Lecture 9 System of Equations

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Lecture 9 System of Equations

• February 13, 2001
• CVEN 302

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Lectures Goals

• Gauss-Jordon
• Tridiagonal Solver
• Iterative Techniques
• Jacobian method
• Gaus-Siedel method
• Relaxation technique

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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.
• 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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Gaussian 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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Band Solver

• The technique can be applied to higher order
  banded matrices.
• The quintdiagonal matrix has a similar algorithm.

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Quintdiagonal 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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Correction Quintdiagonal Method

• There is a correction for the program, the bi
  term will change and cause problems.
• The b term must be modified before being used in
  the updates of a, c and r terms.

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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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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.
• Two iterative techniques were presented.