12 - "Naive Gaussian Elimination System of Equations 2 x1 3...

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2/24/2010 1 “Naive” Gaussian Elimination: System of Equations: 33 6 4 2 5 2 4 9 3 2 3 2 1 3 2 1 3 2 1 x x x x x x x x x System in matrix form (Ax = b): >> A = [2 3 1; 4 2 5; 1 4 6] A = 2 3 1 4 2 5 1 4 6 >> b = [9; 2; 33] b = 9 2 33 First step – form augmented matrix by combining A and b >> C = [A b] C = 2 3 1 9 4 2 5 2 1 4 6 33 In general: [M1 M2 M3 ... Mn] % OK as long as all matrices have same number of rows [M1; M2; M3; ... Mn] % OK as long as all matrices have same number of columns M1 M2 M3 Mn M1 M2 M3 Mn
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2/24/2010 2 2 3 1 9 4 2 5 2 Pivot row Pivot 1 4 6 33 All elements below the pivot are converted to zero by applying row = row – (element below pivot/ pivot) * pivot row To be zeroed In this case row 2 = row 2 – (4/2) * pivot row row 3 = row 3 – (1/2) * pivot row Row 1 is the pivot row, C(1,1) is the pivot: P = C(1,1); Subtract C(2,1)/P times the pivot row from row 2: >> C(2,:) = C(2,:) - (C(2,1)/P) * C(1,:) C = 2 3 1 9 0 -8 3 -20 1 4 -6 33 Subtract C(3,1)/P times the pivot row from row 3: >> C(3,:) = C(3,:) - (C(3,1)/P) * C(1,:) C = 2.0000 3.0000 1.0000 9.0000 0 8 0000 3 0000 20 0000 0 -8.0000 3.0000 -20.0000 0 2.5000 -6.5000 28.5000
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