1 00:00:00,120 --> 00:00:05,939 So previously we have seen that we can actually fit all data using blinds. 2 00:00:06,570 --> 00:00:13,170 And actually, we used polynomials, blinds and these polynomials were of the cubic order. 3 00:00:13,170 --> 00:00:15,900 So the highest term was access to the power of three. 4 00:00:16,500 --> 00:00:22,440 And you see, when we have these individual points here, they are all connected and they are also steady 5 00:00:22,440 --> 00:00:23,940 in their derivative. 6 00:00:24,060 --> 00:00:28,110 So it's a very smooth function and every point is fitted perfectly. 7 00:00:28,920 --> 00:00:35,550 And also, I told you that this is sometimes not what we want physically, but still it is a nice result, 8 00:00:35,550 --> 00:00:36,000 I think. 9 00:00:36,570 --> 00:00:43,830 And here I'm going to show you another method how we can accomplish this, and we will not distinguish 10 00:00:43,830 --> 00:00:50,040 these individual segments here, and we will not try to fit third all of polynomials for these individual 11 00:00:50,040 --> 00:00:54,840 segments, but instead we will fit a single polynomial to all the points. 12 00:00:55,770 --> 00:01:03,030 So you can imagine if you have, for example, 10 points or in our case, I think it was 20 or something 13 00:01:03,030 --> 00:01:09,540 like that, then we need a very high order polynomial to really fit all the points because we need many 14 00:01:09,540 --> 00:01:13,050 minima maxima and it's pretty difficult to fit this. 15 00:01:13,800 --> 00:01:18,060 However, from a mathematical point of view, it's actually not so difficult. 16 00:01:19,020 --> 00:01:25,470 And also, let me tell you that I created this lecture later on after I have created, after I've recorded 17 00:01:25,470 --> 00:01:31,800 even the whole section because I thought it is a nice additional information and especially it introduces 18 00:01:31,800 --> 00:01:35,220 to you another method that's very important for computational physics. 19 00:01:35,610 --> 00:01:39,300 And this is solving a system of linear equations. 20 00:01:40,290 --> 00:01:41,280 So let's get started. 21 00:01:42,030 --> 00:01:46,710 The first thing that we will do is we will reduce our number of points. 22 00:01:47,250 --> 00:01:48,700 So this is just for simplicity. 23 00:01:48,720 --> 00:01:54,780 So because I'm writing down here some equations, and I don't want to make these equations very difficult, 24 00:01:54,780 --> 00:02:01,530 so I will just reduce the number of data points and I decided to use here all the components. 25 00:02:01,530 --> 00:02:10,680 So this will be X and Y, and I will only show you these seven data points and we can even look at them, 26 00:02:10,680 --> 00:02:11,100 of course. 27 00:02:11,730 --> 00:02:18,150 So they will be in the range from X minus one point five to one point five, approximately. 28 00:02:18,630 --> 00:02:20,970 So these will just be these enough points here. 29 00:02:21,840 --> 00:02:27,960 And the reason is now we have seven data points, and this means if we want to fit a polynomial, we 30 00:02:27,960 --> 00:02:32,130 need at least a 6th order polynomial for a perfect interpolation. 31 00:02:32,760 --> 00:02:38,670 So this is because our our sixth order polynomial has seven coefficients. 32 00:02:39,000 --> 00:02:41,220 So we have a zero, which is a constant term. 33 00:02:41,220 --> 00:02:46,170 And then these six other coefficients in front of these x x square and so on. 34 00:02:46,920 --> 00:02:52,950 So this means we have seven degrees of freedom, which we can tune and fit to our data points, and 35 00:02:52,950 --> 00:02:54,180 we have seven data points. 36 00:02:54,660 --> 00:03:00,990 So you see, it's really a differential equation with seven parameters that we have to determine and 37 00:03:00,990 --> 00:03:02,030 seven equations. 38 00:03:02,040 --> 00:03:04,510 So it really makes sense to do this. 39 00:03:05,670 --> 00:03:12,060 And we also note that our polynomial here is supposed to give us the correct value for all the data 40 00:03:12,060 --> 00:03:17,750 points which have to coordinates X one or X, Y and Z. 41 00:03:17,760 --> 00:03:23,430 And why are you and I goes, of course, from zero to six so that it covers all of these data points 42 00:03:23,430 --> 00:03:23,680 here. 43 00:03:24,210 --> 00:03:27,000 And this means we have seven of these equations here. 44 00:03:27,930 --> 00:03:30,980 And um, these are the equations here. 45 00:03:30,990 --> 00:03:39,900 So you can start writing why zero is equal to A0 plus A1 times x zero plus eight, two times x zero 46 00:03:39,900 --> 00:03:41,100 squared and so on. 47 00:03:41,520 --> 00:03:43,980 And this is what they did here in terms of matrices. 48 00:03:44,910 --> 00:03:54,330 So you see here for the first line, we have why zero is equal to one times a zero plus x zero times 49 00:03:54,330 --> 00:03:58,780 A1 plus x zero squared times a two. 50 00:03:58,800 --> 00:04:03,480 So these aren't really all these terms that I have just mentioned to you and we have seven of these 51 00:04:03,480 --> 00:04:04,080 equations. 52 00:04:05,250 --> 00:04:14,010 And so this is a matrix equation, and this means we can now basically just define these individual 53 00:04:14,430 --> 00:04:17,790 things here as a vector, as a matrix and as another vector. 54 00:04:18,510 --> 00:04:24,270 So right now, it's not really clear why I do this, but we will use this in a second for an umpire 55 00:04:24,270 --> 00:04:30,540 routine, which helps us solving this system of linear equations very quickly. 56 00:04:31,140 --> 00:04:36,150 So for this routine, we need the input as a matrix, and these matrices are exactly these that I have 57 00:04:36,150 --> 00:04:36,930 written down here. 58 00:04:37,980 --> 00:04:44,100 So we begin by programming this matrix here, which I call X, because all the terms here are X. 59 00:04:44,670 --> 00:04:51,600 And actually here you could also write x zero to the power of zero x one to the power of zero, because 60 00:04:52,410 --> 00:04:54,510 something to the power of zero is always one. 61 00:04:56,400 --> 00:04:59,490 So we can write that x of a whole matrix here. 62 00:05:00,020 --> 00:05:05,360 Is essentially an array, so MP Todd Parade. 63 00:05:07,580 --> 00:05:16,120 And this array basically consists out of our data points, so Datafolha, but only the X components, 64 00:05:16,130 --> 00:05:25,910 of course, so we write zero and then all the data points like this and then the first term, as I just 65 00:05:25,910 --> 00:05:27,860 mentioned, you could write to the point of zero. 66 00:05:28,130 --> 00:05:32,570 You could actually write also an array of ones, but I will do it like this. 67 00:05:33,020 --> 00:05:36,410 And then the second entry will be the same thing to pull off one. 68 00:05:36,770 --> 00:05:40,040 And then we continue until we are the power of six. 69 00:05:40,670 --> 00:05:44,960 So of course, you could do this more elegantly, but this is just a quick practice. 70 00:05:44,970 --> 00:05:49,790 So I think it's OK if it just do it by hand and doesn't take too long. 71 00:05:51,050 --> 00:05:57,830 But of course, if you would take here 200 points and you would need a polynomial of order 200 approximately 72 00:05:58,460 --> 00:06:03,440 actually one hundred ninety nine, then you would have to write down your many terms. 73 00:06:03,440 --> 00:06:07,610 And then it, of course, would be better to just automate this and not write it down by hand. 74 00:06:08,510 --> 00:06:09,710 But anyway, here's the key. 75 00:06:10,070 --> 00:06:13,550 So this is not our array, and actually we have to transpose it as well. 76 00:06:13,830 --> 00:06:16,970 It's all right and p dot transpose. 77 00:06:19,330 --> 00:06:25,690 And then close it in the end, and maybe we can make this look a bit nicer. 78 00:06:29,650 --> 00:06:31,540 Like this, for example. 79 00:06:33,130 --> 00:06:33,480 OK. 80 00:06:34,600 --> 00:06:37,840 And I will do it like this. 81 00:06:39,130 --> 00:06:39,460 OK. 82 00:06:39,520 --> 00:06:40,230 So I hope you OK. 83 00:06:40,250 --> 00:06:44,620 You see what I did here and I can run this now and we can have a look. 84 00:06:46,690 --> 00:06:53,260 So you see, it's now an array and you have here these individual terms and you see the first column. 85 00:06:53,260 --> 00:06:57,160 So first entry of each of these rows here is always one. 86 00:06:57,490 --> 00:06:59,290 So this is exactly what we have here. 87 00:06:59,590 --> 00:07:05,500 And then for the second entry, we have just the X values of our data, which are these values here 88 00:07:05,500 --> 00:07:07,070 and then so on and so on. 89 00:07:07,180 --> 00:07:10,210 Squared cubed two are four or five six. 90 00:07:10,900 --> 00:07:16,720 So now we have two Matrix X and we want to determine the Vector A with the individual parameters here 91 00:07:16,720 --> 00:07:17,830 a zero to a six. 92 00:07:18,370 --> 00:07:25,810 And of course, we also need to program here the vector of Y, which is basically just this one here. 93 00:07:26,320 --> 00:07:27,790 So that's very easy to program. 94 00:07:28,090 --> 00:07:34,870 Y is equal to data pulley, then the Y component and all the data points. 95 00:07:37,020 --> 00:07:38,820 OK, so then I wrote. 96 00:07:38,880 --> 00:07:45,150 Now we can solve the system of linear equations and linear means linear in a. 97 00:07:45,630 --> 00:07:51,930 Of course, it's not linear in X because you see we have here squared cubed and so on, but it's linear 98 00:07:51,930 --> 00:07:52,350 and eight. 99 00:07:52,920 --> 00:07:58,560 So you see this matrix here, even though you have here squared part two three four five six. 100 00:07:59,100 --> 00:08:04,680 These are all just numbers, as we have just explored when we had had a look at this because these are 101 00:08:04,680 --> 00:08:11,250 really points coordinates and when you calculate the force power of them, they are still just numbers. 102 00:08:11,880 --> 00:08:18,330 So this is here a matrix of numbers, and this is here actually the parameter for which we try to solve 103 00:08:18,330 --> 00:08:19,140 these equations. 104 00:08:19,770 --> 00:08:24,810 So the equations are really linear in the zero eight, one, eight two and so on. 105 00:08:25,500 --> 00:08:31,050 And of course, I think you know this already from high school, you learned, is actually very early 106 00:08:31,050 --> 00:08:36,030 in mathematics how to solve such linear equations or a system of these linear equations. 107 00:08:36,390 --> 00:08:41,760 And a typical way to do this is to use Gaussian elimination methods. 108 00:08:42,030 --> 00:08:44,990 And if you want, you can really program this whole method yourself. 109 00:08:45,000 --> 00:08:50,430 Just remember how you did it in school and then just program this using loops, for example. 110 00:08:51,060 --> 00:08:55,200 But of course, there are also more elaborate methods to solve this. 111 00:08:55,650 --> 00:09:00,900 And if you write it down in terms of a matrix as here, then typically in computational physics, one 112 00:09:00,900 --> 00:09:06,350 uses these so-called l.u decomposition, which means lower upper factorization. 113 00:09:06,360 --> 00:09:14,340 So basically, we try to express this matrix as a product of a matrix that only has a lower triangle 114 00:09:14,910 --> 00:09:16,200 and an upper triangle. 115 00:09:16,890 --> 00:09:19,830 And this leads us then directly to the solution. 116 00:09:20,730 --> 00:09:25,320 So I don't want to go into the details here on how this works, because in this course, we don't really 117 00:09:25,320 --> 00:09:28,530 need to solve systems of linear equations. 118 00:09:28,530 --> 00:09:32,850 But for some problems, a bit more simple problems, it can be useful. 119 00:09:32,850 --> 00:09:38,940 So if you're interested, please just look on Wikipedia, for example, to decomposition, and you could 120 00:09:38,940 --> 00:09:42,450 also try to program this Gorshin elimination method yourself. 121 00:09:43,290 --> 00:09:48,460 However, as I said, we will use here and numpy routine, and it's actually very easy. 122 00:09:48,480 --> 00:09:51,990 It's just and offline arc dot solved. 123 00:09:52,470 --> 00:09:56,970 And then we provide the matrix this one and then the left hand side vector. 124 00:09:57,750 --> 00:10:06,450 So just like this and we want to start a solution which is a vector assay, and now we have already 125 00:10:06,450 --> 00:10:07,080 our solution. 126 00:10:07,080 --> 00:10:10,380 It didn't even take a tenth of a second. 127 00:10:11,280 --> 00:10:14,550 So we can now look at the solutions and these are the solution. 128 00:10:14,550 --> 00:10:18,720 So a zero is 13, a one, two, four and so on. 129 00:10:19,770 --> 00:10:25,140 So we don't really know if this makes sense and if this is correct, but we can, of course, have a 130 00:10:25,140 --> 00:10:26,640 look by plotting. 131 00:10:27,480 --> 00:10:33,690 So let me copy here a few lines of code that I have prepared previously that you can, of course, also 132 00:10:33,690 --> 00:10:34,200 write down. 133 00:10:34,620 --> 00:10:41,160 So you see here, first thing is an X label and then here a specified a limit of the year of the Y axis 134 00:10:41,160 --> 00:10:44,160 goes from zero to 30, as we have learned before. 135 00:10:44,760 --> 00:10:50,700 And then I have a plot similar to our previous plots like this one, for example, very a specified 136 00:10:50,820 --> 00:10:51,510 X list. 137 00:10:51,960 --> 00:10:57,750 And then I plot the function here and then I plots our data points. 138 00:10:58,320 --> 00:11:03,000 But of course, I have adapted this to our current problem, so I have changed these data points to 139 00:11:03,000 --> 00:11:03,300 data. 140 00:11:03,300 --> 00:11:07,680 Polly and I have calculated the why list. 141 00:11:07,680 --> 00:11:13,230 So the Y components of this function, which are actually these values here. 142 00:11:13,650 --> 00:11:20,190 So basically, when you take these functions here or the single function and you take all different 143 00:11:20,190 --> 00:11:24,120 values for X from our X list, then this will be the function. 144 00:11:24,690 --> 00:11:35,340 So a zero times one basically so x list power of zeros one then plus a one times x plus a two times 145 00:11:35,340 --> 00:11:38,970 x squared plus a three times X to the power of three and so on. 146 00:11:39,420 --> 00:11:49,290 Until we are at the last term and we can run this and see maybe we can even go to minus 10 here or minus 147 00:11:49,290 --> 00:11:49,980 20 even. 148 00:11:51,450 --> 00:11:58,410 OK, so now you see, we have now are a seven data points, which are exactly the data points from here 149 00:11:58,410 --> 00:11:59,370 close to zero. 150 00:11:59,370 --> 00:12:01,350 So these are really these data points here. 151 00:12:04,000 --> 00:12:09,140 And you see, we have fitted a sixth order polynomial because we have seven coefficients. 152 00:12:09,520 --> 00:12:13,060 And you see it really fits all the data points perfectly. 153 00:12:14,320 --> 00:12:21,730 However, of course, you can also see that when we get rid of this town far away from these data points, 154 00:12:22,330 --> 00:12:27,670 the polynomial will give totally incorrect results in terms of our data points. 155 00:12:27,970 --> 00:12:34,450 So you see here we have data points in the range of minus 32 30, maybe minus 40 to 40, but here it 156 00:12:34,450 --> 00:12:36,070 goes up to several thousand. 157 00:12:36,820 --> 00:12:43,240 So it's such a method is nice, mathematically speaking, because we are really fitting these data points 158 00:12:43,240 --> 00:12:48,010 perfectly with a single function and not just piece wise, as we did with the spleens. 159 00:12:48,640 --> 00:12:55,480 However, it's not really useful to use this for extrapolation. 160 00:12:55,930 --> 00:13:01,660 So everything that's outside of the range of these data points will give a totally inaccurate result, 161 00:13:02,260 --> 00:13:03,400 and it gets even worse. 162 00:13:03,730 --> 00:13:10,210 We could, for example, fit these data points with the even higher polynomial where we just arbitrarily 163 00:13:10,210 --> 00:13:17,200 choose some of the parameters, and then we could basically get any behavior for these ranges outside 164 00:13:17,200 --> 00:13:18,100 of the data range. 165 00:13:19,240 --> 00:13:19,630 OK. 166 00:13:19,990 --> 00:13:24,130 So but this was just an add on, as I told you, just a nice mathematical method. 167 00:13:24,140 --> 00:13:31,510 And also, I want to tell you how you can use the Nampai module in October solve to solve systems of 168 00:13:31,510 --> 00:13:34,300 linear equations just in case you need it sometimes. 169 00:13:35,320 --> 00:13:37,180 And of course, we will not actually need it. 170 00:13:37,600 --> 00:13:41,020 But yeah, it's still a good tool to have. 171 00:13:41,770 --> 00:13:49,120 And also, it's it's a very nice result that we are not able to fit our data perfectly with just a single 172 00:13:49,120 --> 00:13:51,550 function, which is a higher order polynomial.