Jacobi Method Calculator
Solve a diagonally-dominant linear system iteratively using the Jacobi method.
About Jacobi Method Calculator
The Jacobi method rearranges each equation of a 3x3 system to isolate one variable, then sweeps through all three simultaneously using only values from the previous sweep. That simultaneity is its defining property: every update in a sweep is computed from the same old vector, which makes Jacobi trivially parallelisable but slower to converge than methods that reuse fresh values. Convergence is guaranteed when the matrix is strictly diagonally dominant, meaning each diagonal entry outweighs the sum of the other entries in its row, so the calculator reports that dominance ratio alongside the solution — a ratio above 1 means the iteration is certain to settle, and the further above 1, the fewer sweeps it takes.
Formulas
Jacobi sweep
x_i(new) = (b_i - sum over j not equal to i of a_ij x_j(old)) / a_ii
- x_j(old)
- Values from the previous sweep only
- a_ii
- Diagonal coefficient of row i
Diagonal dominance ratio
ratio = min over rows of |a_ii| / sum of |a_ij| for j not equal to i
- ratio
- Above 1 guarantees convergence
Stopping test
Stop when max |x_i(new) - x_i(old)| < tolerance
Worked example
With the default values below, Jacobi Method Calculator returns x of 1.04326932. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Row 1: coefficient of x
- 10
- Row 1: coefficient of y
- -1
- Row 1: coefficient of z
- 2
- Row 1: right-hand side
- 6
- Row 2: coefficient of x
- -1
- Row 2: coefficient of y
- 11
- Row 2: coefficient of z
- -1
- Row 2: right-hand side
- 25
- Row 3: coefficient of x
- 2
- Row 3: coefficient of y
- -1
- Row 3: coefficient of z
- 10
- Row 3: right-hand side
- -11
- Initial guess x
- 0
- Initial guess y
- 0
- Initial guess z
- 0
- Convergence tolerance
- 0.000001
- Maximum sweepsaccepts 1–5,000
- 200
Results
- x
- 1.04326932
- y
- 2.26923071
- z
- -1.08173067
- Diagonal Dominance
- Strictly diagonally dominant (ratio 3.333) — convergence guaranteed
Step by step
- 1Values used
- 2Jacobi sweep
- 3Diagonal dominance ratio
- 4Stopping test
- 5x= 1.04326932
- 6y= 2.26923071
- 7z= -1.08173067
- 8Diagonal Dominance= Strictly diagonally dominant (ratio 3.333) — convergence guaranteed
Assumptions and limits
- The result is derived only from the 17 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What does the diagonal dominance ratio tell me?
It is the tightest row in the matrix: the diagonal coefficient divided by the sum of the other coefficients in that row. Above 1 the matrix is strictly diagonally dominant and Jacobi is guaranteed to converge from any starting guess. Below 1 it may still converge, but nothing guarantees it.
Why does the initial guess barely change the answer?
For a convergent system Jacobi has a unique fixed point, so every starting vector lands on the same solution. The guess only affects how many sweeps that takes — a good guess can halve the sweep count, which matters for large systems.
When would I prefer Jacobi over Gauss-Seidel?
When you can compute the three updates on separate processors. Jacobi reads only the previous vector, so all updates in a sweep are independent and parallel. Gauss-Seidel converges in fewer sweeps but each update depends on the one before it, forcing sequential execution.