Math Problem Statement
Solution
The system of equations to be solved using the Gauss-Seidel method is:
Initial approximations:
Rearrange the equations for the Gauss-Seidel method:
We need to solve each equation for the variable on the left side. The rearranged system will look like this:
1st Iteration:
Using the initial approximations , , and :
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Update x:
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Update y:
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Update z:
So after the 1st iteration:
2nd Iteration:
Now, use the updated values from the 1st iteration:
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Update x:
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Update y:
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Update z:
So after the 2nd iteration:
Would you like a detailed explanation of the method or help with further iterations?
Related Questions:
- What is the convergence criterion for the Gauss-Seidel method?
- How does the Gauss-Seidel method differ from the Jacobi method?
- Why is the system of equations rearranged for the Gauss-Seidel method?
- Can the Gauss-Seidel method fail to converge for certain types of systems?
- How does the choice of initial approximations affect the iterations in Gauss-Seidel?
Tip: Always check the diagonal dominance of the matrix for better convergence in the Gauss-Seidel method.
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Math Problem Analysis
Mathematical Concepts
Gauss-Seidel Method
Linear Algebra
Iterative Methods
Formulas
x = (4.5 - 2y) / 3
y = (5 - 2x + z) / 3
z = (y + 0.5) / 2
Theorems
Gauss-Seidel Iterative Method
Convergence Theorems for Iterative Methods
Suitable Grade Level
College/University Level
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