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We Helped With This Python Programming Homework: Have A Similar One?
Short Assignment Requirements
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.
