Problem 4a: Gauss-Seidel/Jacobi (10 pts)

Develop a spreadsheet to solve the following system of equations using Gauss-Seidel or Jacobi with relaxation.  Clearly indicate which method you used.  Use l = 0.8 and e_s = 0.001%.  If necessary, make sure to rearrange the equations to achieve convergence.

ENGR 351 Numerical Methods for Engineers
College of Engineering
Southern Illinois University Carbondale

HOMEWORK SET 2: Matrix Methods

(Total: 40 pts.) (Total: 50 pts.)

Due:  Feb. 21, 2003 Feb. 24, 2003

Problem 1: Gauss Elimination  (10 pts)

You must do this problem by hand on engineering paper.

Three carts, interconnected by springs, are subjected to the loads P₁, P₂ and P₃, as shown in the figure.  The displacements of the carts are governed by the equilibrium equations

P₁ - k₁u₁ + k₅(u₃ - u₁) + k₄(u₂ - u₁) = 0

P₂ - k₂u₂ - k₄(u₂ - u₁) + k₆(u₃ - u₂) = 0

P₃ - k₇u₃ - k₈u₃ - k₃u₃ - k₆(u₃ - u₂) - k₅(u₃ -u₁) = 0

Find the displacement of the carts for the following data using Gauss Elimination.

k₁ = 5000 N/m, k₂ = 1500 N/m, k₃ = 2000 N/m, k₄ = 1000 N/m, k₅= 2500 N/m, k₆ = 500 N/m, k₇ = 3000 N/m, k₈ = 3500 N/m, P₁ = 1000 N, P₂ = 2000 N, P₃ = 3000 N

Problem 2: LUD Method (10 pts)

Using the results of the previous problem, perform the LUD method, clearly indicating [L] and [U].  This must also be done by hand.

Problem 3: Use of Inverse Matrix  (10 pts)

• Use Excel to calculate the inverse of [A]
• Using Excel, multiply [A][A^-1].  Indicate whether this is the identity matrix [I]
• Using Excel, determine {x} for {c}^T = { 100  -200  40} and {c}^T = { -100  200  -40}

Submit a hardcopy of your work.

Problem 4a: Gauss-Seidel/Jacobi (10 pts)

Develop a spreadsheet to solve the following system of equations using Gauss-Seidel or Jacobi with relaxation.  Clearly indicate which method you used.  Use l = 0.8 and e_s = 0.001%.  If necessary, make sure to rearrange the equations to achieve convergence.

-5x₁ + 5x₃ = 12

4x₁-x₂ - x₃ = -2

6x₁ -3x₂  = -17

Problem 4b: Gauss-Seidel/Jacobi (10 pts)

Develop a spreadsheet to solve the following system of equations using Gauss-Seidel or Jacobi with relaxation.  Clearly indicate which method you used.  Use l = 0.8 and e_s = 0.001%.  If necessary, make sure to rearrange the equations to achieve convergence.

x₁ + 7x₂ -4x₃ = -51

12x₁ -x₂ +3x₃ = 8

Submit both a hardcopy and an electronic copy of your work.