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COSC 4364 Numerical Methods - Lecture07_Polynomial_Fitting
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00:00 I think it's better that they're I didn't want to see.
00:42 Okay. On the screen. So I'm encouraged. Mm hmm.
00:57 recording is happening. Yes. So to public today on to the next
01:05 in the book at this uh Interpolation. Another way of saying it
01:16 do finding. Okay. The function inter plate or fit. It's a
01:30 fit to accept the data. That's today functions will be polynomial. But
01:41 that. Later on in later lectures talked about other ways of feeling um
01:49 accommodation to accept the data that you . So. Mm hmm.
02:03 So anyway, so this is what will be. It will be about
02:12 keeping data today and talk about the words of doing it. And first
02:19 bit yes, motivations. Who Well, polynomial are quite often used
02:27 terms of instant computer graphics. It's case where to discourage surfaces in a
02:35 sized neighborhood in terms of pulling over so that it doesn't mean that you
02:40 a single polynomial to describe it total but to take pieces of it.
02:46 you often use again following on else describe. Mm hmm. The surface
02:53 for other for rendering what other Um and that's simple for example of
03:04 you may want to fit some analytic to the collection of data. So
03:09 this case it's just somebody took measurements his caustic attitude temperatures and general development
03:16 four data points. And then let's it um sometimes that is convenient to
03:23 analytics description of what might happen. doesn't mean that the underlying physics is
03:32 represented. It just means that So it could be that you
03:36 you get this analytic function that estimate discuss it would be at some other
03:45 than the four temperature subtraction initiative. you may have done that in some
03:51 to some other courses, probably getting the point where the enough so never
03:58 is what its function obviously do. The simplest possible why I was almost
04:05 specifically straight down between the pair of . And if you want to have
04:10 costed it's 8° for instance straight down five and 10. And then you
04:15 the straight time and seem to get eight degrees and rivalries whatever the rather
04:21 on the line at that point. a simple way. Maybe a good
04:26 way, but not necessarily the best . That's a lot. Or do
04:32 want to try to do something where So I have an analytic function that
04:38 all before measurement points. That in case may not be a sequence of
04:44 lines but you have something that also some curvature change and that's supposed to
04:51 collection of straight line segments. So comes into polynomial with straight line.
04:57 a simple polynomial of degree what? But then it can have other polynomial
05:07 . It's all four points for another case. It's not like to
05:11 a degree one because to be a one, that would mean that the
05:16 time will actually perfectly fit all four points which is not like that.
05:23 on. So another version is that a polynomial and as I mentioned you
05:31 you can do a collection of straight between this pair of points and that
05:37 them the principal collection of falling They won't talk about that today But
05:44 difficult on as spikes and there will in Chapter six we'll talk about saturday
05:50 of full installments instead of a Mhm Yeah. So you can also
06:00 but the functions pulling almost no means only set of analytic functions that is
06:06 for fitting to the contrary other functions actually I will say more frequently use
06:14 polynomial except in terms of computer garden possible. So so and the I'll
06:26 to that much at some future class on the issues so these functions that
06:34 used to um trying to model data some other function but generally on a
06:42 basis functions and the trajectory is too basis functions that captures a lot of
06:56 properties of the data that they So that means you can use found
07:03 in a few of these basic functions that's kind of was the basis for
07:12 it for compression. So in that the older the institutes cause I understand
07:19 and the impact standards and the point for a lot of different. I
07:25 make this mostly in that case it take that many. So the call
07:30 platforms to capture the essence of the . So that's the largest. And
07:36 the only thing you need to communicate the coefficients um for the basis
07:43 So the senator and the receiver agree what the basis function is. So
07:47 you get the coefficients, you can the data based on just the
07:53 we'll talk much more about that So just to point out that today
07:57 will be enormous. That's by no the only functions we used to produce
08:07 and the other thing one needs to if one wants to exactly ah at
08:15 match. So the data points that may have or for that matter is
08:20 try to make the simple representation of complex function if you want to have
08:25 . Exactly. That's the favor on . And the point is if this
08:31 like and this suggests the case is measurement data measurements are really perfect for
08:38 areas. So you may not necessarily to perfectly that's the point. Have
08:44 measure of goodness of fit instead of and that's uh what comes up in
08:50 guest tonight. So all this. I think I'm pretty much said what
08:59 on the next few slides in this uh documentation. But this is that
09:04 can get up to design. you the perfect exactly or not or if
09:12 want an approximate trip and then it's notion of quite simple Sylvia is again
09:19 make something that captures the essence of data were function and that this simple
09:29 whichever means you find simple belief for particular application targeting. So So what
09:41 said one, can you polynomial polynomial not necessarily what it says in this
09:48 should be an example of an exercise suffering is simple to write down simple
09:54 evaluate but they are not necessarily the for you approximating the second. As
10:01 said in the next slide, I'm to sure what may happen. So
10:07 is kind of a very simple Kind of a nice bell curve if
10:13 like. And what this slow tends show is what happens to think that
10:22 higher degree polynomial you have the better of the data. So and this
10:30 just trying to show that that's not the case. So this black dots
10:37 equally spaced points In the interval between and one. And this application here's
10:43 function and the just kind of follow dots and that shows what the function
10:50 actually are. And then it's uh uh attend to gratefully. Normally the
10:59 that you don't use all the black . You just use a subset of
11:02 . Then the black dots are basically the case to use 40 of them
11:08 . Yeah, hopefully very good approximation now you're trying to make sure that
11:13 polynomial matches all Than 40 black dots that case. But if you do
11:20 to get the polynomial. So when look at the polynomial values between the
11:26 where you make the fitting as you see in these areas, they're highly
11:33 there is nowhere close to what the function that saw if you move away
11:37 the interpolation points. So that's one the drawbacks that high degree polynomial.
11:45 have a very prospect or behavior and why the reason why you may not
11:51 calling on me having him money. wants some other function. The betting
11:55 pass to the essence of this So this was kind of the preamble
12:03 the getting into. Yeah. The of using pulling normal approximations. Any
12:10 or comments on the behavior of So this is amusing polynomial for that
12:22 not be as effective for functions that passed. Right. Right. because
12:29 if they're higher than zero or then must either approach me in a deeper
12:34 more and right. And it's also that we'll talk about in a future
12:43 . So it turns out a little of this social cultural behavior is also
12:46 by the notion that is straightforward Take your interpolation points as equally spaced
12:56 in the interval. So if you it on and choose different ah distances
13:04 the population points, you can keep same number. But they move them
13:08 the village. And then you can find out that yes you do.
13:12 you can avoid some of the So it's just a yeah melanomas are
13:21 many ways great. But it tends be best when you have a limited
13:26 of points. So yes, you see the blue ghost that swings off
13:32 it never behaves as badly as the . So it tends to be that
13:38 of having a high degree polynomial it's sometimes better to use the daughter
13:46 build degree polynomial to approximate the And that's spine centers. Think I'm
13:52 popular and we'll talk about spines in later venture. So that's instead of
13:57 a high, high degree polynomial to many points to use the collection all
14:05 . And then we'll come back to initial collection. The first thing then
14:12 just to go through the exercise and , oh what? Okay. Now
14:16 going to look at the mechanics of polynomial approximations. And this mechanics will
14:24 used for the gun When we talk spines. It's just the difference is
14:27 spine, once they have a law from enormous. But the mechanics
14:32 That's right. So, um I guess that's mostly introducing vocabulary points
14:45 nodes. The points in this case this collection of terrorized values that you
14:52 in your little table and have an variable presumably to the X. And
14:57 dependent points up for every X. , I have a corresponding. So
15:03 paradigm is known as the points um you want to find a polynomial that
15:10 exactly for today's manager. Now the files themselves, it's known as
15:21 So many times you can have, know, not just single division,
15:26 have a surface, you can have Z coordinates every next time. So
15:33 are there's three the coordinate values for you want to do an approximation of
15:40 function value in this case. It's to have distinction. Clear points and
15:52 And that's the one. So now is the trivial and talking ridiculous.
15:59 , it's doing the interpolation of the for instance, that's the concept.
16:03 no wonder what you can do. that's the degree of this polynomial of
16:08 zero. And of course then the thing that is also simple. You
16:12 two points and you can do a line between two points Polynomial of degree
16:23 . So and of this land investment reopened it in two different ways.
16:31 1 1 is this form of an . So you can see here by
16:37 minute X and the variable it's free access to X. one. Then
16:45 thing turns into zero and X one X zero. So this parenthesis becomes
16:50 one. So that's one This on normal evaluates the white one and you
16:56 plug in zero. Then this term and this becomes the one and then
17:01 value is Or the point of almost . So it clearly perfectly represents the
17:09 points that we're giving. Excellent 41 . 01 Singapore. Another way of
17:16 the same thing is just can't be it. Perhaps more common to look
17:21 this. The equation for the line in this case X660 then disappear.
17:30 we get the same thing again there on the polynomial zero is 0.
17:36 if you put in X one then this one, chancellor and you've got
17:41 two things so that what's left is . So it's just two different ways
17:45 writing the same polynomial in the sense pleased to ways of writing the polynomial
17:55 in the same value for any given . So in that sense they were
18:01 same even though they're written in two ways and that's what an important.
18:07 talk about that a little bit more . So that's um uh the other
18:16 yes. Yes. Perhaps reflect about that we have one point. The
18:24 has degree zero. Now we have points and the polynomial is everyone.
18:32 the whole the role is that if have endpoints, the degree of the
18:40 that to use green for its opponents of order N -1 or no
18:47 It can be a higher degree but we can put the higher order polynomial
18:53 goes around that you don't have enough to make what you need. So
18:58 only way you can get a unique is selected degrees at most one lower
19:05 the number of points. It may lower if all the points fall on
19:10 street. So um Yeah. And see it's all. So this is
19:20 at that. Right? So now a couple of names here. So
19:25 way of writing the polynomial is known the formulation and this way of writing
19:31 political is known as a good Yes, we'll talk about the constructive
19:39 in the event and questions on So I think an example concrete next
19:51 to do this thing. So it a few points straight time and um
19:55 what was photographed from the previous One thing uh huh was the notion
20:06 this brilliant. So it looks like would think perhaps that would write things
20:15 left to right and increasing X But in this case, you
20:19 this was used as the first 2nd point and it's in the lower
20:26 value. So that's what you find this thing do that. That's the
20:31 as for the two data points from previous draft. So it's one point
20:36 come back to later. Is that interpolations. The results should be independent
20:42 which order they happen to write in faith. So that's an important aspect
20:48 the approximations. So, um um this is just working it down in
20:56 of the the ground formulation has the point, That was x 1.4 minus
21:04 distance to the other point. So have this question like it's so in
21:11 denominator is the difference between The interpolation in this case that still wouldn't explain
21:18 next zero so that we will see both of them, but they're ordered
21:23 defense of they've got for the first point X ah was the next one
21:37 very welcome. 25 and then we the X value and the function
21:43 It's zero Mostel 3.7 that comes from . And the other one is the
21:49 point to three point. So it's pretty straightforward just plugging in X zero
21:55 X one and the correspondent by values this regard for relations or you can
22:04 in the needs of what and and of which way you write it,
22:09 can simplify it. Uh this for the expressions. So um any questions
22:21 that. Yeah. So now a bit more kind of formal and generally
22:35 the first simple taste with us two but in general dig out some polynomial
22:46 that looks like this in terms of growing formulation, in terms of
22:50 it kind of ends up generalizing in what we are going to use both
22:57 them future and past for other chapters the book but to this significant differences
23:06 I want to point out so and like our formulation, one of its
23:15 features is that they um the grinch multipliers or expressions the health. They
23:31 totally independent of the function doctors. you can construct this and this totally
23:39 of what the interpretation points are. they only need to know the
23:47 They need to all the nodes So you don't need anything about my
23:54 function. But in order to be to construct that's not true in the
24:01 formulations, Newton's formulation will actually have obviously part of the city of the
24:10 and the suppression of. So you also see it in this little two
24:16 version here that you have a function is so this is kind of a
24:21 and this is a one. So can see that coefficients in this whole
24:26 expression. Then in fact it becomes on the function. So in that
24:32 you can have two. The construction's the polynomial without knowing function values.
24:38 this case you can construct the polynomial you need the function. That is
24:42 they want it. Okay. Not . This is right. So this
24:52 what I said already. Mhm. So now I guess it was the
24:59 we're talking about that obviously the ground works and this was just well these
25:09 and almost Sorry? No, I ah these are what these panels are
25:17 and they are constructed in such a that one there at the corner of
25:25 for the same. No, to the index of the polynomial, then
25:32 one and all the other notes it's and maybe let's let's go back.
25:43 . It's always said So again this kind of zero and then you can
25:51 that it's a plug in ah Say one and the first one that becomes
25:58 And the X. one and the one that becomes about. So if
26:03 want to evaluate this next one and got to talk to me. So
26:09 just going to see that depending on term you have in this corner of
26:18 Is either zero or 1. The interpolation points. So they've got for
26:28 words is sincere. So when you 100 this expression and one of the
26:38 points, the access for which quantifying . It's only one. It is
26:49 . but it is at the east whole point and then the function then
26:55 the value is one. And so the plus P of X. I
27:00 too um the function value or the values, we wanted to interpolate exactly
27:08 that. And if you then plug a different thanks value Then this is
27:14 . So basically, yes. Um term in the summation is only equal
27:22 the function bezel. Um at the you are safe in simple.
27:34 Okay so yeah it's on this machine plug in any pleasure for ex um
27:47 is fine. It's all about an ah for the east polynomial most firstly
27:56 -1 terms because to take the distance the I don't know to respect each
28:06 or the other ones but not to because that will be division zero and
28:12 them from the nominator you have X the distance from X to the respective
28:20 contextual 13 discipline. So it's easy see regarding what thanks noble values.
28:30 plug in. The only point where again it's it's not equal to
28:37 I. Then. And one of Factors in the expression is going to
28:44 zero. So pickaxe through our X and visible on the screen and this
28:52 be zeros of the whole product. it's true for any one of their
28:57 except thanks bye. Because except hiring not present in any one of the
29:06 . So that's about it. Except then all the factors a lot.
29:14 that's the worry one. And it a disease that this is oh and
29:19 of the cardinal doctors is only one one of the nodal points and
29:28 Okay so mhm. So there's some what I said and then I have
29:34 complete example of this a little bit it looks like the different forms of
29:41 . So zero make it completely depends the next slide. So and the
29:53 you have a different factors and we're down all this way. Her
30:02 That was the first one was like was my smart. I think so
30:09 was. And the -5. So mm hmm. Ex miners and Sarah
30:22 mm hmm. Right. Right. I looked at the slides before the
30:30 but hopefully mm hmm. The restaurant . No. Anyway, I'm sorry
30:45 got confused now no one can help . This is supposed to be ah
30:56 somewhere now for risers for the four points to expect the ones we are
31:05 looking at. And then what's already confused at this point. Yeah,
31:16 right. Okay. You're buying Right. Right. You can start
31:22 the back. So this is the one. The next one is one
31:25 then six points. That's my next . And so they exclude the first
31:31 . So that's why it's for Zero. We excluded zero points.
31:36 they start with the first one. . So that's what it is similar
31:40 is to go to um the ones enterprising there um Second Novel Point Sort
31:48 X. one. We have it's here which is finest X. Zero
31:53 X. 01 minus one plus Go on with this. Just
32:00 There you go. A little point which the paranormal is supposed to be
32:05 question. Yeah. So so anyway another thing to look at and that
32:15 come back to the center point is this case the corner polynomial. Czar
32:22 um particularly to him. But you see some of them are kind of
32:30 So they became other ones that are hostile territory but they're always down there
32:35 plus and minus one and we can that that's the case it seems.
32:43 they have bounded isolation even though between . So that's one other good
32:52 The cardinal points. So how many ? So it's fairly straightforward how to
33:01 them. The good point again is the only depends on the nodal
33:09 It does not but then on the violence just gives you a way of
33:15 together in some sense the functional value pondering at some point between okay points
33:25 approximation. No no no. Okay in America example now three points.
33:35 see if we can write the So Again start with zero. So
33:42 just um The distance to the X and X two. So X one
33:50 minus a quarter Or -11 is the and X two is the one.
33:57 it's and then in the denominator you that's zero. Um Two. It's
34:05 song and it's zero and it's one next zero. And it's two from
34:09 against you. The government expression a cardinal polynomial that is designed to
34:20 And the function value at zero And the next time not paranormal is to
34:26 the function at X one. So under this will have x minus X
34:32 and x minus X two. And the correspondent thinks the denominator and the
34:39 between that. Thanks for X one should say in X zero and X
34:48 and X two. We got a one and similar To the other
34:54 So now you can write it So these are the three corner
35:00 Is this for capture the function value functions in the function value attacks
35:06 And they have these expressions and now can write down the whole pulling normal
35:15 . So four the function value Um that's gonna record another polynomial for
35:26 zero X one X two. And It's obvious so that this kind of
35:32 a look at the functional value was . And function earlier is too so
35:37 what it takes -36 and constant transfer value seven. I think that the
35:46 ones and it's just trying to straightforward too. Yes or no, probably
35:53 . Well That exactly matches the function of these three. I gotta go
36:00 . Okay. Mm hmm. So now the next thing I want
36:08 talk about community formulation. I want right. So one. Oh yes
36:19 the book, come on makes It's kind of incremental. So it's
36:26 like started the lecture things Just one . But it's a concept and then
36:32 points and the strength time and three will probably get a second order polynomial
36:39 it happens to be kind of So this is basically the kind of
36:45 the curse is broken and that the of the polynomial. So at some
36:50 you have a calling on your friend and then you have one more.
36:54 for this expression suppose you have a organization just it's the perfect fit thing
37:01 k points. Take plus one points and you add one more point and
37:08 you can that's the right thing done you have the following normal you have
37:17 it and then you add the new , that is basically taking the new
37:24 whole point. But if it's k one and its distance to all the
37:30 total points. And if you plug the value now on the polynomial at
37:40 no notable point, then it should the new in population at that
37:46 That is okay. Plus one. because so this polynomial yes, basically
38:03 preserves the property of perfectly matching all previous points that you have. Because
38:13 if you replace in this equation Xscape was one where any one of
38:20 previous no level points. one of terms in this one of these factors
38:28 this term is going to be So for any one of the bridges
38:32 points. Ah this extra term is to be zero. So basically says
38:39 interpolation property. Always construction work that done before adding this new point is
38:46 . And if you don't make this stick then obviously your computer. See
38:52 this is the question is true Then does seem to police 20.6 outfits.
39:01 I think that's what I guess just to say. I'm just that
39:07 So our business is business fallen on for the party. A simple example
39:17 how to do this now. five points. So they can do
39:25 and they didn't success the white soul this incremental or inductive way that's We
39:31 from one point. So the polynomial is a constant Then it's best to
39:39 because They wanted for X0 To be . It's actually for any extra minus
39:46 . And that's certainly true for the points That stuck with me now.
39:52 that was produced and now we want include one more point. So then
39:59 form is it has the previous polynomial some suitable constant. Times the distance
40:05 the new point through the previous interpolation this case. Now we know what
40:13 is. That's mine, it's And now we have the condition to
40:17 also the new points. So somehow one x equals one Day one at
40:26 was one is supposed to be my tsunami thing was plugged in and this
40:31 so I've got to write it in opposite order was minus five plus
40:36 Times X equals one. Then the should be three. So how do
40:41 have an equation? It's just is the computer see then we now have
40:48 pulling over the congress to park and we just keep repeating that I
40:57 So it can start from to you . That was the thing we have
41:01 we want to The next point x two. So now we have the
41:05 polynomial. Now there are the term then this one and then in terms
41:10 the factor is the product of the between them. And then um Kalla
41:17 or the X values too. Each of the previous interpolation. That was
41:22 the factor because this guarantees that this disappears for any one of the previously
41:29 points. So in this case for see what plus You're plugging in at
41:37 . And it's one and the expression zero or zero. So that just
41:41 the X. And the next time the X. One this one.
41:47 now we have also the condition and new polynomial P two should have the
41:53 minus 15 if they can gain XP minus one. So now we have
41:59 conditions are not known to see from . Yes. And so we can
42:03 it in and it should be solution minus force. And now you have
42:08 Newport. We also know that in the degree increases by one each
42:16 But now we have three points of ingredients two years before the at
42:23 Then the number of points for And that means you keep the
42:30 All right. And I don't do the steps but testimony is all said
42:35 done and I have five points the order polynomial. And that means you're
42:41 . And it turns out to be will be specific to the fourth.
42:50 so but this is one way right the hole. No money. You
42:57 write it in this kind of honor nested formal talked about turning up uh
43:03 then just evaluated. That's one way you can also Right then. And
43:11 other negative forms of the form is necessarily but it's the same principle.
43:18 the same polynomial. Then call this . You picked value of this polynomial
43:26 in this way or this way or way is all the same. It's
43:31 different way of writing it. But principle it's the same for you know
43:38 it may not be obvious but they where to take this expression for instance
43:45 unravel. All the multiplication is hidden the parenthesis and collect the terms of
43:53 powers of X. You got an that looks like this. So this
43:58 kind of perhaps symbols. Common way writing the polynomial, different powers of
44:05 with the different topical efficient for And if you have it in this
44:10 can sort of fairly easy to see this this expression is a nested version
44:16 this 3X. That comes from the order and then you have the
44:22 So that means this expression is multiplied X. So the third to get
44:26 right answer. Yeah, they are . And then The -70 seconds or
44:32 , you know, I didn't tell in terms of so just again,
44:38 ways of right things all along. is values for exactly the same.
44:50 hmm. No. Now the next of clients are totals in some different
44:58 . Okay. But I think the is actually following this Warner's idea and
45:04 more complicated his shoes and have to they get the coefficients of the book
45:14 the standard procedure to and remember from newton formulations, the coefficients are dependent
45:20 the function the construction that was So, um, also, so
45:36 was just the general form. It a previous slide and one can essentially
45:44 do it step by step. So can start during the evaluation that was
45:56 some of the previous slides and get one of the conditions and use it
46:02 get more and more of the conditions or evaluated in the nested form once
46:08 have the coefficients best unravel it from innermost expression in his domestic point that's
46:19 we did early on in her previous . Um one thing that they will
46:25 back to ah towards the end of class today if I get there are
46:30 ones next time is if you look now this expression that is basically multiplying
46:39 coefficients. Um so this is the of does not have the function values
46:47 it. Um They they have behaved differently compared to the cardinal pulling owners
46:55 the laboratory expression. So as I commented by one of you more
47:03 Right so the higher order polynomial you see that it's got to be
47:07 Ah that's so nicely. Okay substitution up the lower order ones are I
47:14 behave but the highest degree then they become difficult to be able to potentially
47:20 America what? So this is just simple routine as domestic evaluation center.
47:32 . To the same in terms of homeowner So now the thing that has
47:39 principal also fairly simple just following the that they actually live in the examples
47:47 at one Point. Ah that is because of the the constant districts afterwards
47:55 there function value you want to approximately first point and once you have that
48:01 can move on as they did they one more point. Now you know
48:04 one and then you can evaluate the from the next equation and jim
48:12 So um brother mhm. To you kind of have wherever you are
48:21 so this is something that you have previous conditions and figure out this
48:27 take the new functions. Uh There's difference basically everything except the new coefficients
48:36 you're trying to value. So this is done and moving to the left
48:39 side and then divide by this thing and then finds out the partition.
48:46 that's kind of one way of doing . And there is another way that
48:53 described in the book, has certain that will use later on. So
49:00 something um Yes, yes, I so. It's mhm Yes um known
49:13 divided difference and that's the and describing idea of the differences. Obviously this
49:25 that these coefficients and they don't polynomial kind of food as a function of
49:37 the previous or previous including the current point. In fact both the local
49:46 the X values but also the function is remember this is supposed to
49:51 So this just stresses the facts and and the general notation instead of having
49:58 proficiently doesn't display just simple that doesn't the dependence on nodes and function values
50:09 that use this kind of notation instead shows how these professions is actually dependent
50:17 um all the interpolation points that you been working on until that point.
50:22 that's what I'm going to be seeing some of the coming slides. Um
50:29 . In terms of the simple it's A K. But it's also
50:34 dependencies in a different way. Alright, bye. Okay.
50:43 Okay. And this notion of dividing on the bottom side until they come
50:52 . But I think first of what call the direct evaluation, which was
50:57 what was used in the previous So now take the successive points and
51:04 before day zero was 3. The equations 13 and takes a constant
51:11 Next turn ah uh expand. So difference between these two which is a
51:20 of five. So that's what for proper science. So just keep going
51:24 you get a one a zero a a two. And now in this
51:28 of organization that meets That one is and one Class. Now with two
51:34 1 and four. And the coefficient a two was a one was punished
51:43 . And then we have the state symbol to get a three and a
51:47 or a two, sorry, and was etcetera. So it's just linking
51:53 notations. Very complete example. But same procedure as before. Ah
52:05 Okay. Well, yes. So think the only point of this particular
52:11 . Yes, essentially. Um looking the product in disease and in this
52:21 it may be the case of the index is lower than lower index of
52:27 . Starch. The world starts on standard assumption that the survivor, I
52:34 basically what? So it doesn't divide into service. And that was the
52:41 of do we agree that it's this the product of and one is one
52:48 so against agreed again increases by one every no point. So I'm going
53:00 let's see what's going on with this . Okay, so now the soviet
53:06 of divided differences. So here's something a one uh first of 080.
53:19 the next term is to compute a Therefore X 1 0. And
53:27 but it is getting these two nodal . Yeah. And a zero was
53:33 fact that felt like sarah. so in this sense, what part
53:39 beginning this notion of providing difference as is basically expressing the slope of the
53:47 between 2.60. They forget zero on . And but so for me
53:58 it's kind of nice late there aren't want to talk about association.
54:07 I can think of this again. just, it's the second they can
54:12 to me about it and they will be used for food finding or otherwise
54:16 approximation of the derivative as a So that's kind of the difference gives
54:25 something of emotional stop. Mhm mm , okay, that's what's that.
54:34 then it's been out and go back say, oh come forward. And
54:39 think the next to his expression then can rewrite it. So manipulate this
54:48 and that's what they have. This The best for the slope of the
54:54 between these two points 10 0 and their corresponding function values. And this
55:01 now the slope which means ah X and X two solvents. Thanks for
55:08 space stone, first between the First points and then take steps on between
55:13 2nd 2 points. And then So that's the difference between the two
55:19 down slopes And then divided by the now between the two endpoints. So
55:29 it's that's why this notion again divided you take in this case difference between
55:36 slopes between the neighboring segments if you if you were to have straight line
55:42 and then take kind of the average these. So this is also why
55:52 about America differentiation and approximation of This is kind of the way also
55:57 time construct an approximation of the second derivative of the curvature. So that
56:05 like to me it helps a little have intuition of looking at interpreting what
56:13 kind of this question tried to model capture. So, so this is
56:26 I guess I didn't manipulate this expression everyone wants to follow what I
56:30 I have to come from one to thank you. So this is something
56:37 they have now trying to do this I guess is what's going to happen
56:44 . So I started lately that was using this petition together square brackets is
56:51 and points On interpolation. So that just one and the next one is
56:58 jam packed two points of one under and that was going to stay forward
57:04 didn't have before and we get a that's the famous. Mm hmm.
57:11 then the next one here I guess just keep doing it the same thing
57:16 the expression that was derived in terms the divided difference notion and yes,
57:26 eventually To this now 8-7 and mhm , hopefully. So when we did
57:39 more direct version and not trying to it uses divided different notion of deriving
57:47 . Um as you can see when get the same results, it's a
57:50 way of getting to it that disposes a little bit more of properties of
57:58 points are related to the subject because notion of first and secondary looking for
58:04 trying to tell the truth curvature and this blindly thinking about the point should
58:11 corrected. Mhm I don't see what is more no. Mm hmm.
58:28 . Um that perhaps shows a little again the defendants and the expressions of
58:36 is Yeah and right, zero Day which is like a two with the
58:45 for the truck. Um so let's . So this is. Yeah.
59:00 so if one does start this expression ah Finding the coefficients from 1 to
59:12 other. Ah yeah. In fact is. And to show that it
59:20 on this particular form. So a bit talk about of the figure a
59:33 . This is certainly in that case was just looking at the slopes between
59:37 adjacent segments and then taking Kind of average between the endpoints zero and 2
59:45 that face. So the broadcast the of these functions, You can see
59:54 differences here that shifted by one. They started and they started zero.
60:02 it's kind of intuitively communities But taking two segments but now they're over that
60:08 but they're shifted by one and then kind of the average by the lion
60:14 the total distance covered by all the that's the way you get. Can
60:23 coefficients? So, so if you one of the proficiency investments, take
60:30 difference between these two proficient and divide transformer. Mm hmm. What?
60:47 . and uh I guess one this one and that kind is used in
60:54 of it. So I said, is that? And made the comment
61:01 on today. And there was that very first example of X0 and two
61:05 in a straight line between and I but this and do it In increasing
61:15 of importance. six zeros. That there was 1.25 and 1.4 and then
61:21 x four tasted before the X mm One that was 1/25 cents X zero
61:29 zero. And the lower right hand X. One Y. One Closer
61:34 the X. equals to zero. the straight line between the two doesn't
61:39 on which order it happened. So is just trying to explore that which
61:46 you happen to label the points. should get the same and that's what
61:52 invariants trying to say the the procedure all of these things. That's the
61:58 thing. Regardless of they have put labor reform and what's so I'm not
62:12 . So what's the difference formula and example here. And then I think
62:18 have it actually kind of a real of how to do these things.
62:24 we need to do this divided different . Weren't convinced to happen. Find
62:29 simple steamer and like the council and of these point is historic kind of
62:41 four. And such other points. then you have the corresponding function
62:48 So in this case yeah you can with them all the interpolation In this
62:56 like 0 - 60 in the correspondent values. And then you take this
63:01 differences. So now you take them wise and bear wise and then they
63:05 to the next column that's opening up difference between events and divided by the
63:12 range of all the time. So just what I'm going to do.
63:19 , So basically we got 30 this . You use these two points.
63:24 get the next one here. You're to use these two points. That's
63:29 . And then you keep going the of this to have at least divided
63:34 and invested the slope between reversal The next one we want to try
63:39 get one more to get captured curvature then the best can take Again,
63:46 adjacent shifted by one and the They won't have the distance between the
63:53 points. And then again in the months from now the best they have
64:00 the coefficients in the polynomial company. than this for the simple steel.
64:11 , so that's what sin. Ah , no. I think that being
64:21 concrete example. Next let's stop and if there's questions. So, so
64:34 see first. Not using expressive. , let's see what I am.
64:47 we have the difference right between um hmm. Function value as on the
64:56 points. It was zero and So this is uh, function function
65:04 . And the distance between them was . Um Sorry, it's um
65:15 and nine. So I'm just gonna retracted it and then you take success
65:20 neighboring points and the distance between It was zero and three halves.
65:26 this one and then another 10. then you move on to trying to
65:34 the bracketed ones the next. So basically the difference between these two
65:40 comfortably used um to this one and one and the extreme points industrious.
65:48 you don't have to And similar for next one is between expanding nicely.
65:57 you guys can keep doing nothing for head on ingredients side. Yeah.
66:05 now let's they want one has the is actually going to use and the
66:14 for the hard work. I thought would have been the next slide.
66:18 this is just taking what the yeah that these were here. The function
66:25 is on the table and this is using the schema, I don't know
66:33 on the previous slide and these are this chapter once and then you can
66:40 down so this is a gun. coefficients for this is the constant.
66:45 is a proficient for the first longer . That was Yes, it's Sarah
66:50 that's one and that is the next order ah coefficient in the new Kampala
66:58 . All shows up on the diagonal . So, but in order to
67:03 the diagonal the investor needs to and back and have the kind of triangular
67:10 and figure out what's included in order get those points. So what kind
67:16 property from the left in this then I had it. So then
67:22 have it and then you can write , they got an expression for the
67:29 , but it's just sort of divided . If one has this kind of
67:34 sets up its service simple and straightforward figure out. You gotta take the
67:41 between the function values and the distance and all points for those function values
67:49 uh yeah uh to progress basically between points, the first one is the
67:56 value itself, The next one that on the difference between these coordinates and
68:02 fairways. And then they get to one mr defected from the students and
68:08 for the distance from the holding. , by construction. So the order
68:16 which you happen to write down is don't have to order them in terms
68:20 you know Professor the increasing or decreasing family, they can write them down
68:26 several water and it still comes Okay, so um now this is
68:41 the sooner called it down there and it. So so this was,
68:49 the doctor has a pseudo code to the coefficients based on adoption values.
68:54 And then earlier we talked about the of using this next the corner first
69:02 have to complete proficiency and then they for whatever experiment they want, right
69:10 . And this has pretty much been I've said three things. Oh,
69:19 let's see um yeah, I'm gonna kind of an example here. Any
69:28 so far on this. Sure, little bit more complicated. Again,
69:33 cost function. Um the construction involving the trunk and now it's an example
69:46 computing an approximation of the same function a polynomial. And then this gets
69:54 . Let's and so the idea is you for this example that you introduced
70:02 number of points um or I understand Points between zero and 1.6875 that they
70:16 to have the exact representation. And we kind of tried this. That's
70:21 good is it by looking at what's difference between the same function the polynomial
70:29 other points than the interpolation points. I picked a few points between the
70:34 points and have a look at what pulling over value and what is the
70:38 value that gives you something? They estimating the error in the approximation for
70:44 points of the population. So remember supposed of the bell curve in the
70:52 of the lecture And and the purple isolated the lots in that case the
70:57 between the polynomial and actually function value some places were quite dark. So
71:05 is kind of a standard checked when tried to figure out the paranormal.
71:08 this was something that says that this looks at it from this place for
71:14 example that we're I guess the point between each barrel insulation points of
71:21 the four times as many the violation as interpolation points, it turns out
71:29 not every loss one of these That was not too bad. Just
71:36 single position influence in terms of divided the test points. If you like
71:48 three points in between the interpolation it doesn't mean that any one of
71:53 three points was actually one of the for if you have taken 100
71:59 they would have not gotten a larger different sampling in this case someone needs
72:08 , mm hmm. Okay. My is never get a chance to talk
72:15 that. So this is the saying the thing I'm trying to say and
72:19 is over. Is that the different way you're right velma in a sense
72:26 mean it's a different falling over in sense that regardless of what was written
72:32 , if the value is to the value requirement. But any X values
72:38 about this. The forum looks But argument seems to a polynomial to
72:45 the same boring America approximations and around . If you have infinite position you
72:56 get this thing and this is what says. And it's just a simple
73:01 . The argument is supposed they were different for the same interpolation points.
73:06 will be their friends and okay, they may actually difference. But it
73:13 if you look at taking the difference these two points, it's the agreed
73:20 difference is also having The same degree either one of the two. But
73:26 also means that this polynomial, The polynomial is zero, it's all the
73:34 . And since any paranormal cannot have roads are busier and more points ah
73:46 the I guess one degree less of paranormal degrees. So that means effectively
73:51 forces the coefficients in the polynomial is in the proper way and to be
73:56 same. So because of the number routes, because the difference polynomial is
74:01 . As all the speculation points, forces us to the pediatricians in the
74:09 up a lot of sense. And I guess, I don't know,
74:17 just have this as into two the lecture. So I just said in
74:26 beginning of all enormous are by no the only functions one use. So
74:31 general what problems to do is to some combination of some as mentioned basis
74:40 . Just fired on the stones. there's some linear combinations of things 19
74:47 maybe approximate or it perfectly represents a of points like we did today.
74:56 was trying to have the funding over equally. They function by the closing
75:03 points. Um but one and potentially to pick any other type of function
75:16 gave the example of the compression. You're better anyone else or if zero
75:26 pretty much almost identical to the function . Um That for the rest of
75:32 thing as a restaurant and you get basis function. So if you have
75:39 example where they sine function. So if your function F is in
75:48 trigger event of functions sine and cosine your basis function happens to be signed
75:53 call sign, They only need one function and the coefficient just needs to
76:00 the magnitude and face right except on head. And then they have
76:06 It takes quite a few terms to the variable approximation of the project and
76:13 can get it. Take a serious gives you an idea of what the
76:17 you need and the polynomial. But the places function wouldn't is that it
76:22 a trick function and a couple which just been treatments. So so normally
76:31 ends up doing it. The general we have a bunch of coefficients then
76:41 to the value of the basis activities, population points ends up being
76:46 matrix arrived in the question for each of the double points um correspondent function
76:53 unpardonable points. So then to find professions, you have to solve the
76:58 of equations and I will talk more that lady And here's a little bit
77:09 in principle I can think of the as the basis functions being what's known
77:15 the more no meals. That is various parts of that. Mhm So
77:21 have first the constant and x itself X square. So these are the
77:27 basis functions Uh That was effectively used I was just calling over approximation.
77:36 here we can see the behavior of . Well, normal is only into
77:40 became 11 very nicely and some not much and couple more minutes. So
77:50 in this case if one looks at Manami als um So here's the questions
77:56 Uh constant, the first degree 2nd so on. And for the different
78:02 points and the correspondent, its values you get the matrix of this type
78:07 is known as the Mhm. Then mold matrix. And it turns out
78:11 that's not something you really want to with. Unfortunately the constant tends to
78:16 that this matrix is very conditioned sort things. Getting numerical accuracy in doing
78:21 this way. It's not so And we'll come back to that in
78:27 future lecture. So there's something for . And so here we saw the
78:33 Poland almost. That is also true but it's about the oscillation and that's
78:40 true for the newton. And then all males in the interval is starting
78:44 headbutt. Not so good. Next we talk about other basis functions and
78:52 want to get a drink functions. that's when you're on the other polynomial
78:59 nice behavior. seven. That's two great questions. Okay, thank
79:19 so much. Thank
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