Numerical Analysis: Chapter 2 Exercises

General Properties, Norms, and Condition Number

• Review Problems Pages 93 - 99: #2.2, 2.3, 2.5, 2.22, 2.23, 2.25, 2.52, 2.53, 2.54, 2.55, 2.57, 2.58, 2.59, 2.61, 2.62, 2.65

• Exercise #2.33 page 99.

• Find the condition number of each of the following matrices (find it using both the 1-norm and the infinity-norm):

           3   4   3                A =    .780   .563
    A =    1   5  -1                       .913   .659
           6   3   7
       
  

Residuals

• Review Problem #2.63, page 95.

• Let Ax = b be the following system:

  
        A =   .780   .563          b =  .217
              .913   .659               .254
  
  

  1. Compute the residual for the "solution" [ .999, -1.001 ]^t.

  2. Compute the residual for the "solution" [ .341, -.087 ]^t.

  3. Which solution is better? How do you know? Can you tell just from the residual?

  4. What does the condition number of A tell you?

Basic Gaussian Elimination

• Use Gaussian Elimination to solve the following system and find the LU decomposition of the coefficient matrix.

  
           3x + 4y + 3z = 10
            x + 5y -  z =  7
           6x + 3y + 7z = 15
  
  

• Use Gaussian Elimination with 4-digit arithmetic (rounding to nearest after each operation) to solve the following system.

          0.1036x + 0.2122y = 0.7381
          0.2081x + 0.4247y = 0.9327