1. See This Maple Help Sheet to learn how to use Maple to calculate norms and the condition number.

2. A system  Ax = b   is given by:

   24 7 –5 2 8 –30 15 3 0 –4 12 6 3 2 7 18

   x1 x2 x3 x4

   =

   36 375 24 132

   .

   Suppose the vector

   x̃   =   2 –14 –10 13

   is the computed solution (with truncation error) or is an approximate solution obtained by an approximating method.
   1. Calculate Ax̃.    Answer:  [ 26, 325, 14, 142]^T
   2. Calculate the residual vector.    Answer:  r = [ 10, 50, 10, 10]^T
   3. If x̃ were a reasonable approximation of the exact solution x*, then the residual vector r would be close to what?
   4. Calculate || r ||₁.    Answer:  80
   5. Calculate || r ||₂.    Answer:  52.915
   6. Calculate || r ||_∞.    Answer:  50

3. Comment:   The correct (true) solution of the system in Problem 2 above is

   x*   =   3 –15 –9 12

   .

   1. Determine the error vector:  e = x* – x̃.
   2. What is || e ||₁ ?    Answer:  4
   3. What is || e ||₂ ?    Answer:  2
   4. What is || e ||_∞?    Answer:  1

4. Given the linear system:

   0.835 0.667 0.333 0.266

   x1 x2

   =

   0.168 0.067

   1. Solve the system for x. Use Maple to check your answer.
   2. Determine A^-1. Use Maple to check your answer: Inverting A Matrix.
   3. Use the infinity norm to determine the condition number of A.    Answer:  1.754336 x 10⁶
   4. Is A ill-conditioned or not?

5. Using the same cofficient matrix A as in Problem 4, solve the following system with a slightly different input vector.

   0.835 0.667 0.333 0.266

   x1 x2

   =

   0.167 0.067

   Notice that a very small change in one digit of input vector b produces a very large change in the solution. It is because A is ill-conditioned.

6. Answer the following as True or False, where x̃ is the computed solution of Ax = b.
   1. The relative residual is always a good approximation of the relative error.
   2. If the norm of the residual vector is small, then the computed solution is very close to the exact solution.
   3. If Ax̃ ≈ b, then x̃ is very close to the exact solution.

7. When is the relative residual a good approximation of the relative error?

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