Universidad Industrial de Santander
Numerical Analysis
Department of Computer Science
PhD. Henry Arguello Fuentes
Undergraduate Students
Homework #3
Solution of Linear Systems AX = B
DATE : 16th June 2020 D UE : 5th July 2020
Name: Scholar ID:
1 I NDICATIONS
• Write down the solution process for all problems in this assignment sheet.
• Answers with no process are not valid.
• Make all calculations with 5 decimal places of precision.
2 S OLUTION OF L INEAR S YSTEMS AX = B
1. (0.8 points) For the following linear system, start with P0 = 0 and use Gauss-Seidel iteration to find Pk for
k = 1, 2, 3, 4, 5. Will Gauss-Seidel iteration converge to the solution?
a)
2x + 8y − z = 11
5x − 12y + z = 10
−x + y + 14z = 3
Process:
P0 = P1 = P2 = P3 = P4 = P5 =
Ph.D. Henry Arguello Fuentes Page 1 of 6
2. (0.8 points) Suppose that three computers, A, B and C , are working in parallel in three different tasks, T1 , T2
and T3 . Table 1 shows the consumed time per task per computer and the total of instructions required per
task.
Table 1. Time consuming per task per computer
Task A B C Instructions
T1 9[s] 20[s] 5[s] 139’500,000
T2 30[s] 150[s] 30[s] 894’000,000
T3 0.01[s] 0.002[s] 0.3[s] 2’372,800
Distribution of instructions per task per computer. Example: the 1390 500, 000 instructions required by T1 were distributed such that, computer A
spend 9[s], computer B spend 20[s], and, computer C spend 5[s] processing the asigned instructions for T1 .
a) Determine a linear system of equations AX = B, such that, it allows to find the processor speed,
VA , VB and VC , of computers A, B and C respectively.
Process and Answer:
b) Determine an equivalent upper-triangular system UX = Y for the linear system of equations AX = B
found in literal a).
Process and Answer:
c) Use backsubstitution method over the upper-triangular system UX = Y found in literal b) to determ-
ine VA , VB and VC .
Process:
Answer:
VA = VB = VC =
2
3. (0.8 points) For the following linear system, start with P0 = 0 and use Jacobi iteration to find Pk for k =
1, 2, 3, 4, 5. Will Jacobi iteration converge to the solution?
a)
15x − y + z = 12
2x + 8y − z = 11
−x + y + 4z = 3
Process:
P0 = P1 = P2 = P3 = P4 = P5 =
π
4. (0.8 points) The vectors X ∈ Rn and Y ∈ Rn are said to be orthogonal if the angle between them is .
2
a) Prove that X and Y are orthogonal if and only if X · Y = 0.
Process and Answer:
3
b) Find two different vectors, Y and Z, that are orthogonal to
i. X = (4, −7, 5, 9)
Process: Process:
Answer: Answer:
Y= Z=
ii. X = (6, 2, −3, −3)
Process: Process:
Answer: Answer:
Y= Z=
5. (0.8 points) In a movie store there are movies in 4 different categories: Thriller, Comedy, Romance and
Action. The total amount of movies in the store is 700. Besides, the sum betweeen the amount of Thriller
movies and the double of the amount of Comedy movies is equal to the 30% of Romance movies. Also you
know that, the 10% of Thriller movies plus the 30% of Comedy movies, plus the 84% of Romance movies,
plus the 40% of Action movies is equal to 480. Finally you know that the amount of Comedy movies plus
985 is equal to the double amount of Romance movies plus the 35% of Action movies.
a) Determine a linear system of equations AX = B with the given information.
Process and Answer:
b) Determine the LU factorization for the matrix A found in literal a).
Process and Answer:
4
c) Find the solution X of the linear system of equations AX = B by using the LU factorization found in
literal b).
Process and Answer:
6. (1.0 point) Let
7 −6 3
−5 6 10 12 14 8 1
−6 2 11
A = 15 3 28 1 ,B = ,C = 5 4 1
10 −12 −20
10 −12 −20 −24
2 25 6
6 17 3
a) Determine AB + C
Process and Answer:
b) Determine BA
Process and Answer:
5
c) Determine CA
Process and Answer:
d) Find the determinant of A, B and C if it exists.
Process and Answer:
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