How to use the Jacobi Iteration Method Calculator
- 1. Enter the equationsOne equation per line as coefficients then the right-hand side: 10 1 1 12.
- 2. Set tolerance and startThe default start is all zeros. Change the tolerance or iterations if needed.
- 3. Read the solutionThe table lists the values after each iteration and the largest change.
How the Jacobi method works
Rewrite each equation for its diagonal unknown: x = (b₁ − a₁₂y − a₁₃z)/a₁₁, y = (b₂ − a₂₁x − a₂₃z)/a₂₂, and so on. Start from a guess, usually zeros, and compute all new values using only the previous iteration's values. That is why Jacobi is also called the method of simultaneous displacements. Repeat until no value changes by more than the tolerance.
The method is guaranteed to converge when the coefficient matrix is diagonally dominant: in each row the diagonal entry is at least the sum of the other entries in size. Reorder the equations to achieve this before you start.
Worked example
Solve 10x + y + z = 12, 2x + 10y + z = 13, 2x + 2y + 10z = 14. The formulas are x = (12 − y − z)/10, y = (13 − 2x − z)/10, z = (14 − 2x − 2y)/10.
Iteration 1 from (0, 0, 0): x = 1.2, y = 1.3, z = 1.4. Iteration 2: x = (12 − 1.3 − 1.4)/10 = 0.93, y = (13 − 2.4 − 1.4)/10 = 0.92, z = (14 − 2.4 − 2.6)/10 = 0.9. Iteration 3: (1.018, 1.024, 1.03). The values close in on x = y = z = 1, and the tool stops after 9 iterations for a tolerance of 0.0001.
Jacobi or Gauss-Seidel?
Gauss-Seidel uses each new value straight away, so it usually needs about half as many iterations. Jacobi's updates are independent, which makes it easy to run in parallel. Compare both with the Gauss-Seidel calculator.
Frequently asked questions
When does the Jacobi method converge?
It always converges for a strictly diagonally dominant matrix. It may also converge otherwise, but that is not guaranteed.
Why must diagonal entries be non-zero?
Each update divides by the diagonal entry. Swap equations so none of them is zero.
What does the tool do if the values blow up?
It stops and tells you the method is diverging, so you can reorder the equations.
How accurate is the answer?
It stops when no unknown changes by more than your tolerance. The residual |Ax − b| is shown as a check.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.