__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

- First rearrange the equations so the coefficient matrix is
diagonally dominant.
- Perform the first two iterations of the Jacobi method (on the
re-arranged system) starting with the vector x = 0, y = 0, z = 0.
- Perform the first two iterations of the Gauss-Seidel method for this system (again, rearranged and starting with the 0 vector).