Linear Systems Gauss Seidel

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Linear Systems Gauss Seidel

  1. 1. Iterative Methods for Systems of Equations <ul><li>Linear Systems of Equations </li></ul><ul><ul><li>Jacobi Iteration </li></ul></ul><ul><ul><li>Gauss-Seidel </li></ul></ul><ul><ul><ul><li>Convergence and diagonal dominance </li></ul></ul></ul><ul><ul><ul><li>Relaxation </li></ul></ul></ul><ul><li>Nonlinear Systems of Equations </li></ul><ul><ul><li>Newton-Raphson </li></ul></ul>
  2. 2. Model Problem
  3. 3. Jacobi Iteration Algorithm <ul><li>Initial guesses </li></ul><ul><li>Jacobi iteration </li></ul><ul><li>Iterate until convergence criterion is met </li></ul>
  4. 4. Gauss-Seidel Method <ul><li>The Gauss-Seidel method is the most commonly used iterative method for solving linear algebraic equations </li></ul><ul><li>Improve guesses for x 2 , x 3 , …, x n using the most recently calculated values from the present iteration </li></ul><ul><li>Example iteration for 3x3 system: </li></ul>
  5. 5. Gauss-Seidel Jacobi
  6. 6. Convergence <ul><li>Convergence of an iterative method can be calculated by determining the relative percent change of each element in { x }. </li></ul><ul><li>For example, for the i th element in the j th iteration, </li></ul><ul><li>The method is ended when all elements have converged to a set tolerance. </li></ul>
  7. 7. Model Problem Solution How does operation count compare to Gauss Elimination?
  8. 8. Model Problem Rearranged What is different?
  9. 9. Diagonal Dominance <ul><li>The Gauss-Seidel method may diverge </li></ul><ul><li>Diagonal dominance guarantees convergence, but is not necessary for convergence. </li></ul><ul><li>Diagonal dominance means: </li></ul>
  10. 10. Relaxation <ul><li>Used to enhance convergence. </li></ul><ul><li>Value at a particular iteration is made up of a combination of the old value and the newly calculated value: </li></ul><ul><li>λ is a weighting factor that is assigned a value between 0 and 2. </li></ul><ul><ul><li>0 < λ < 1: underrelaxation </li></ul></ul><ul><ul><li>λ = 1: no relaxation </li></ul></ul><ul><ul><li>1 < λ ≤ 2: overrelaxation </li></ul></ul>

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