Jacobi Iterative Method Calculator
Enter the coefficients for a diagonally dominant 2×2 or 3×3 linear system and choose how many Jacobi iterations to run. The solver reports the latest approximation, highlights divergence issues, and shares a copy-ready summary.
| Iteration | x₁ | x₂ | x₃ | ∞-norm change |
|---|
Spectral Sync Challenge
Click to pulse when each orbital update aligns with the focus window. Keep the residual error low and finish the iteration arc before the clock expires.
Aim with your pointer or arrow keys. Tap, click, or press space/enter to pulse. Capture each variable while the window glows to imitate Jacobi updates and keep the error under control.
What the Jacobi Method Computes
The Jacobi algorithm solves a linear system by repeatedly updating each unknown with the latest values from the previous iteration. For the -th row, the update rule is
, which mirrors isolating a single variable in the original equations. Each iteration rebuilds the vector using only information from the prior step, making the method easy to implement but sensitive to diagonal dominance.
Monitoring Convergence
The preview table samples the early, middle, and final iterations so you can confirm that the solution stabilizes. The change column reports the infinity norm between successive vectors—if it stops shrinking, consider reducing the iteration count or switching to a faster method such as Gaussian Elimination. Persistent divergence may indicate the matrix is poorly conditioned. Comparing condition numbers with the Matrix Condition Number Calculator highlights troublesome systems before you iterate.
Because the updates treat each variable independently, the Jacobi approach parallelizes well. Large sparse matrices in scientific computing often use accelerated variants or upgrade to Gauss–Seidel and Successive Over-Relaxation methods for faster convergence. After experimenting here, you can dive deeper with the Matrix Inverse Calculator to contrast direct versus iterative techniques.
Worked 3×3 Example
Suppose you solve with . Starting from the zero vector, the first iteration produces . By iteration ten the values converge near , matching the analytic solution. Watching the infinity norm shrink confirms the matrix is safely diagonally dominant.
Tips for Reliable Results
- Reorder rows so the diagonal entries dwarf the off-diagonal terms whenever possible.
- Scale the equations to avoid very large or tiny coefficients that can amplify rounding error.
- Increase the iteration count gradually and monitor the norm change column to catch stagnation early.
- Use the copy button to paste the summary into lab notes or classroom solutions without retyping.