Lecture 9 Gaussian Elimination

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Lecture 9 - Gaussian Elimination

• CVEN 302
• September 12, 2001

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Lectures Goals

• Discuss how to solve systems
• Gaussian Elimination
• Gauss-Jordan
• Tridiagonal Solver
• Problems with the technique
• Examples
• Iterative Techniques

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Gauss-Jordan Method

• The Gauss-Jordan Method is similar to the
  Gaussian Elimination.
• The method requires almost 50 more operations.

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Gauss-Jordan Method
The Gauss-Jordan method changes the matrix into
the identity matrix.
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Gauss-Jordan Method

• There are one phases to the solving technique
• Elimination --- use row operations to convert the
  matrix into an identity matrix.
• The new b vector is the solution to the x values.
  

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Gauss-Jordan Algorithm

• Ax b
• Augment the n x n coefficient matrix with the
  vector of right hand sides to form a n x (n1)
• Interchange rows if necessary to make the value
  a11 with the largest magnitude of any coefficient
  in the first row
• Create zero in 2nd through nth row in first row
  by subtracting ai1 / a11 times first row from ith
  row

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Gauss-Jordan Elimination Algorithm

• Repeat (2) (3) for first through the nth rows,
  putting the largest magnitude coefficient in the
  diagonal by interchanging rows (consider only row
  j to n ) and then subtract times the jth row
  from the ith row so as to create zeros in all
  positions of jth column and the diagonal becomes
  all ones
• Solve for all of the equations, xi ai,n1

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Example - Gauss Jordan

• X1 3X2 5
• 2X1 4X2 6

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Example - Gauss-Jordan

• -3X1 2X2 - X3 -1
• 6X1 - 6X2 7X3 -7
• 3X1 - 4X2 4X3 -6

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Band Solver

• Large matrices tend to be banded, which means
  that the matrix has a band of non-zero
  coefficients and zeroes on the outside of the
  matrix.
• The simplest of the methods is the Thomas Method,
  which is used for a tridiagonal matrix.

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Advantages of Band Solvers

• The method reduce the number of operations and
  save the matrix in smaller amount of memory.
• The band solver is faster and is useful for large
  scale matrices.

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Thomas Method

• The method takes advantage of the bandedness of
  the matrix.
• The technique uses a two phase process.
• The first phase is to obtain the coefficients
  from the sweep.
• The second phase solves for the x values.

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Thomas Method

• The first phase starts with the first row of
  coefficients scales the a and r coefficients.
• The second phase solves for x values using the a
  and r coefficients.

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Thomas Method

• The program for the method is given as
  demoThomas(a,d,b,r)
• The algorithm is from the textbook, where a,d,b,
  r are vectors from the matrix.

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Chapter 4Iterative Techniques
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Iterative Techniques

• The method of solving simultaneous linear
  algebraic equations using Gaussian Elimination
  and the Gauss-Jordan Method. These techniques
  are known as direct methods. 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

• 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 1(Jacobi Example)

• 4X1 - 2X2 3
• - X1 5X2 -2

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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
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Jacobi Example
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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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Summary

• Gauss-Jordan method is more computational intense
  and does not improve the round-off errors.
  However, it is useful for finding matrix
  inverses.
• Banded matrix solvers are faster and use less
  memory.
• Convergence conditions need to be met in-order
  for iterative techniques to converge
• Jacobi method upgrades the values after each
  iteration

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Homework

• Check the Homework webpage