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Short Assignment Requirements

It is a math course using python to solve engineering problems. I have attached the homework file. Please add comments to show why you have programmed it that way. Please solve the problems so that I could change the variables and solve similar problems.

Assignment Description

ME581 Homework 2

Due: 4:15pm September 26, 2017

The following problems are to be documented, solved, and presented in a Jupyter notebook.

On-Campus students: Save the notebook as a single PDF, then print and return a hard copy in class.

Off-Campus students:  Save the notebook as a single PDF, then upload and submit the PDF in Blackboard. The name of the file should be SURNAME-HW2.pdf.

Problem 1

A system of equations 𝐴𝒙 = 𝒃 is given as

2.01    1.99           4

[                    ] 𝒙 = [ ] .

1.99    2.01           4

The solution to this system of equations is

1

𝒙 = [ ] , 1

and one approximate solution is

2

𝒙̃ = [ ] .

0

(i)                Compute the error 𝒆 = 𝒙̃ − 𝒙.

(ii)              Compute the residual 𝒓 = 𝐴𝒙̃ − 𝒃.

(iii)           Using the 𝑙 norm, compute the relative error 𝒙𝒆‖ .

(iv)            Using the 𝑙 norm, compute the condition number 𝜅.

(v)              Using the 𝑙 norm, compute the relative residual 𝒃𝒓‖ .

(vi)            Compute the product of the condition number and the relative residual.

(vii)          Compare the relative error to the product of the condition number and the relative residual.

Problem 2

Let

5.1    8.7                      9.48

𝐴 = [               ]   and   𝑏 = [        ] .

2.4    4.1                      4.48

(i)                Using the 𝑙 norm, compute the condition number 𝜅(𝐴).

(ii)              Solve the system of equations 𝐴𝒙 = 𝒃 for 𝒙.

(iii)           Perturb the coefficient matrix 𝐴 and the right-side vector 𝒃 by

−0.001    0                          0.05

𝛿𝐴 = [                    ]   and   𝛿𝒃 = [           ]

0.001      0                        −0.05

and solve the resulting perturbed system of equations (𝐴 + 𝛿𝐴)𝒙̃ = (𝒃 + 𝛿𝒃) for the approximate solution 𝒙̃ .

(iv)            Using the 𝑙 norm, compute the actual value of the relative change in the

‖𝛿𝒙‖solution,   for the perturbation in part (iii).

‖𝒙‖

(v)              Using the 𝑙 norm, compute the theoretical upper bound of the relative

‖𝛿𝒙‖change in the solution,  for the perturbation in part (iii). ‖𝒙‖

(vi)            For the perturbation in part (iii), compare the actual value of 𝛿𝒙‖ to its

‖𝒙‖theoretical upper bound.

(vii)          Perturb the original coefficient matrix 𝐴 and the original right-side vector 𝒃 by

0.001      −0.001                        −0.1

𝛿𝐴 = [                              ]   and   𝛿𝒃 = [         ]

−0.001      0.001                           0.1

and solve the resulting perturbed system of equations (𝐴 + 𝛿𝐴)𝒙̃ = (𝒃 + 𝛿𝒃) for 𝒙̃ .

(viii)       Using the 𝑙 norm, compute the actual value of the relative change in the

‖𝛿𝒙‖solution,   for the perturbation in part (vii).

‖𝒙‖

(ix)            Using the 𝑙 norm, compute the theoretical upper bound of the relative

‖𝛿𝒙‖change in the solution,  for the perturbation in part (vii). ‖𝒙‖

(x)              For the perturbation in part (vii), Compare the actual value of 𝛿𝒙‖ to its

‖𝒙‖theoretical upper bound.

Problem 3

Solve the augmented matrix

3    1      4    −1    7

2    −2 −1     2     1 ]

[                           |

5    7    14    −9 21

1    3      2      4    −4

By means of

(i)                Gaussian Elimination with Partial Pivoting.

(ii)              Gaussian Elimination with Scaled Partial Pivoting.

Problem 4

(i)                Solve the augmented matrix by means of Gaussian Elimination with Partial Pivoting in double precision:

(ii)              Using the 𝑙 norm, estimate the condition number of the coefficient matrix based on your result. The exact solution for this problem is given by

𝒙 = [1    −1    1    −1    1]𝑇.

Problem 5

Determine the member and reaction forces within the plane truss shown in Figure 1 when the truss is subjected to each of the following loading configurations.

(a)  500-pound forces directed vertically downward at nodes#3 and #5, and a 1000pound force directed vertically downward at node#4.

(b)  A 500-pound force directed vertically downward at nodes#3, a 1000-pound force directed vertically downward at node#4, a 1500-pound force directed vertically downward at node#5.

(c)   A 1500-pound force directed vertically downward at nodes#3, a 1000-pound force directed vertically downward at node#4, a 500-pound force directed vertically downward at node#5.

(d)  500-pound force acting at node#4, and a 1000-pound force acting at node #3, both forces acting horizontally to the right.

(e)  500-pound force acting at node#4, and a 1000-pound force acting at node #5, both forces acting horizontally to the left.

Solve the problem using your GE code with partial pivoting. Show the augmented matrix and the resulting forces for each case.

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