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Linear Systems: Iterative Methods in Physics

Learn about Jacobi and Gauss-Seidel methods for solving simultaneous linear algebraic equations, including algorithmic steps, iteration results, and convergence analysis. Explore examples and applications in physics.

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Linear Systems: Iterative Methods in Physics

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  1. Linear Systems – Iterative methods EvarLutfallaSadraddin College of Science Physics Department Salahaddin University-Erbil Evar Lutfalla Sadraddin

  2. Linear Systems – Iterative methods • Jacobi Method • Gauss-Siedel Method

  3. Jacobi iteration

  4. Use the latest update Gauss-Seidel (GS) iteration

  5. Gauss-Seidel Method Algorithm A set of n equations and n unknowns: If: the diagonal elements are non-zero Rewrite each equation solving for the corresponding unknown ex: First equation, solve for x1 Second equation, solve for x2 . . . . . .

  6. Gauss-Seidel Method Algorithm Rewriting each equation From Equation 1 From equation 2 From equation n-1 From equation n

  7. Gauss-Seidel Method Algorithm General Form of each equation

  8. Gauss-Seidel Method Algorithm General Form for any row ‘i’ How or where can this equation be used?

  9. Graphical depiction of the difference between (a) the Gauss-Seidel and (b) the Jacobi iterative methods for solving simultaneous linear algebraic equations.

  10. Jacobi Iterative Technique Consider the following set of equations.

  11. Convert the set Ax = b in the form of x = Tx + c.

  12. Start with an initial approximation of:

  13. Results of Jacobi Iteration:

  14. Gauss-Seidel Iterative Technique Consider the following set of equations.

  15. Results of Gauss-Seidel Iteration: (Blue numbers are for Jacobi iterations.)

  16. The solution is: x1= 1, x2 = 2, x3 = -1, x4 = 1 It required 15 iterations for Jacobi method and 7 iterations for Gauss-Seidel method to arrive at the solution with a tolerance of 0.00001. While Jacobi would usually be the slowest of the iterative methods, it is well suited to illustrate an algorithm that is well suited for parallel processing!!!

  17. EXAMPLE Gauss-Seidel Method Problem Statement. Use the Gauss-Seidel method to obtain the solution for Note that the solution is Solution. First, solve each of the equations for its unknown on the diagonal:

  18. By assuming that x2 and x3 are zero This value, along with the assumed value of x3 =0, can be substituted into Eq.(E11.1.2) to calculate

  19. The first iteration is completed by substituting the calculated values for x1 and x2 into Eq.(E11.1.3) to yield For the second iteration, the same process is repeated to compute

  20. Gauss-Seidel Method: Example 2 Given the system of equations The coefficient matrix is: With an initial guess of Will the solution converge using the Gauss-Siedel method?

  21. Gauss-Seidel Method: Example 2 Checking if the coefficient matrix is diagonally dominant The inequalities are all true and at least one row is strictly greater than: Therefore: The solution should converge using the Gauss-Siedel Method

  22. Gauss-Seidel Method: Example 2 Rewriting each equation With an initial guess of

  23. Gauss-Seidel Method: Example 2 After Iteration #1 Substituting the x values into the equations After Iteration #2

  24. Gauss-Seidel Method: Example 3 Given the system of equations Rewriting the equations With an initial guess of

  25. Gauss-Seidel Method The Gauss-Seidel Method can still be used The coefficient matrix is not diagonally dominant But this is the same set of equations used in example #2, which did converge. If a system of linear equations is not diagonally dominant, check to see if rearranging the equations can form a diagonally dominant matrix.

  26. Gauss-Seidel Method Not every system of equations can be rearranged to have a diagonally dominant coefficient matrix. Observe the set of equations Which equation(s) prevents this set of equation from having a diagonally dominant coefficient matrix?

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