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Gauss Seidel Method Calculator

No of equations:

x_{1 +}

x_{2 +}

x_{3 =}

x_{1 +}

x_{2 +}

x_{3 =}

x_{1 +}

x_{2 +}

x_{3 =}

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Gauss Seidel Method Calculator

The online Gauss Seidel method calculator helps you solve systems of linear simultaneous equations using an iterative approach. This tool computes approximate solutions efficiently. You may also explore other numerical techniques such as the power method calculator for eigenvalue problems.

What is the Gauss Seidel Method?

The Gauss-Seidel method is an iterative numerical technique used to solve systems of linear equations. It works best when the system is arranged in diagonally dominant form.

This method, sometimes called the method of successive substitution, improves solution estimates step by step by immediately using newly calculated values within the same iteration.

General Formula

The matrix form of the Gauss-Seidel iteration is:

x^(k+1) = L*^-1 (b − Ux^(k))

L* = Lower triangular matrix (including diagonal elements)
U = Strictly upper triangular matrix
b = Constant vector

Lower Triangular Matrix

A matrix in which all entries above the main diagonal are zero is called a lower triangular matrix.

Example:

$$ A = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 5 & 0 \\ 1 & -1 & -2 \end{bmatrix} $$

Upper Triangular Matrix

A matrix in which all entries below the main diagonal are zero is called an upper triangular matrix.

Example:

$$ A = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 0 & 0 & -2 \end{bmatrix} $$

Gauss Seidel Iteration Algorithm

Follow these steps to apply the Gauss-Seidel method manually:

Start the process.
Rewrite the system in diagonally dominant form (if possible).
Choose a tolerance level (error threshold) e.
Rewrite each equation to isolate its leading variable.
Assume initial approximations (x₀, y₀, z₀, etc.).
Update each variable sequentially using the most recent values.
Check convergence: if |x_k+1 − x_k| < e for all variables, stop.
If not, repeat the iteration.
Display the final approximated solution.

The Gauss Seidel calculator performs these steps automatically and delivers quick results.

Gauss Seidel Method Example

Solve the system:

$$ \begin{cases} x_1 + 2x_2 = 7 \\ 8x_1 + 9x_2 = 7 \end{cases} $$

Solution Outline:

Upper triangular matrix U: $$ \begin{bmatrix} 0 & 2 \\ 0 & 0 \end{bmatrix} $$
Lower triangular matrix L*: $$ \begin{bmatrix} 1 & 0 \\ 8 & 9 \end{bmatrix} $$
Inverse of L*: $$ L*^{-1} = \begin{bmatrix} 1 & 0 \\ -0.89 & 0.11 \end{bmatrix} $$
Iteration matrix: $$ T = -L*^{-1}U = \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \end{bmatrix} $$
Constant vector: $$ C = L*^{-1}b = \begin{bmatrix} 7 \\ -5.44 \end{bmatrix} $$

Applying the iterative formula:

$$ x^{(k+1)} = Tx^{(k)} + C $$

Repeated iterations generate increasingly refined approximations of the solution.

For direct solution methods, you may use the Gaussian elimination calculator.

How the Gauss Seidel Calculator Works

Inputs:

Select the number of equations (2 or 3).
Enter coefficients and constants.
Click Calculate.

Outputs:

Lower triangular matrix inverse (L*^-1)
Iteration matrix (T)
Constant vector (C)
Step-by-step iterative updates
Final approximated solution

FAQs

Is the Jacobi method iterative?

Yes, the Jacobi method is iterative. However, it updates all variables simultaneously after each iteration, unlike Gauss-Seidel.

What is the difference between Jacobi and Gauss-Seidel?

The Jacobi method uses only values from the previous iteration. Gauss-Seidel immediately applies updated values, usually resulting in faster convergence.

Which method converges faster?

In most cases, the Gauss-Seidel method converges more rapidly than the Jacobi method.

What is the difference between Gauss elimination and Gauss-Seidel?

Gauss elimination is a direct method that produces an exact solution in finite steps, whereas Gauss-Seidel is an iterative approximation method.

When can the Gauss-Seidel method be applied?

It is most effective when the coefficient matrix is diagonally dominant or symmetric positive definite.

Conclusion

The Gauss-Seidel method is an efficient and widely used iterative technique for solving systems of linear equations. This online calculator simplifies the computation process and provides quick, reliable approximations.

References

Wikipedia: Gauss–Seidel Method
ScienceDirect: Iterative Methods for Solving Linear Algebraic Systems

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