Lab 6: Linear Algebra

Transcription

1 6.1 Introduction Lab 6: Linear Algebra This lab is aimed at demonstrating Python s ability to solve linear algebra problems. At the end of the assignment, you should be able to write code that sets up and solves linear algebra problems for both a single set of coefficients and solutions as well as an array of situations. You should also be able to determine the condition of the linear system and understand how that relates to the accuracy of the results. 6.2 Resources The additional resources required for this assignment include: 0 Books: Chapra 0 Pratt Pundit Pages: Python:Linear Algebra 6.3 Getting Started 1. Log into one of the PCs in the lab using your NetId. Be sure it is set to log on to WIN. 2. In Windows: (a) Mount your CIFS drive. For instructions, start a browser and point it to: To Get Work Done Then click the link for File Access under Your Own Windows Computer. Note the comments on Pundit modifying what the OIT page instructions say. (b) Start a terminal with graphics. For instructions, point a browser to Check out the Using the PCs to Run Unix Programs Remotely section for how to get connected and make sure the connection is working. (c) Start Spyder. In the Windows menu, there is a group for Anaconda3; Spyder is in that group. If the PC asks for network permissions, click cancel. If the PC asks about updating, do not update. 3. In the terminal window (i.e. in UNIX; note the represents a space): (a) Change into your folder for EGR 103L, create a lab6 folder inside it, then switch into that folder: cd /EGR103 mkdir lab6 cd lab6 (b) Copy the tar file from Dr. G s space into your lab6 folder and expand the tar file: wget people.duke.edu/ mrg/egr103public/lab6files.tar tar -kxvf Lab6Files.tar (c) Make a personal copy of the lab report skeleton: cp -i Lab6Sample S19.tex lab6.tex 6 1

2 6.4 Ungraded 0 Chapra Chapter 8 problems 2, 3, 4 0 Chapra Chapter 11 problems 1, Assignment Based on Chapra Problem 8.3, p. 244 First, re-cast the problem as a matrix equation: A x = b??? x 1???? x 2 =????? then write a Python script to solve for the unknowns. Compute both the transpose and inverse of the [A] matrix. Also, calculate the condition number of the system using the 1-norm, the 2-norm, the Frobenius norm, and the -norm - read Sections through in Chapra for information on norms and condition numbers. Typeset the matrix equation, the transpose, and the inverse in your lab report. Also report the values of the unknowns and the condition numbers in some professional-looking way. State what the values of the condition numbers mean with respect to the accuracy of your answer relative to the accuracy of the coefficients in the matrix (Hint: Seriously - read of Chapra ). For all non-integer values, use four significant figures (scientific notation preferred but not required). Note that L A TEX with the amsmath package gives you some excellent tools for typesetting matrix equations. The equations above, for example, were typeset with: \ begin { align *} \ begin { bmatrix }{\ mathbf A}\ end { bmatrix } \ begin { bmatrix }{\ mathbf x}\ end { bmatrix }&= \ begin { bmatrix }{\ mathbf b}\ end { bmatrix }\\ \ begin { bmatrix }? &? &? \\? &? &? \\? &? &? \ end { bmatrix } \ begin { bmatrix } ~ x_1 ~ \\ x_2 \\ x_3 \ end { bmatrix }&= \ begin { bmatrix }? \\? \\? \ end { bmatrix } \ end { align *} x 3 6 2

3 6.5.2 Chapra 8.10, pp Typo notification: Figure P8.10 incorrectly has the force at node 1 as 1000 N down; it should be 2000 N down. Be sure to use N for F 1 v when solving. Rewrite the equations symbolically in matrix form as: cos(30 o ) 0 cos(60 o ) F 1 F 1 h F 2? F 3? = H 2? V 2?? Typeset these symbolic equations in your lab report, leaving the six external forces F 1 h through F 3 v as symbols rather than the specific values in the program. Note: in your lab report you can have italicized subscripts like F 1 h and F 3 v. The first equation has been done for you. Next write a Python script that will solve for the six unknowns in this problem. Your program should print the results to the screen with the format below (these numbers come from the same truss but using different external force values and thus are not what your program will get): F1: e +02 N F2: e +03 N F3: e +03 N H2: e +03 N V2: e +03 N V3: e +02 N Note that the format command is not especially fond of 2-D arrays. Somewhere in your code you will likely have a line like: s o l n v e c = s o l n [ :, 0 ] to copy your 2-D solution array into a 1-D array and then will use soln_vec in your format() commands. If you get an error unsupported format string passed to numpy. ndarray. format you are trying to give an array, rather than a value, to the string. See if you can figure out how to print these six items in a loop rather than with six different print commands. Look at the following code to see how it works: names = [ alpha, beta, gamma, d e l t a ] v a l s = [ 3, 1, 4, 2 ] for k in range ( l e n ( names ) ) : print ( { : 5 s } : { : 2 d}. format ( names [ k ], v a l s [ k ] ) ) and adapt it as needed. Finally, add code to your script that uses open, write, and close to write a table like the one above into a text file called truss_data.txt. If you are printing in a loop, you should only need to open the file before the loop, close the file after the loop, and write to the file with one command in the loop. The L A TEX code to bring the text from that file into your lab report is in the skeleton (but commented out). The matrix equations and the table should be in the body of the lab report while the code will be in the appendix. V 3 6 3

4 6.5.3 Based Loosely on Chapra 8.16, pp Important Note: Your Python function will not be doing the plotting - your function will instead just be returning a set of rotated coordinates. The main part of your script will be doing the plotting To rotate a set of coordinates, you will pre-multiply a block matrix with the coordinates by the rotation matrix. Imagine you have some original point at (x y). That point will be at some distance r from the origin and will make some angle φ from the x axis such that: x = r cos(φ) y = r sin(φ) Now imagine you rotate the point some positive angle θ along the circle described by r from the original coordinates to some new set of coordinates (x r y r ). Figure 6.1 on page 6 6 shows both the original and rotated points. The new coordinates are at an angle φ + θ from the x axis. The coordinates for the new point can be calculated as: x r = r cos(φ + θ) y r = r sin(φ + θ) x r = r(cos(φ) cos(θ) sin(φ) sin(θ)) x r = x cos(θ) y sin(θ) y r = r(cos(φ) sin(θ) + sin(φ) cos(θ)) y r = x sin(θ) + y cos(θ) This can be written in matrix form as: x r = cos(θ) sin(θ) y r sin(θ) x = R cos(θ) y x y where the rotation matrix R is given by: cos(θ) sin(θ) sin(θ) cos(θ) To expand this to rotating multiple points simultaneously, imagine you have 1-D arrays of x and y coordinates: x = [x 1 x 2 x 3...] y = [y 1 y 2 y 3...] If you build a block matrix (i.e. a 2-D array) from them: Coords = x = x 1 x 2 x 3... y y 1 y 2 y 3... you can rotate the entire set of coordinates by pre-multiplying the coordinates matrix by a rotation matrix: New Coords = R Coords New Coords = cos(θ) sin(θ) x 1 x 2 x 3... sin(θ) cos(θ) y 1 y 2 y 3... You will then just need to extract each row from the new coordinates into an array. 0 In a script called chapra_08_16.py, define a function called rotate_2d that takes three arguments - an angle (in degrees), a 1-D array of x coordinates, and a 1-D array of y coordinates, and returns two 1-D arrays: a 1-D array of rotated x coordinates and a 1-D array of rotated y coordinates. Test your program with some easily-calculated rotations. For instance, if you rotate a point by 0 or 360 or 360*n degrees where n is an integer, you should end up at the same location. You should also be able to figure out what happens when you rotate (x 0) or (0 y) by 90*n degrees so test with those. You can put your tests in the main if tree in your script. 0 Run the test_chapra_08_16.py file to make sure your function is correctly rotating the coordinates defining the square in the problem. The tester code will rotate the square by integer multiples of 30 o. It will make a plot with all the rotated squares shown. Note that the color kwarg takes a list of three numbers between 0 and 1; these numbers represent how much red, green and blue to put in the plot. 6 4

5 0 Make art by coming up with the coordinates for some shape then using code similar to the test_chapra_08_16.py code (call the script creative_chapra_08_16.py) to draw several copies of your shape rotated around a full circle in some number of steps. Get creative You can even change the colors of each copy if you would like The example in Fig. 6.2 is a skewed octagon copied 12 times with the colors determined by the angle. Include your chapra_08_16.py and creative_chapra_08_16.py files in the lab report along with the art you made Parameter Sweep 1: Changing...Constant For this problem, you will be taking the code from Case Study 8.3 in Chapra and writing a script that graphs how current flowing from node 5 to node 2 changes as the voltage drop between nodes 1 and 6 changes. Your program will sweep through 101 linearly spaced voltages between 0 V and 400 V, solving for all the currents each time and storing the solution for i 52 in either an array or a list. After performing all the calculations, you will produce a graph of the current as a function of the voltage. Use a solid purple line for the voltage (I use purple for voltages in my classes). Add a grid and proper axis labels, including units. See Chapra Figure 1.2 on page 8 for how to label a graph with a variable name and units. You do not need a title or a legend. As a quick check, the current in index 50 of your list or array should match the i 53 current in the case study since the 50th voltage value in the linearly spaced array is 200 V; that current is as given in the second entry of the current array on p In the body of the lab report you will describe generally how the voltage value impacts the current Parameter Sweep 2: Changing Coefficient For this problem, you will be taking the code from Case Study 8.3 in Chapra and writing a script that graphs how current flowing from node 5 to node 2 changes as the resistance between nodes 5 and 2 changes. You will use an applied voltage of 200 V for this as given in the original problem. Your program will sweep through 201 (i.e. 100 more than 101) linearly spaced resistances between 0 Ω and 100 Ω, solving for all the currents each time and storing the solution for i 52 in either an array or a list. You will also be calculating the 2-norm based condition number of the coefficient matrix and storing that in an array or a list. After performing all the calculations, you will produce a graph of the current as a function of the resistance and another graph of the base-10 logarithm of the condition number as a function of the resistance. Use a solid orange line for the current (I use orange for current in my classes) and a solid black line for the condition numbers. For each graph, add a grid and proper axis labels, including units. You can get the symbol for Ω in your axis labels by putting the L A TEX command for capital omega between dollar signs in the label command; for instance, p l t. x l a b e l ( 1\Omega1 ) will put the Ω all by itself. See Chapra Figure 1.2 on page 8 for how to label a graph with a variable name and units. Note that the condition number is dimensionless. You do not need a title or a legend. As a quick check, the current in index 20 of your list or array should match the current in the case study since the 20th resistance value in the linearly spaced array is 10 Ω. In the body of the lab report you will describe generally how the resistance value impacts the current and the condition number. 6 5

6 Figure 6.1: A point and a rotated point Figure 6.2: Something creative 6 6