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Gaussian Elimination

# Gaussian Elimination - GaussianElimination Major:...

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

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Na ï ve Gauss Elimination      http://numericalmethods.eng.usf.edu
Na ï ve Gaussian Elimination A method to solve simultaneous linear  equations of the form [A][X]=[C] Two steps 1.  Forward Elimination 2. Back Substitution

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Forward Elimination = 2 . 279 2 . 177 8 . 106 1 12 144 1 8 64 1 5 25 3 2 1 x x x The goal of forward elimination is to transform the  coefficient matrix into an upper triangular matrix - = - - 735 . 0 21 . 96 8 . 106 7 . 0 0 0 56 . 1 8 . 4 0 1 5 25 3 2 1 x x x
Forward Elimination A set of  n  equations and  n  unknowns 1 1 3 13 2 12 1 11 ... b x a x a x a x a n n = + + + + 2 2 3 23 2 22 1 21 ... b x a x a x a x a n n = + + + + n n nn n n n b x a x a x a x a = + + + + ... 3 3 2 2 1 1 . . . . . . (n-1) steps of forward elimination

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Forward Elimination Step 1  For Equation 2, divide Equation 1 by        and  multiply by     . ) ... ( 1 1 3 13 2 12 1 11 11 21 b x a x a x a x a a a n n = + + + + 1 11 21 1 11 21 2 12 11 21 1 21 ... b a a x a a a x a a a x a n n = + + + 11 a 21 a
Forward Elimination 1 11 21 1 11 21 2 12 11 21 1 21 ... b a a x a a a x a a a x a n n = + + + 1 11 21 2 1 11 21 2 2 12 11 21 22 ... b a a b x a a a a x a a a a n n n - = - + + - ' 2 ' 2 2 ' 22 ... b x a x a n n = + + 2 2 3 23 2 22 1 21 ... b x a x a x a x a n n = + + + + Subtract the result from Equation 2. ________________________________________________ _ or

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Forward Elimination Repeat this procedure for the remaining  equations to reduce the set of equations as 1 1 3 13 2 12 1 11 ... b x a x a x a x a n n = + + + + ' 2 ' 2 3 ' 23 2 ' 22 ... b x a x a x a n n = + + + ' 3 ' 3 3 ' 33 2 ' 32 ... b x a x a x a n n = + + + ' ' 3 ' 3 2 ' 2 ... n n nn n n b x a x a x a = + + + . . . . . . . . . End of Step 1
Step 2 Repeat the same procedure for the 3 rd  term of  Equation 3. Forward Elimination 1 1 3 13 2 12 1 11 ... b x a x a x a x a n n = + + + + ' 2 ' 2 3 ' 23 2 ' 22 ... b x a x a x a n n = + + + " 3 " 3 3 " 33 ... b x a x a n n = + + " " 3 " 3 ... n n nn n b x a x a = + + . . . . . . End of Step 2

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Forward Elimination At the end of (n-1) Forward Elimination steps, the system  of equations will look like ' 2 ' 2 3 ' 23 2 ' 22 ... b x a x a x a n n = + + + " 3 " 3 3 " 33 ... b x a x a n n = + + ( 29 ( 29 1 1 - - = n n n n nn b x a . . . . . . 1 1 3 13 2 12 1 11 ... b x a x a x a x a n n = + + + + End of Step (n-1)
Matrix Form at End of Forward  Elimination = - ) (n- n " ' n ) (n nn " n " ' n ' ' n b b b b x x x x a a a a a a a a a a 1 3 2 1 3 2 1 1 3 33 2 23 22 1 13 12 11 0 0 0 0 0 0 0

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Back Substitution Solve each equation starting from the last equation Example of a system of 3 equations - =
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Gaussian Elimination - GaussianElimination Major:...

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