WEBVTT 00:01.120 --> 00:05.100 So hello, welcome back to the Python course for the vehicle 00:05.100 --> 00:11.860 technicians and we are now here in the part Tools and Packages, in the 00:11.860 --> 00:14.220 fifth part at SymPy. 00:18.530 --> 00:22.370 And yes, SymPy is a library with which you can do symbolic 00:22.370 --> 00:22.710 calculations. 00:23.830 --> 00:28.170 Maybe you have already worked with other programs that can solve 00:28.170 --> 00:30.630 mathematical problems on a computer symbolically. 00:31.350 --> 00:36.630 There are, for example, Mavcat or Maple, these are the most famous, I 00:36.630 --> 00:36.770 think. 00:37.310 --> 00:41.150 Of course there are also other programs or extensions. 00:43.670 --> 00:47.430 At MATLAB, for example, there is also such an extra tool that you can 00:47.430 --> 00:47.870 turn on. 00:50.230 --> 00:54.590 So symbolic computing is also possible in Python with this library and 00:54.590 --> 01:00.570 it offers a lot of possibilities and we will just take a look at what 01:00.570 --> 01:01.370 you can do with SymPy. 01:03.270 --> 01:06.090 The nice thing about SymPy is that you can simply integrate it into 01:06.090 --> 01:10.510 your other programming projects and then don't have such a break in 01:10.510 --> 01:12.390 the tools, so that you have to switch to a different place for certain 01:12.390 --> 01:13.310 calculations. 01:15.190 --> 01:18.550 So you can simply install that as a package and then embed it directly 01:18.550 --> 01:20.050 into your tasks. 01:23.670 --> 01:29.710 There is a live demonstrator where you can find the help very quickly. 01:30.510 --> 01:36.170 It can be found at live.sympy.org and it looks like this. 01:36.610 --> 01:44.110 On the left side you have a console where you can type in commands and 01:44.110 --> 01:47.350 on the right side there are also a few small example sessions. 01:47.350 --> 01:50.630 The help is really good. 01:50.910 --> 02:01.090 You can look at small example tasks and then try it out interactively. 02:01.550 --> 02:05.570 If you don't want to put a Python installation on your computer and 02:05.570 --> 02:09.410 can learn SymPy step by step. 02:09.550 --> 02:14.270 But let's take a look at a few features in this part. 02:15.270 --> 02:20.270 First, let's take a look at how to import it and a few simple things. 02:21.770 --> 02:25.690 If you have installed it, you can do it with conda or pip, simply with 02:25.690 --> 02:27.150 conda install sympy. 02:27.750 --> 02:30.430 Then you can import it as normal as any other package. 02:31.610 --> 02:35.570 Here I did import sympy as sp, so that it is a bit shorter. 02:36.610 --> 02:41.790 The second line is for the notebooks, so that you can see the plots 02:41.790 --> 02:42.030 directly. 02:43.570 --> 02:45.550 You can define symbols. 02:46.270 --> 02:51.550 Of course, you have to tell the Python interpreter that an assignment 02:51.550 --> 03:00.070 is not a variable or a pointer that points to an address in the 03:00.070 --> 03:05.010 computer's memory, but that it should be a symbol with completely 03:05.010 --> 03:05.770 different properties. 03:06.550 --> 03:09.790 This works with the symbols function. 03:10.610 --> 03:16.430 You can also define several symbols at once, for example separated 03:16.430 --> 03:16.430 with a comma. 03:17.070 --> 03:18.810 That's how it works on the right side. 03:18.910 --> 03:21.830 You see a high comma there, you can define that. 03:22.250 --> 03:25.910 And then this x, y and t has certain properties. 03:30.850 --> 03:33.890 Such symbols are also rendered specifically. 03:33.890 --> 03:42.090 If you work in Jupyter Notebooks, you get a math font. 03:43.950 --> 03:51.270 The font is rendered and displayed. 03:51.510 --> 03:55.490 With such a display command you can work in notebooks. 03:56.110 --> 03:58.630 The x is shown in such a math font. 04:01.450 --> 04:08.130 Then you can define such expressions and equations. 04:09.390 --> 04:11.370 Equations have a left and right side. 04:14.250 --> 04:17.010 Equations can be read from left to right or from right to left. 04:17.430 --> 04:21.190 That's a difference to such an expression. 04:22.310 --> 04:27.350 Now let's look at an expression that you can define. 04:29.530 --> 04:32.510 Such an expression is equal to x squared. 04:33.790 --> 04:42.510 That means we have such an x squared in math font, if you execute the 04:42.510 --> 04:43.230 object there. 04:43.910 --> 04:47.410 And here is such an equation shown, how it works. 04:48.050 --> 04:52.970 That's this simple dot, capital E and small q. 04:53.630 --> 04:56.870 You see in the brackets that it is separated by a comma. 04:59.730 --> 05:05.390 The first argument is the left side of an equation and the second the 05:05.390 --> 05:05.870 right side. 05:06.590 --> 05:12.290 You see that when it is executed, 3x is equal to minus 10. 05:12.610 --> 05:16.590 That's how you can define equations in Symbi. 05:21.010 --> 05:23.010 There are already built-in plot functions. 05:23.010 --> 05:27.190 You don't necessarily need an extra plot library. 05:28.990 --> 05:34.030 You can also make some settings there, set the value range and so on. 05:35.850 --> 05:37.850 We wrote such an expression before. 05:39.650 --> 05:44.130 That was the abbreviation expression is equal to x squared. 05:45.910 --> 05:50.790 If you call this Symbi plot and type in the expression there, then you 05:50.790 --> 05:55.630 can specify the second argument in which value range the x should be 05:55.630 --> 05:55.630 executed. 05:56.710 --> 05:58.350 This is very practical. 05:58.670 --> 06:03.850 In the previous part or in the NumPy part, we also made similar plots, 06:04.170 --> 06:04.890 that we had generated a time series. 06:06.310 --> 06:16.430 It was important to create a time vector, for example with numpy 06:16.430 --> 06:16.950 .arrange. 06:18.390 --> 06:25.810 Then apply a mathematical function to it, for example x squared. 06:26.490 --> 06:31.670 Then you can plot these two time series with a plot command. 06:31.970 --> 06:37.110 Here it is a bit shorter and easier because it is calculated 06:37.110 --> 06:37.110 symbolically. 06:37.210 --> 06:43.170 You just have to say in which value range this expression should be 06:43.170 --> 06:43.650 executed. 06:43.650 --> 06:46.190 Then you get such a plot with one line. 06:48.070 --> 06:51.050 You can also tune the axis label. 06:51.790 --> 06:54.830 You can already see a small x on the x-axis. 06:55.510 --> 07:00.270 On the y-axis you can see a f of x. 07:02.210 --> 07:06.390 You get plots of mathematical functions pretty quickly. 07:07.350 --> 07:09.150 This is the first advantage. 07:09.390 --> 07:13.610 If you have larger functions that you want to type in and also with a 07:13.610 --> 07:19.630 display or a print command, or you just have to call the expression, 07:20.810 --> 07:26.370 then you can see what you typed in as mathematical functions in such a 07:26.370 --> 07:27.310 mathematical font. 07:28.230 --> 07:30.310 And then just plot it. 07:30.770 --> 07:35.630 That means you probably get to the goal faster with SymPy, if you type 07:35.630 --> 07:42.110 a certain formula from a book or a paper, just to show it graphically. 07:46.550 --> 07:50.610 That would be the first reason why you should calculate symbolically. 07:51.350 --> 07:54.510 But the main reason why you should look at it, the topic of 07:54.510 --> 07:58.730 calculating symbolically on a PC, is the accuracy. 07:59.250 --> 08:00.710 We talked about it before. 08:01.950 --> 08:06.090 It was about comparisons of floating point numbers. 08:06.770 --> 08:11.550 That was two parts before in the lecture. 08:12.390 --> 08:16.090 That you can't make precise comparisons. 08:17.950 --> 08:20.650 It's only possible with certain tolerance ranges. 08:21.610 --> 08:26.770 Simply because the values of the floating point numbers differ 08:26.770 --> 08:29.990 minimally in the comma places and in the decimal places. 08:32.490 --> 08:34.490 That has nothing to do with Python. 08:34.490 --> 08:38.350 That's just the computer architecture and how floating point numbers 08:38.350 --> 08:38.910 are represented on a computer. 08:40.390 --> 08:43.110 And of course you can do that with symbolic, if you do such 08:43.110 --> 08:47.770 conversions with symbolic mathematics, then again exactly. 08:49.230 --> 08:50.170 This is shown here. 08:50.630 --> 08:56.490 We get the mathematics module from the Python Basics Library and just 08:56.490 --> 08:57.930 calculate the root of 8. 08:59.010 --> 09:01.770 And that's where a floating point number comes out. 09:04.770 --> 09:11.240 So a floating point number. 09:11.720 --> 09:19.960 And if we do that with sympi, then the equation is solved correctly 09:19.960 --> 09:21.640 and the two are drawn in front of the root. 09:22.260 --> 09:24.280 And then you have the exact solution. 09:25.200 --> 09:26.980 This may not be very interesting for a one-liner. 09:29.180 --> 09:33.600 But if you imagine that you have a longer algorithm, then you can do 09:33.600 --> 09:33.600 that. 09:33.600 --> 09:37.320 Where you perform many of these mathematical operations one after the 09:37.320 --> 09:43.000 other and then use them again and have such intermediate results, that 09:43.000 --> 09:43.880 makes a difference. 09:44.700 --> 09:50.060 And then it gets even more exciting if you have an algorithm that you 09:50.060 --> 09:50.960 call up again and again. 09:51.800 --> 09:53.760 This is also the case with optimizers, for example. 09:55.580 --> 09:58.500 That you have such a preliminary loop that iterates up to a certain 09:58.500 --> 09:59.440 termination criterion. 10:00.580 --> 10:05.420 And if you can calculate that symbolically, then you have a much 10:05.420 --> 10:12.360 higher accuracy and only have to convert the result into such a 10:12.360 --> 10:20.380 floating point number As if you were always working with floating 10:20.380 --> 10:24.160 point numbers and then make such a numerical error again and again, 10:24.380 --> 10:26.440 which accumulates over time. 10:28.740 --> 10:34.680 So that would be a reason why you should take a look at this, what you 10:34.680 --> 10:37.060 can do with symbolic mathematics on a computer. 10:42.280 --> 10:49.560 Of course, you can simplify such expressions and equations, also pull 10:49.560 --> 10:50.280 them apart again. 10:50.500 --> 10:52.540 There are a few built-in commands. 10:53.360 --> 10:57.260 Here is an expression shown. 10:57.660 --> 10:59.900 So sin²x and cos²x. 11:00.820 --> 11:06.420 And if you just make a simplification, then the known 1 comes out. 11:11.080 --> 11:19.440 Here is another expression, that you use a bracket and you can 11:19.440 --> 11:20.520 calculate that with the expand. 11:22.040 --> 11:24.260 These are of course very, very simple examples. 11:24.820 --> 11:26.600 It's pretty clear what comes out. 11:26.640 --> 11:30.840 But if you calculate bigger problems and then have very, very long 11:30.840 --> 11:35.380 formulas, then such commands are of course very helpful, before you 11:35.380 --> 11:36.420 calculate it all on paper. 11:40.460 --> 11:46.160 And with the factor command, you can then pull the multiplier out 11:46.160 --> 11:46.160 again. 11:51.380 --> 11:54.640 Of course, it is also built-in to differentiate and integrate. 11:58.020 --> 11:59.800 Here we have a very simple example. 12:01.940 --> 12:05.200 You can also see that in SymPy there are of course all these 12:05.200 --> 12:06.160 functions, i.e. 12:06.220 --> 12:09.020 sine and exponential function and root and so on. 12:09.520 --> 12:14.000 Of course, you also have to use them from SymPy, so that it all works. 12:15.420 --> 12:23.960 And then you can differentiate and also integrate, and you always have 12:23.960 --> 12:30.780 the expression in such a mathematical type and can then work very well 12:30.780 --> 12:31.080 with it. 12:37.920 --> 12:44.580 Of course, you can also do this with a fixed interval and then get an 12:44.580 --> 12:46.580 exact solution for it. 12:47.200 --> 12:53.840 There are also functions for converting this into a floating point 12:53.840 --> 12:57.080 number and so on. 12:57.940 --> 13:00.980 You can use it in other scripts or in other parts of your program. 13:02.420 --> 13:06.860 If you do such an integrated symbolic calculation in the middle, for 13:06.860 --> 13:07.080 example. 13:11.470 --> 13:14.830 Then you can also make such substitutions, so that you put some 13:14.830 --> 13:17.450 expressions with a certain value. 13:19.470 --> 13:20.710 Here it is done. 13:20.810 --> 13:26.870 You simply take this expression and take the command subs and simply 13:26.870 --> 13:27.330 enter a 2 for the x. 13:29.910 --> 13:35.570 And what you can also do is simply solve equations. 13:36.070 --> 13:38.310 Here we have the x² plus 3x is 10. 13:39.370 --> 13:40.890 Of course, you can still do that on paper. 13:41.270 --> 13:43.090 These are all very, very simple examples. 13:43.770 --> 13:46.730 If it gets more complicated, it is of course worth starting the 13:46.730 --> 13:47.650 computer for it first. 13:48.690 --> 13:50.610 And you can do that with the spsolv. 13:51.790 --> 13:54.110 Here you have to pull everything to the left. 13:54.890 --> 13:59.190 That is, the 10 is equal to 10 becomes a minus 10. 13:59.790 --> 14:02.630 And then you get the solutions minus 5 and minus 2. 14:11.140 --> 14:14.240 Then of course there are also possibilities that you can solve 14:14.240 --> 14:14.540 equations. 14:16.340 --> 14:17.360 This is shown here. 14:17.840 --> 14:22.200 There is an equation e1 and an equation e2. 14:23.240 --> 14:25.400 There is left and right side again. 14:25.400 --> 14:31.060 The first argument that comes in here, this 3x plus 4y, is the left 14:31.060 --> 14:31.780 side of the equation. 14:32.100 --> 14:34.920 And then after the comma comes the right side. 14:35.460 --> 14:37.460 Here at equation 2 exactly the same. 14:38.040 --> 14:42.580 And then you can simply build this equation system, that you put both 14:42.580 --> 14:44.020 equations together in such a list. 14:45.340 --> 14:49.420 And then there are such solvers at SymPy that solve the equation 14:49.420 --> 14:49.980 system. 14:52.220 --> 14:57.500 And then you get the solution with the command linsolv. 15:05.550 --> 15:09.370 Of course, you can also work with ODEs and solve differential 15:09.370 --> 15:09.830 equations. 15:14.260 --> 15:17.980 You can do that with two lines. 15:18.720 --> 15:21.060 First you have to define a function. 15:23.140 --> 15:26.980 SymPy knows that y is a function of t. 15:27.600 --> 15:29.640 And then you can solve an equation. 15:30.060 --> 15:34.080 The command to solve it yourself is this dsolv differential equation. 15:34.760 --> 15:38.160 And then in the middle of the bracket there is an equation command. 15:39.880 --> 15:49.460 And then you can differentiate y of t twice to t minus y of t comma. 15:49.660 --> 15:53.940 Then comes the right side, the e function. 15:55.000 --> 15:57.980 And then this should be solved to y of t. 15:58.380 --> 16:01.520 And then you get it here with the integration constants. 16:03.260 --> 16:08.220 So you can solve such differential equations very quickly. 16:08.620 --> 16:13.120 And there is of course also the possibility to enter boundary 16:13.120 --> 16:18.100 conditions, so that the integration constants are also solved. 16:23.910 --> 16:26.290 Then you can also work with matrices in SymPy. 16:27.650 --> 16:30.210 There is also the matrix command. 16:32.090 --> 16:35.710 You type it in very similarly as in NumPy. 16:37.310 --> 16:39.930 That means you work with these right-angled brackets. 16:40.450 --> 16:42.950 There is a two-dimensional matrix. 16:43.210 --> 16:47.130 That means there is an outer right-angled bracket and in there you can 16:47.130 --> 16:47.970 find the lines. 16:48.790 --> 16:52.190 I wrote it with a comma and a line break so that you can see it right 16:52.190 --> 16:52.410 away. 16:52.410 --> 16:56.370 I think it's pretty clear how the syntax works. 16:56.450 --> 16:58.650 It's just like in NumPy. 16:59.310 --> 17:07.550 There are also special matrices like the unit matrix with SymPy i or 17:07.550 --> 17:08.670 such a zero matrix. 17:08.870 --> 17:12.310 You can specify the dimensions and so on. 17:21.150 --> 17:26.410 You can of course also do matrix decompositions and such inverts. 17:27.190 --> 17:31.390 You can either write that with the high minus 1 or so that it is 17:31.390 --> 17:33.630 perhaps a little more readable with the .inf. 17:35.170 --> 17:37.450 Here is the unit matrix built up. 17:37.690 --> 17:38.570 It is multiplied by 4. 17:39.210 --> 17:40.570 You have the values 4 on the main diagonal. 17:42.230 --> 17:43.790 The diagonal elements are 0. 17:44.670 --> 17:46.270 Then it is inverted. 17:48.550 --> 17:53.710 Then you can also overwrite the elements. 17:54.730 --> 17:58.150 Here is the second last entry. 18:00.250 --> 18:01.670 An X is simply written in there. 18:02.270 --> 18:04.670 Then we have such an X on the side diagonal element. 18:06.730 --> 18:11.310 You can of course also use other symbols in such a matrix and maybe 18:11.310 --> 18:17.610 define a state-space model and use certain physical units. 18:18.230 --> 18:25.590 Then later with substitution you can replace such symbols with 18:25.590 --> 18:26.730 numerical values in matrices. 18:31.520 --> 18:37.180 It is a very, very powerful algebra package and you can calculate 18:37.180 --> 18:38.120 symbolically with it. 18:38.680 --> 18:41.220 I think the most exciting thing is that you can simply install it as a 18:41.220 --> 18:45.100 package and install it in your Python environment. 18:46.780 --> 18:50.920 If you have a part where you want to calculate symbolically, then you 18:50.920 --> 18:56.860 don't have to switch to another tool or install another software and 18:56.860 --> 19:01.620 then export and import data and values from one tool to another. 19:02.660 --> 19:06.140 You can do it all integrated. 19:06.960 --> 19:10.260 That was the reason why I wanted to briefly introduce SymPy. 19:11.560 --> 19:13.920 Just a few features of it. 19:14.520 --> 19:17.900 You can find many more examples in the documentation. 19:18.280 --> 19:19.760 You can just have a look. 19:19.880 --> 19:22.260 You can try to solve a differential equation. 19:24.120 --> 19:30.280 Maybe you have some tasks in another course or solve a equation 19:30.280 --> 19:30.760 system. 19:33.280 --> 19:37.900 I think you will find interesting tasks where you can try out SymPy. 19:39.180 --> 19:44.940 If you already work with another tool and, for example, use Maple or 19:44.940 --> 19:51.060 something like that, then you can compare it and see what the commands 19:51.060 --> 19:55.520 are like and how it feels when you work directly in Python in a 19:55.520 --> 19:58.180 Jupyter notebook with SymPy. 20:00.360 --> 20:01.840 Thank you for listening. 20:01.980 --> 20:03.360 That was a bit of a shorter part. 20:03.480 --> 20:07.980 I just wanted you to see SymPy and know that it also works in Python. 20:07.980 --> 20:10.220 That's Symbolic Computing.