Enter Linear System Values
Example Data Table
This sample system is already loaded in the form.
| Equation | Linear Form | Initial Guess |
|---|---|---|
| 1 | 10x1 - x2 + 2x3 = 6 | x1 = 0 |
| 2 | -x1 + 11x2 - x3 = 25 | x2 = 0 |
| 3 | 2x1 - x2 + 10x3 = -11 | x3 = 0 |
Formula Used
- Linear system:
Ax = b - Gauss Seidel update:
xᵢ(k+1) = [bᵢ - Σaᵢⱼxⱼ]/aᵢᵢ - Use latest values for
j < iand old values forj > i. - Relaxed update:
xᵢ(new) = (1 - ω)xᵢ(old) + ωxᵢ(GS) - Residual vector:
r = Ax - b - Update error:
||x(k+1) - x(k)||
How to Use This Calculator
Enter the coefficient matrix in the first box. Put each row on a new line. Enter the constant vector in the second box. Enter the starting guess in the third box. Set tolerance, maximum iterations, norm type, and relaxation factor. Use one as the relaxation factor for the standard method. Press the calculate button. The result appears above the form and below the header section.
Why Gauss Seidel Iteration Matters
Gauss Seidel iteration is a practical numerical method for solving linear equation systems. It is useful when direct elimination becomes slow, repetitive, or difficult to manage by hand. The method starts with an initial guess. Then it improves each unknown one by one. Each new value is used immediately in the same iteration. This makes the method faster than simple Jacobi iteration in many classroom and engineering problems.
How The Method Works
The calculator separates the coefficient matrix, constant vector, and starting vector. During every pass, it solves one row for one unknown. Values already updated in the current pass are reused. Values not yet updated still come from the previous pass. This staged update often reduces the error quickly when the system is well conditioned.
Convergence And Accuracy
Convergence is not guaranteed for every matrix. A diagonally dominant matrix usually behaves well. Symmetric positive definite systems also commonly converge. The calculator checks row dominance and reports warnings when the diagonal is weak. These checks help users decide whether row reordering, scaling, or another method may be better.
Advanced Controls
The tolerance field controls when the iteration stops. A smaller tolerance gives a tighter answer, but it may require more iterations. The maximum iteration field prevents endless loops. The relaxation factor changes the update strength. A value of one gives the standard method. Values below one damp the changes. Values above one can speed convergence, but poor choices may cause instability.
Reading The Results
The result table shows every iteration, each estimated variable, the update error, and the residual norm. The graph tracks convergence over time. A falling curve means the estimates are stabilizing. CSV export helps with spreadsheets. PDF export helps with reports and assignments.
Best Use Cases
This tool is best for learning, homework checking, and quick numerical experiments. It also helps compare initial guesses and relaxation settings. Always verify important results with the original equations. A small residual means the calculated solution closely satisfies the system. For larger systems, review conditioning before trusting the output. If errors grow, try row swaps. Use better starting values. Choose another stable solver for safety too.
FAQs
1. What does this calculator solve?
It solves square systems of linear equations using the Gauss Seidel iteration method. It estimates each unknown repeatedly until the selected tolerance is reached or the maximum iteration limit is met.
2. What is the best starting guess?
A zero vector is often fine for learning examples. A better guess can reduce iterations. When you already know approximate variable values, enter them as the initial guess.
3. Does Gauss Seidel always converge?
No. Convergence depends on the coefficient matrix. Diagonal dominance and positive definite structure usually help. Weak diagonal values can make the process slow or divergent.
4. What is the relaxation factor?
The relaxation factor changes update strength. A value of one gives standard Gauss Seidel. Lower values damp updates. Higher values may speed convergence, but they can also create instability.
5. What tolerance should I use?
Use 0.000001 for many study problems. Use a smaller tolerance for tighter accuracy. Very small tolerances may need more iterations and better numerical conditioning.
6. What is residual norm?
The residual norm measures how closely the current solution satisfies Ax = b. A smaller residual means the computed values fit the original equations more closely.
7. Why is diagonal dominance shown?
Diagonal dominance is a useful convergence clue. If each diagonal value is large compared with other row values, Gauss Seidel is more likely to converge well.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a clean report containing the solution, iteration table, and convergence details.