's sort of on
62:54 risk investments like this. No How High Class one And want to
63:00 them. This disappears and just So yes, but having the constant
63:06 the interpolation points then it satisfies the conditions for the people. And so
63:18 that case you have any interpolation one of the intervals ah and we
63:28 so let's see the next one that have summarized and then have it.
63:35 they interglacial conditions that was and they me for many of the conditions on
63:43 first derivative that was satisfied by construction by that too. And that's for
63:49 internal points. There's N -1 of . And this is very simple Examples
63:55 three intervals how the best teams are for they published. So we have
64:04 know what and plus and -1. we have two N -1 conditions.
64:12 the immediate conditions. So now the is what was not, I will
64:27 the continuity of the delivery. I think the prevention Prevention four.
64:48 , I will not do this. continuity condition on the polynomial.
64:54 Not to have the continuity on the delivered in that cannot have interpolation in
65:03 midst of not understanding ensuring that. rightfully so did you ever think of
65:11 same continuity? So that's send one well. Okay. And that things
65:22 . Mm hmm. That somehow you . Okay. Sylvester talking about.
65:31 we have where there is continuity on Tuesday birthday go. Internal points to
65:40 up with three N -2 in And then we have the incorporation of
65:44 two endpoints. But we did not yesterday, inc in the middle.
65:49 that's now that's two more. So they end up opinion and I think
65:56 so big let's say in terms of to, I had to put down
66:02 now look the continuity of rightful and one of the knots and then get
66:12 yeah, that is joining knots from left and the right side. That
66:15 supposed to be the same. And lots of expressions for the solo normals
66:20 then you that's great enough for And the right polynomial, they're supposed
66:28 be the same. Mm hmm. knowing what towers that that midpoint it
66:36 simplify expressions in terms of these and mill around and help with this
66:43 your best. Again. Any question looks up. So it's nothing.
66:49 the point is the continuity conditions and the left and right for the normal
66:59 but then we only have unknowns in case, You know, we have
67:04 wise are known because we know why the super and we know tao and
67:10 things you don't know or disease. we get an equation here. See
67:15 that put conditions obviously it should be order to satisfy the continuity conditions for
67:22 point on itself and it comes So what's known as these things as
67:33 values in the intervals that you suggested and we know about the length of
67:39 intervals are the edges. So the that are not announced or disease.
67:47 and I have this for Yeah and equations. But it's a tool of
67:58 And -1 sees for the internal So in principle make sure you have
68:05 equations. Mhm. Figure out Now one thing to looking for a
68:14 thing, it's essentially diagonal system of questions there. We unknown disease.
68:26 do you think of it as a spectrum type formulation for going over their
68:33 ? And that's basically three non 0 in a row that corresponds to the
68:40 scenes and they are kind of sliding um okay, it comes guests will
68:49 up from the slight soon. So have these conditions and then Yeah,
68:55 he's not the additional conditions to and . Before the specificity. Where's
69:03 Yeah. Yeah. So um That's for the first one. And
69:19 similar to the endpoint. Mm So just 10 points value where they
69:31 because of this form for the first . This The question and their natural
69:39 101 t zero. The left stand . Do you have an expression?
69:49 the now involves non value and to this situation? Yes, this one
69:56 then you can give you the right environment interpolation condition. So now you
70:04 this relationship. So kind of in middle to have the strike there
70:11 Uh huh at the bottom of this and for I equals zero equals
70:25 Any questions for that. Oh this . No we're not. And that's
70:34 . So that's it. So so what I was thinking of respect
70:39 This is from the interpolation on the hand point and then we have this
70:47 journal points and then we have the question that comes from the population at
70:54 right hand point. So the best this now we have the china system
71:01 the question and you have enough Nothing can solve it for a lot
71:07 difference. Mm hmm. So it's little bit more complicated machinery to get
71:13 this space. The derivative values the Internals. # two in points.
71:26 it comes from the interpolation conditions in first. So that part of
71:35 So in the spines and events and they talked about um directed questions.
71:41 it was way back a lot of pay attention to try the ignition systems
71:46 you're going to use them to talk there's an ex alcoholic you end up
71:52 triangle systems in order. Yeah, following defines that's not fully known as
71:59 differences. Address to what they So once you have disease.
72:11 What? Mm hmm. Mhm That's . Any questions on that benefits.
72:31 one thing is that we have thursday between the degree of the spine and
72:47 to smooth best foundation 6 25 so Okay so so and so as to
72:59 it says at least in terms of have the first risk fine. I
73:09 to by the conditions on the derivative wouldn't end up there saying in every
73:20 and that obviously doesn't. So there a connection between how many derivatives thank
73:29 , forced to be continuous in respect the order of the So it is
73:37 to what distance sensors that can have driver thinks of an order up to
73:46 less than the order as part of most blessed condition. And that's probably
73:54 than two. In terms of the polynomial they worked out how to have
74:00 the function or the themselves to form continuous function but also derivatives was continuous
74:08 different thing boss. That was the order. They didn't try to do
74:15 . So that was an example of an example. So for the first
74:20 the function itself but no derivative. order fine function. The first one
74:28 everything but not trying to push it the so what this set of science
74:37 to tell you that if you try insist that even in the face,
74:42 of them but eric falling over the relative, it is the same but
74:50 knots left and right then in fact to push it to become one across
74:57 hall. So this is what it . So yeah, that's all for
75:04 one already in the sense of So besides a few look at the
75:15 learn from approaching and the correct or . So the african right. And
75:19 supposed to be the same for the conditions. And if you force it
75:27 B, that makes sense. These for the normal CS functions and forward
75:34 and right for normal P and Yeah. There's the other derivatives From
75:43 up to whatever and order community. want to be the same other
75:52 Well, that's that's coming continuity. . So now yes, I tried
75:59 push it. So it also includes empty of everything then the local potato
76:06 expansion. That's where all the derivatives the same. And in fact ends
76:13 , he told them them terrific than are in fact the same paranoia.
76:20 that's best for us. All one to stay with him. So one
76:25 to increase the degree of the polynomial the different intervals. Ah Such
76:31 there are always the one degree higher and everything that's a lot of this
76:43 . So I think that's what it's . Right, okay. No,
76:50 didn't, the second point talked about degree expands next time. So you're
77:02 among the most widely used the 2nd in response from the country function
77:10 . That's what we used to 2nd was not that sounds wrong, but
77:21 inside the views as well. And of the integration some of the
77:32 And as an aside computer graphics to some surfaces and low complexity, some
77:43 our genetic offspring desolate destruction and divisive domains. You lose your spine,
77:50 sub domain. This is the same breakfast part of the differential equations.
77:59 . We will construct their spending an that much of the main of the
78:10 and the whole system of equations based no these supplements from approximately or we
78:21 find out too. That's the final of finance and that's assembly lines,
78:27 formulation and also it speaks first approximation . Okay, thank you.
78:41 so they just hopefully the inquest later . Great. The radio report.
78:53 break for the midterm is gonna be monday after. Right. That's the
78:59 it's scheduled them. If there's an with that. But you know,
79:05 to the preference not to do it but have adapted as the usually is
79:08 case in this class. And then be the ring before spring break.
79:14 trying to give you some examples. one. Mm hmm. Uh
79:21 Okay, interesting. Thank you. you. Thank you. Thank
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