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Take Home Midterm Exam Numerical Methods – ME428 Spring 2021 Copyright
2021, University of Illinois at Chicago, all rights reserved. No
distribution without express written permission. 2/6 1. FINITE DIFFERENCE OF A LARGE DATA SET (25 POINTS) You are given the following experimental dataset for position (x) as a function of time (t). t (s) x (m) 0.000 6.000 0.050 5.988 0.100 5.951 0.150 5.890 0.200 5.804 0.250 5.693 0.300 5.559 0.350 5.399 0.400 5.215 0.450 5.007 0.500 4.774 0.550 4.516 0.600 4.234 0.650 3.928 0.700 3.597 0.750 3.241 0.800 2.861 0.850 2.456 0.900 2.027 0.950 1.573 1.000 1.095 1.050 0.592 1.100 0.065 WRITE A CODE TO DO THE FOLLOWING: A. (7 points) Calculate the velocity, = !”!# [m/s], for the entire data set using a second-order accurate method. Show your equations for your derivatives (do not need to substitute numbers, just specify the equations). *Note: the first and last datapoints will require different numerical derivatives than the central points. Take Home Midterm Exam Numerical Methods – ME428 Spring 2021 Copyright
2021, University of Illinois at Chicago, all rights reserved. No
distribution without express written permission. 3/6 B. (7 points) Calculate the acceleration, = !!”!#! [/$], for the entire data set using a second-order accurate method. Show your equations for your derivatives (do not need to substitute numbers, just specify the equations). *Note: the first and last datapoints will require different numerical derivatives than the central points C. (6 points) Present your results in a table of the form: t (s) x (m) v (m/s) a (m/s2) 0.000 6.000 0.050 5.988 0.100 5.951 0.150 5.890 0.200 5.804 0.250 5.693 0.300 5.559 0.350 5.399 0.400 5.215 0.450 5.007 0.500 4.774 0.550 4.516 0.600 4.234 0.650 3.928 0.700 3.597 0.750 3.241 0.800 2.861 0.850 2.456 0.900 2.027 0.950 1.573 1.000 1.095 1.050 0.592 1.100 0.065 D. (5 points) Plot the results for position, velocity, and acceleration on three graphs. The position, velocity, and acceleration should be on the y-axis, and the time should be on the x-axis. All lines and axes should be clearly labeled. Take Home Midterm Exam Numerical Methods – ME428 Spring 2021 Copyright
2021, University of Illinois at Chicago, all rights reserved. No
distribution without express written permission. 4/6 2. SOLUTION OF A LARGE SYSTEM OF EQUATIONS (25 POINTS) Given the system of equations: 9 5 -2 3 4 8 4 6 1 2 4 8 -12 -3 -3 ” 20 4 10 -5 3 6 7 1 2 -4 6 -3 -3 2 -7 -1 # 11 1 8 11 1 1 -1 6 -3 -2 1 -4 -1 6 -7 -4 $ 82 3 4 7 12 2 -3 5 -2 -3 2 -5 1 5 6 -5 % 83 4 3 8 -3 10 -2 2 -1 -4 3 -1 -2 4 -5 -6 & 74 2 2 4 5 3 9 3 -3 -6 -4 -2 2 -6 2 2 ‘ 56 1 6 6 6 4 -4 -9 -2 -7 -3 -3 -3 -5 1 3 ( 47 6 4 2 2 5 -5 4 -12 -1 -2 -2 3 -4 3 1 ) = 43 7 9 1 1 6 -6 5 -1 11 -4 -1 -4 -3 2 2 * 72 -5 1 9 9 7 -1 6 -4 -3 11 2 4 2 -5 -3 “+ 76 -6 2 8 -6 8 -3 7 -5 -2 1 -15 -5 1 1 4 “” 85 -4 3 6 -2 3 -2 1 -6 -4 2 3 -12 2 2 2 “# 56 -3 4 3 -3 -4 1 2 -2 -5 3 4 5 9 3 3 “$ 81 2 5 4 -1 -5 -3 2 -4 -3 -2 -8 -3 -5 8 5 “% 57 5 -4 2 -3 -4 2 -6 -4 -2 -6 -7 -2 -4 -3 11 “& 33
A. (20 points) Solve the system using your choice of Gaussian
Elimination, Gauss-Jordan Elimination, Jacobi Iteration, or Gauss Seidel
iteration via a code that you wrote. You must submit your functioning
code, that is executable and produces the correct result. Present your
results as ! = , ” = , # = , … B. (5 points) Compare your answer
against RREF or a similar function (that is already built in) to check
your result. This should be included in your code to verify your result.
END OF EXAM