ChE 250 Numeric Methods

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ChE 250 Numeric Methods

• Lecture 11, Chapra Chapter 9
• 20070209

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Gauss Elimination

• Background
• Systems of linear equations
• Determinates, Cramers Rule
• Calculate with Matlab/Scilab
• Naïve Gauss Elimination
• Pitfalls
• singularity
• Improving the Method
• Pivoting
• Scaling
• Non-linear systems
• Gauss-Jordan

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Linear Equations

• For a set of n equations and n unknowns, the
  equations are said to be linear if they can be
  expressed as a sum of coefficients and the
  variable only
• It is convenient to use matrix arithmetic to
  represent the set of equations

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Linear Equations

• Call the square matrix of a coefficients A and
  the column vector of variables X and the column
  vector of b coefficients B
• To solve for the variable values given all of the
  constants we can multiply both sides of the
  equation by the inverse matrix of A-1

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Linear Equations

• The only problem with this neat solution is that
  generally it is difficult to calculate the
  inverse matrix, especially of the matrix is large
• Scilab/Matlab can easily calculate inverse
  matrix, and will indicate if it is singular or
  nearly singular, but it will not detect round off
  error
• Matlab limit?

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Two Equations

• For two equations and two unknowns the graphical
  solution is available
• This can easily verify the matrix solution
• Matlab is very easy to use for this type of
  problem

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Cramers Rule

• Cramers rule relies on the evaluation of the
  determinate of the coefficient matrix as well as
  that of a modified matrix
• Because this is computationally intensive for
  large n, this method is only used for small
  systems

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Gauss Elimination

• Gauss Elimination systematically removes one
  variable at a time from the set until all are
  solved
• No determinate or inverse matrix is required
• This is important because calculation time is
  proportional to the cube of the order, n

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Gauss Elimination

• The idea is to eliminate one variable from all
  subsequent rows by subtracting a ratio of the
  current row
• The prime indicates how many subtractions the row
  experienced
• On the last row, that variable will be solved
• Then back substitute all the way up

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Pitfalls

• There can be the case were two of the equations
  in the system are singular, they do not have a
  solution
• The three examples below show no solution (a),
  infinite solution (b), and ill-conditioned (c)

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Pitfalls

• The matrix may contain zero values on the
  diagonal, which cause a division by zero error
  in a program
• Also the constants may be dissimilar in
  magnitude, which can cause round off errors in
  the calculations and loss of significant figures

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Improving Gauss

• Two techniques to reduce errors are scaling and
  pivoting
• Pivoting
• Select the largest value of aij for the current
  column j, igtj and use that as the pivot
• Scaling
• Used with pivoting, this uses a scaled value of
  the row constant for comparison

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Gauss with Scaled Pivots

• First, scale all rows
• Then decide the pivot row order
• Then go back to the original matrix and perform
  the pivot elimination
• The scaled equations also may show any
  singularities

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Gauss with Scaled Pivots

• To visualize the elimination, we can actually
  reorder the set to show the diagonal pivot
• This step is not necessary in a computer
  algorithm, just follow the pivot order
• Once the elimination is complete, solve for xn
  and then substitute this value into row n-1 and
  solve for xn-1, etc.
• Questions?

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Gauss and Non-Linear Systems

• The first step is to make a linear Taylor
  expansion of the equations, and use the
  multi-dimensional Newton-Raphson method
• Using matrix notation our previous equation
  reduces to a linear set.

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Systems of Nonlinear Equations
Taylor Expansion for two variables
Rearrange and group
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Non-Linear Equations
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Non-Linear Systems

• For n equations the partials make a square matrix
• This can be solved using the Gauss method

• The problem comes up of how to calculate all
  those partials.frequently a finite difference
  method must be used

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Gauss-Jordan

• Gauss-Jordan method eliminates forward and
  backward at every step
• This produces a solution without the back
  substitution set which may be easier to program
• However it involves approximately 50 more
  calculations which adds time and error to large
  sets

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Preparation for Feb 12th

• Reading
• Chapra Chapter 10 LU Decomposition