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1. The question below pertains to the coefficient matrix portion of the augmented matrix given in Problem 137:34.

   The coefficient matrix portion is  (circle all classifications that apply):

   • square
   • diagonal
   • diagonally dominant
   • sparse
   • dense

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   Convergence Test for Jacobi and Gauss-Seidel

2. In Problem 137:34, are the Jacobi method the Gauss-Seidel method guaranteed to converge to the solution? Explain your answer using the appropriate convergence test.

3. In Problems 140:78–79, are the Jacobi method the Gauss-Seidel method guaranteed to converge to the solution? Explain your answer using the appropriate convergence test.

4. In Problem 140:81, are the Jacobi method the Gauss-Seidel method guaranteed to converge to the solution? Explain your answer using the appropriate convergence test.

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5. Do Problem 140:78. Perform 4 iterations.
   • Compare the results after 4 iterations with the true solution.

6. Do Problem 140:79. Perform 4 iterations.
   • Compare the results after 4 iterations with the true solution.
   • Compare the results after 4 iterations with those obtained in Problem 78 when using Jacobi iteration.
   • Which method, Jacobi or Gauss-Seidel, gives faster convergence?

7. Do Problem 140:81. Perform 4 iterations.
   
   • Compare the results after 4 iterations with the true solution.

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   The system given here is typical of those that arise in heat transfer problems (except they would be much larger systems).

8. Use the Jacobi method to approximate the solution of the system given in Problem 137:34.
   Start with approximate solution [50,100,100,100,100,50], and perform 4 iterations.

9. Use the Gauss-Seidel method to approximate the solution of the system given in Problem 137:34.
   Start with approximate solution [50,100,100,100,100,50], and perform 4 iterations.

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