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NaiveGaussianElimination

# NaiveGaussianElimination - Solving Sets of Linear Equations...

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Solving Sets of Linear Equations using Naïve Gaussian Elimination Given: Ax = b 11 12 13 14 11 21 22 23 24 22 31 32 33 34 33 41 42 43 44 44 aaaa x b x b x b x b ⎡⎤ ⎡⎤⎡⎤ ⎢⎥ ⎢⎥⎢⎥ = ⎣⎦⎣⎦ ⎣⎦ Step 1: Augment A with b 11 12 13 14 1 21 22 23 24 2 31 32 33 34 3 41 42 43 44 4 b b b b Step 2: Starting with i =1, set the pivot at a ii (so pivot is a 11 ) a) - multiply 1 st row by 21 21 11 a m a =− - add result to second row Æ 21 0 a = 22 22 21 12 aam a =+ 23 23 21 13 a = + 24 24 21 14 a = + 22 2 1 bbm b 1 b) -multiply 1 st row by 31 31 11 a m a -add result to 3 rd row c) -multiply 1 st row by 41 41 11 a m a -add result to 4 th row

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11 12 13 14 1 22 23 24 2 32 33 34 3 42 43 44 4 0 0 0 aaaa b aaa b b b ⎡⎤ ⎢⎥ ⎣⎦ Step 3: Pivot is a 22 a) -multiply 2 nd row by 32 32 22 a m a =− -add result to 3 rd row Æ 31 0 a = 32 0 a = 33 33 32 23 aam a =+ 34 34 32 24 a = + 33 3 2 bbm b 2 b) -multiply 2 nd row by 42 42 22 a m a -add result to 4 th row 11 12 13 14 1 22 23 24 2 33 34 3 43 44 4 0 00 b b aa b b Step 4: Pivot is a 33 a) -multiply 3 rd row by 43 43 33 a m a -add result to 4 th row Æ 41 0 a = 42 0 a = 43 0 a = 44 44 43 34 a = + 44 4 3 b 3 = +
11 12 13 14 1 22 23 24 2 33 34 3 44 4 0 00 000 aaaab
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NaiveGaussianElimination - Solving Sets of Linear Equations...

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