Gaussian Elimination - Gaussian Elimination Major: All...

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Unformatted text preview: Gaussian Elimination Major: All Engineering Majors Author(s): Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates Nave Gauss Elimination http://numericalmethods.eng.usf.edu Nave Gaussian Elimination A method to solve simultaneous linear equations of the form [A][X]=[C] Two steps 1. Forward Elimination 2. Back Substitution 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 . 21 . 96 8 . 106 7 . 56 . 1 8 . 4 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 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 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 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 . ....
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This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.

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Gaussian Elimination - Gaussian Elimination Major: All...

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