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3. 2x₁ - x₂ 2 xj - 3x2 + X; = -2 -X1 + X2 - 3x3 = -6

3. 2x₁ - x₂ 2 xj - 3x2 + X; = -2 -X1 + X2 - 3x3 = -6

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.5: Iterative Methods For Solving Linear Systems
Problem 6EQ
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Please answer number 7
Gauss-Seidel Method: Example 2 Rewriting each equation With an initial guess of [12 3 -5 a, 15 189 3 a2 =28 1-0 3 76 X₁ x₂ = X₂ 7 13 a 1-3x₂ + 5x₂ 12 28-x₁ - 3x₂ 5 76-3x, -7x₂ 13 X₂ x₁ = 1-3(0)+5(1) 12 x₂ = 28-(0.5)-3(1) 5 = 4.9000 76-3(0.50000)-7(4.9000) 13 = 0.50000 = 3.0923
Transcribed Image Text:Gauss-Seidel Method: Example 2 Rewriting each equation With an initial guess of [12 3 -5 a, 15 189 3 a2 =28 1-0 3 76 X₁ x₂ = X₂ 7 13 a 1-3x₂ + 5x₂ 12 28-x₁ - 3x₂ 5 76-3x, -7x₂ 13 X₂ x₁ = 1-3(0)+5(1) 12 x₂ = 28-(0.5)-3(1) 5 = 4.9000 76-3(0.50000)-7(4.9000) 13 = 0.50000 = 3.0923
In Exercises 1-4, apply the Jacobi method to the given system of linear equations, using the initial approximation (x₁, x2,...,x₁) = (0, 0, . . . , 0). Continue performing iterations until two successive approximations are identical when rounded to three significant digits. 1. 3x₁ x₂ = 2 2. - 4x₁ + 2x₂ = -6 3x₁ - 5x₂ = 1 x₁ + 4x₂ = 5 3. 2x₁ - x₂ 2 4. 4x₁ + x₂ + x3 - x₁ = 3x₂ + x3 = -2 X1 - 7x₂ + 2x3 -x₁ + x₂-3x3 = -6 3x₁ 5. Apply the Gauss-Seidel method to Exercise 1. 6. Apply the Gauss-Seidel method to Exercise 2. 7. Apply the Gauss-Seidel method to Exercise 3. 8. Apply the Gauss-Seidel method to Exercise 4. || = = + 4x3 7 2 11
Transcribed Image Text:In Exercises 1-4, apply the Jacobi method to the given system of linear equations, using the initial approximation (x₁, x2,...,x₁) = (0, 0, . . . , 0). Continue performing iterations until two successive approximations are identical when rounded to three significant digits. 1. 3x₁ x₂ = 2 2. - 4x₁ + 2x₂ = -6 3x₁ - 5x₂ = 1 x₁ + 4x₂ = 5 3. 2x₁ - x₂ 2 4. 4x₁ + x₂ + x3 - x₁ = 3x₂ + x3 = -2 X1 - 7x₂ + 2x3 -x₁ + x₂-3x3 = -6 3x₁ 5. Apply the Gauss-Seidel method to Exercise 1. 6. Apply the Gauss-Seidel method to Exercise 2. 7. Apply the Gauss-Seidel method to Exercise 3. 8. Apply the Gauss-Seidel method to Exercise 4. || = = + 4x3 7 2 11
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