ECE3040 Chapter 4.pdf - Chapter 4 The lecture notes are selected from the public Powerpoint lecture documents provided by Authors Autar Kaw Sri Harsha

ECE3040 Chapter 4.pdf - Chapter 4 The lecture notes are...

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Chapter 4 The lecture notes are selected from the public Powerpoint lecture documents provided by Authors: Autar Kaw, Sri Harsha Garapati
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Naïve Gauss Elimination
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Naïve 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
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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
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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
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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 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)
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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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