### More Chapter 2 Exercises

Gaussian Elimination. LU Decomposition, and Pivoting Strategies
• Review Problems Pages 93 - 96: # 2.14, 2.18, 2.19, 2.34, 2.35, 2.37, 2.38, 2.39, 2.40, 2.41, 2.48, 2.64.

• Exercises Page 97 - 99: #2.7, 2.11, 2.17.

Residual and Iterative Improvement (Iterative Refinement)

• Review Problems Page 96 # 2.66, 2.79

• Using Gaussian elimination with naive pivoting on a 4-digit machine gives the solution x = 0, y = 1 for the following system:
```        10^-5 x + y = 1
x + y = 2
```
Compute the residual for the solution and use one iteration of iterative improvement to get a better solution.

• Using Gaussian elimination with naive pivoting on a 4-digit machine gives the solution x = 5, y = 2.002, and z = 0 to the system
```       -0.002x +     4y +     4z = 7.998
-2x + 2.906y - 5.387z = -4.481
3x - 4.031y - 3.112z = -4.143
```
Use one iteration of iterative improvement to get a better solution.

Jacobi and Gauss-Seidel Iterative Methods

Consider the following system:

```      6x - 2y +  z = 11
x + 2y - 5z = -1
-2x + 7y + 2z = 5
```
1. First rearrange the equations so the coefficient matrix is diagonally dominant.

2. Perform the first two iterations of the Jacobi method (on the re-arranged system) starting with the vector x = 0, y = 0, z = 0.

3. Perform the first two iterations of the Gauss-Seidel method for this system (again, rearranged and starting with the 0 vector).