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Gauss-Seidel method is used to solve the following equations (as per the given order).
x1 + 2x2 + 3x3 = 5
2x1 + 3x2 + x3 = 1
3x1 + 2x2 + x3 = 3
Assuming initial guess as x1 = x2 = x3 = 0, the value of x3 after the first iteration is ________
x1 + 2x2 + 3x3 = 5
2x1 + 3x2 + x3 = 1
3x1 + 2x2 + x3 = 3
Assuming initial guess as x1 = x2 = x3 = 0, the value of x3 after the first iteration is ________
(Important - Enter only the numerical value in the answer)
Correct answer is '-6'. Can you explain this answer?
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Gauss-Seidel method is used to solve the following equations (as per t...
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Solution:
To solve the given system of equations using the Gauss-Seidel method, we start with an initial guess and iteratively update the values of the variables until convergence is achieved.
Given equations:
1. x1 + 2x2 + 3x3 = 5
2. 2x1 + 3x2 + x3 = 1
3. x1 + 2x2 + x3 = 3
Let's denote the initial guess as x1^(0) = x2^(0) = x3^(0) = 0.
Step 1: Substitute the initial guess into the equations and solve for x1, x2, and x3.
1. x1^(1) = (5 - 2x2^(0) - 3x3^(0))/1 = 5/1 = 5
2. x2^(1) = (13 - 2x1^(1) - x3^(0))/3 = (13 - 2(5) - 0)/3 = 1
3. x3^(1) = (3 - x1^(1) - 2x2^(1))/1 = (3 - 5 - 2(1))/1 = -6
Step 2: Update the values of x1, x2, and x3 using the values obtained in step 1.
x1^(1) = 5
x2^(1) = 1
x3^(1) = -6
Therefore, after the first iteration, the value of x3 is -6.
Explanation:
- The Gauss-Seidel method is an iterative algorithm used to solve a system of linear equations.
- It starts with an initial guess and updates the values of the variables one by one, using the most recently updated values in each iteration.
- In this problem, we are given an initial guess of x1 = x2 = x3 = 0.
- In the first iteration, we substitute these values into the given equations and solve for x1^(1), x2^(1), and x3^(1).
- The updated values are then used to update the values of x1, x2, and x3 in the next iteration.
- After the first iteration, we find that x3^(1) = -6.
- This means that if we continue the iterations, the value of x3 will converge to -6 as the algorithm progresses.
- The negative sign indicates that the value of x3 is decreasing with each iteration.
- The correct answer is indeed -6, as stated in the question.
To solve the given system of equations using the Gauss-Seidel method, we start with an initial guess and iteratively update the values of the variables until convergence is achieved.
Given equations:
1. x1 + 2x2 + 3x3 = 5
2. 2x1 + 3x2 + x3 = 1
3. x1 + 2x2 + x3 = 3
Let's denote the initial guess as x1^(0) = x2^(0) = x3^(0) = 0.
Step 1: Substitute the initial guess into the equations and solve for x1, x2, and x3.
1. x1^(1) = (5 - 2x2^(0) - 3x3^(0))/1 = 5/1 = 5
2. x2^(1) = (13 - 2x1^(1) - x3^(0))/3 = (13 - 2(5) - 0)/3 = 1
3. x3^(1) = (3 - x1^(1) - 2x2^(1))/1 = (3 - 5 - 2(1))/1 = -6
Step 2: Update the values of x1, x2, and x3 using the values obtained in step 1.
x1^(1) = 5
x2^(1) = 1
x3^(1) = -6
Therefore, after the first iteration, the value of x3 is -6.
Explanation:
- The Gauss-Seidel method is an iterative algorithm used to solve a system of linear equations.
- It starts with an initial guess and updates the values of the variables one by one, using the most recently updated values in each iteration.
- In this problem, we are given an initial guess of x1 = x2 = x3 = 0.
- In the first iteration, we substitute these values into the given equations and solve for x1^(1), x2^(1), and x3^(1).
- The updated values are then used to update the values of x1, x2, and x3 in the next iteration.
- After the first iteration, we find that x3^(1) = -6.
- This means that if we continue the iterations, the value of x3 will converge to -6 as the algorithm progresses.
- The negative sign indicates that the value of x3 is decreasing with each iteration.
- The correct answer is indeed -6, as stated in the question.
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Gauss-Seidel method is used to solve the following equations (as per the given order).x1 + 2x2 + 3x3 = 52x1 + 3x2 + x3 = 13x1 + 2x2 + x3 = 3Assuming initial guess as x1 = x2 = x3 = 0, the value of x3 after the first iteration is ________(Important - Enter only the numerical value in the answer)Correct answer is '-6'. Can you explain this answer?
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