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Numerical Methods – ME428
创建日期：2024-04-06 17:12:14
所属分类：数学mathematics
标签： ME428
Numerical Methods

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Numerical Methods – ME428
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.

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.

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.

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