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Gaussian Elimination

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Holistic Numerical Methods Institute Other titles: Tahoma Arial Times New Roman Verdana Default Design Microsoft Equation 3.0 Gaussian Elimination Na ve ... – PowerPoint PPT presentation

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Title: Gaussian Elimination


1
Gaussian Elimination
  • Major All Engineering Majors
  • Author(s) Autar Kaw
  • http//numericalmethods.eng.usf.edu
  • Transforming Numerical Methods Education for STEM
    Undergraduates

2
Naïve Gaussian Elimination
A method to solve simultaneous linear equations
of the form AXC
Two steps 1. Forward Elimination 2. Back
Substitution
3
Forward Elimination
The goal of forward elimination is to transform
the coefficient matrix into an upper triangular
matrix
4
Forward Elimination
A set of n equations and n unknowns
. . .
. . .
(n-1) steps of forward elimination
5
Forward Elimination
Step 1 For Equation 2, divide Equation 1 by
and multiply by .
6
Forward Elimination
Subtract the result from Equation 2.

- ________________________________________________
_
or
7
Forward Elimination
Repeat this procedure for the remaining equations
to reduce the set of equations as


. . .
. . .
. . .
End of Step 1
8
Forward Elimination
Step 2 Repeat the same procedure for the 3rd term
of Equation 3.

. .
. .
. .


End of Step 2
9
Forward Elimination
At the end of (n-1) Forward Elimination steps,
the system of equations will look like


. .
. .
. .


End of Step (n-1)
10
Matrix Form at End of Forward Elimination
11
Back Substitution
Solve each equation starting from the last
equation
Example of a system of 3 equations
12
Back Substitution Starting Eqns


. .
. .
. .


13
Back Substitution
Start with the last equation because it has only
one unknown
14
Back Substitution
15
  • THE END
  • http//numericalmethods.eng.usf.edu
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