Lecture 7 Systems of Equations

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Lecture 7 - Systems of Equations

• CVEN 302
• June 17, 2002

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Lectures Goals

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

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Computer Program

• The program GEdemo(A,b) does the Gaussian
• elimination for a square matrix (nxn). It does
• not do any pivoting and works for only one
• b vector.

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Test the Program

• Example 1
• Example 2
• New Matrix
• 2X1 4X2 - 2 X3 - 2 X4 - 4
• 1X1 2X2 4X3 - 3 X4 5
• - 3X1 - 3X2 8X3 - 2X4 7
• - X1 X2 6X3 - 3X4
  7

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Problem with Gaussian Elimination

• The problem can occur when a zero appears in the
  diagonal and makes a simple Gaussian elimination
  impossible.
• Pivoting changes the matrix so that it will
  become diagonally dominate and reduce the
  round-off and truncation errors in the solving
  the matrix.

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Example of Pivoting

• 2 X1 4 X2 - 2 X3 10
• X1 2 X2 4 X3 6
• 2 X1 2 X2 1X3 2
• Answer X1 X2 X3 -3.40, 4.30, 0.20

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Computer Program

• GEPivotdemo(A,b) is a program, which will do a
  Gaussian elimination on matrix A with pivoting
  technique to make matrix diagonally dominate.
• The program is modification to handle a single
  value of b

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Question?

• How would you modify the programs to handle
  multiple inputs?
• What is diagonal matrix, upper triangular matrix,
  and lower triangular matrix?
• Can you do a column exchange and how would you
  handle the problem if it works?

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Gaussian Elimination

• If the diagonal is not dominate the problem can
  have round off error and truncation errors.
• The scaling will result in problems

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Question?

• What happens with the following example?
• 0.0001X1 0.5 X2 0.5
• 0.4000X1 - 0.3 X2 0.1
• What happens is the second equation becomes
  0.4000X1 - 2000 X2 -2000

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Question?

• What happens with the following example for
  values with two-significant figures?
• 0.4000 X1 - 0.3 X2 0.1
• 0.0001 X1 0.5 X2 0.5

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Scaling

• Scaling is an operation of adjusting the
  coefficients of a set of equations so that they
  are all of the same magnitude.

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Scaling

• A set of equations may involve relationships
  between quantities measured in a widely different
  units (N vs. kN, sec vs hrs, etc.) This may
  result in equation having very large number and
  others with very small , if we select pivoting
  may put numbers on the diagonal that are not
  large in comparison to other rows and create
  round-off errors that pivoting was suppose to
  avoid.

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Scaling

• What happens with the following example?
• 3X1 2 X2 100X3 105
• - X1 3 X2 100X3 102
• X1 2 X2 - 1X3 2

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Scaling

• The best way to handle the problem is to
  normalize the results.
• 0.03X1 0.02 X2 1.00X3 1.05
• - 0.01X1 0.03 X2 1.00X3 1.02
• 0.50X1 1.00 X2 - 0.50X3 1.00

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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 1

• X1 3X2 5
• 2X1 4X2 6

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Example 2

• -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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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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Summary

• Scaling of the problem will help in the
  convergence.
• 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.

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Homework

• Check the Homework webpage