1. Solve the following system:
2x1- 6x2- x3 = -38
-3x1–x2 +7x3 = -34
-8x1 +x2 – 2x3 = -20
By:
a. LU Factorization
b. Gauss-Siedel Method, error less that10-4
Hint (pivoting is needed, switch rows).
Consider the linear system of equations below
3x1 − x2 + x3 = 1
3x1 + 6x2 + 2x3 = 0
3x1 + 3x2 + 7x3 = 4
i. Use the Gauss-Jacobi iterative technique with x
(0) = 0 to find
approximate solution to the system above up to the third step
ii. Use the Gauss-Seidel iterative technique with x
(0) = 0 to find
approximate solution to the third step
Consider the following linear optimization model.
Z = 3x1+ 6x2+ 2x3
st 3x1 +4x2 + x3 ≤2
x1+
3x2+ 2x3 ≤ 1
X1, x2, x3 ≥0
(10) Write the optimization problem in standard form with the
consideration of slack variables.
(30) Solve the problem using simplex tableau method.
(10) State the optimal solution for all variables.
Given the following LP
max z = 2x1 + x2 + x3
s. t.
3x1 - x2 <= 8
x2 +x3 <= 4
x1,x3 >= 0, x2 urs (unrestricted in sign)
A. Reformulate this LP such that
1)All decision variables are non-negative.
2) All functional constraints are equality constraints
B. Set up the initial simplex tableau.
C. Determine which variable should enter the basis and which
variable should leave.
4-Consider the following problem:
max − 3x1 + 2x2 − x3 + x4
s.t.
2x1 − 3x2 − x3 + x4 ≤ 0
− x1 + 2x2 + 2x3 − 3x4 ≤ 1
− x1 + x2 − 4x3 + x4 ≤ 8
x1, x2, x3, x4 ≥ 0
Use the Simplex method to verify that the optimal objective
value is unbounded. Make use of the final tableau to construct an
unbounded direction..
Solve the following system of
equations using LU factorization without partial
pivoting:
2x1 - 6x2 - x3 =
-38
-3x1 - x2 + x3 =
-34
-8x1 + x2 - 2x3 =
-20
Consider the following LP model.Max Z = 3x1 - 4x2 +
x3
subject to x1 + x2 + x3 >= 9
2x1
+ x2 + x3<= 12
x1 + x2 = 5
x1, x2, x3 >= 0
Change it to standard form.
Obtain all the basic solutions and indicate which ones are basic
feasible solutions and write down the corresponding corner points.
For each basic solution, you have to obtain the values of all the
variables.
Obtain the solution of the LP...
Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}.
(a) Prove that U is a subspace of F4.
(b) Find a basis for U and prove that dimU = 2.
(c) Complete the basis for U in (b) to a basis of F4.
(d) Find an explicit isomorphism T : U →F2.
(e) Let T as in part (d). Find a linear map S: F4 →F2 such that
S(u) = T(u) for all u ∈...