Lecture 10 System of Equations

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Lecture 10 System of Equations

• February 15, 2001
• CVEN 302

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Lectures Goals

• Iterative Techniques
• Jacobian method
• Gaus-Siedel method
• Relaxation technique

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Iterative Techniques

• The method of solving simultaneous linear
  algebraic equations using Gaussian Elimination.
  Problems can arise from round-off errors and zero
  on the diagonal.
• One means of obtaining an approximate solution to
  the equations is to use an educated guess.

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Iterative Methods

• We will look at three iterative methods
• Jacobi Method
• Gauss-Seidel Method
• Successive over Relaxation (SOR)

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Convergence Restrictions

• There are two conditions for the iterative method
  to converge.
• Necessary that 1 coefficient in each equation is
  dominate.
• The sufficient condition is that the diagonal is
  dominate.

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Jacobi Iteration

• If the diagonal is dominant, the matrix can be
  rewritten in the following form

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Jacobi Iteration

• The technique can be rewritten in a shorthand
  fashion, where D is the diagonal, A is the
  matrix without the diagonal and c is the
  right-hand side of the equations.

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Jacobi Iteration

• The technique solves for the entire set of x
  values for each iteration.
• The problem does not update the values until an
  iteration is completed.

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Example (Jacobi Iteration)

• 4X1 2X2 2
• 2X1 10X2 4X3 6
• 4X2 5X3 5
• Solution (X1 , X2 , X3 ) (0.41379, 0.17241,
  0.86206)

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Jacobi Example

• Formulation of the matrix

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Jacobi Iteration

• Iteration 1 2 3 4 5 6 7
• X 1 0.5 0.2 0.45 0.324 0.429 0.376 0.42
• X 2 0.6 0.1 0.352 0.142 0.248 0.16 0.204
• X 3 1 0.52 0.92 0.718 0.886 0.802 0.872

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Jacobi Program

• The computer program is setup to do the Jacobi
  method for any size square matrix
• Jacobi(A,b)
• The program can has options for maximum number of
  iterations, nmax, and tolerance, tol.
• Jacobi(A,b,nmax,tol)

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Gauss-Seidel Iterations

• The Gauss-Seidel / Seidel technique is similar to
  the Jacobi iteration technique with one
  difference.
• The method updates the results continuously. It
  uses the new information from the previous
  iteration to accelerate converge to a solution.

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Gauss-Seidel Model

• The Gauss-Seidel Algorithm
• The combined vector is upgraded ever term.

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Example (Gauss-Seidel Iteration)

• 4X1 2X2 2
• 2X1 10X2 4X3 6
• 4X2 5X3 5
• Solution (X1 , X2 , X3 ) (0.41379, 0.17241,
  0.86206)

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Gauss-Seidel Example

• Formulation of the matrix problem

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Gauss-Seidel Iteration

• Iteration 1 2 3 4 5 6 7
• X 1 0.5 0.25 0.345 0.384 0.401 0.408 0.411
• X 2 0.5 0.31 0.231 0.197 0.183 0.177 0.175 X
  3 0.6 0.75 0.815 0.842 0.854 0.858 0.858

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Gauss-Seidel Model

• The computer program is setup to do the
  Gauss-Seidel method for any size square matrix
• Seidel(A,b)
• The program can has options for maximum number of
  iterations, nmax, and tolerance, tol.
• Seidel(A,b,nmax,tol)

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Example - Iteration with no diagonal domination

• 3X1 - 3X2 5X3 4
• X1 2X2 - 6X3 3
• 2X1 - X2 3X3 1
• Solution (X1 , X2 , X3 ) (1.00, -2.00, -1.00)

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Using the GS algorithm

• Using the Gauss-Seidel Program with the
  following A matrix and b vector.
• Solution (X1 , X2 , X3 ) (1.00, -2.00, -1.00)

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Using a Gauss-Seidel Iteration

• Iteration 1 2 3 4 5 6
  7
• X 1 1.33 2.283 3.425 2.057 2.765 1.474 2.023
• X 2 1.5 1.825 -0.243 -1.953 -5.622 -4.95 -7.91
  5
• X 3 0.33 -0.16 -0.58 -2.031 -1.689 -3.384 -0.3
  34

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Successive over Relaxation

• The technique is a modification on the
  Gauss-Seidel method with an additional parameter,
  w, that may accelerate the convergence of the
  iterations.
• The weighting parameter, w, has two ranges 0 lt w
  lt1, and 1lt w lt2. If w 1, then the problem is
  the Gauss-Seidel technique.

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SOR Method

• The SOR algorithm is defined as
• The difference is the weighting parameter, w.

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The Weighting Parameter

• If the parameter, w is under 1, the residuals
  will be under-relaxed.
• If the parameter, w 1, the residuals are equal
  to a Gauss-Seidel model.
• If 1lt w lt 2 the residuals will be over-relaxed
  and will general help accelerate the convergence
  of the solution.

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Example of SOR

• 4X1 2X2 2
• 2X1 10X2 4X3 6
• 4X2 5X3 5
• Solution (X1 , X2 , X3 ) (0.41379, 0.17241,
  0.86206)

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SOR Example

• Formulation of the SOR Algorithm

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SOR Example
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Effects of w Parameter

• Using the SOR program SOR(A,b,w,nmax,tol) with
  nmax50 and tol 0.000001

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Summary

• Convergence conditions need to be met in-order
  for iterative techniques to converge
• Jacobi method upgrades the values after each
  iteration.
• Gauss-Seidel upgrades continuously through
  method.
• SOR (Successive over Relaxation) uses the
  residuals to accelerate the convergence.