Gauss-Jacobi Iterative Method Calculator

The Gauss-Jacobi method (also called the Jacobi method) is an iterative algorithm for solving systems of linear equations. Starting from an initial guess, each variable is updated simultaneously using values from the previous iteration, repeating until the solution converges within a specified tolerance. It was developed by mathematician Carl Gustav Jacobi.

A matrix is diagonally dominant when the absolute value of each diagonal element is greater than the sum of the absolute values of all other elements in that row. Diagonal dominance guarantees that the Jacobi iteration will converge to a solution. If your matrix is not diagonally dominant, the method may diverge or fail to converge.

In the Gauss-Jacobi method, all variables are updated simultaneously using only values from the previous iteration. In the Gauss-Seidel method, each variable is updated using the most recently computed values. Gauss-Seidel typically converges faster, but Gauss-Jacobi is easier to parallelize and is a fundamental iterative technique.

A common starting point is to use all zeros (x₁⁰ = x₂⁰ = x₃⁰ = 0), which works well for diagonally dominant systems. If you have prior knowledge about approximate values, using those can reduce the number of iterations needed to reach convergence.

A tolerance of 0.0001 is suitable for most practical problems, giving solutions accurate to about 4 decimal places. For higher precision (engineering or scientific applications), use 0.000001 or smaller. The iteration stops once the maximum change in any variable between consecutive iterations falls below your chosen tolerance.

If the method reaches the maximum number of iterations without the error dropping below the tolerance, the system is reported as not converged. This usually means the matrix is not diagonally dominant or is ill-conditioned. Try rearranging rows to increase diagonal dominance, or consider using a direct method like Gaussian elimination instead.

This calculator supports 2×2 and 3×3 systems of linear equations. Select the system size from the dropdown, then fill in the coefficient matrix A and constants vector b accordingly. For larger systems, the same algorithm applies but requires specialized software.

No. The Jacobi method is only guaranteed to converge for strictly diagonally dominant matrices. It may also converge for some non-dominant systems, but this is not guaranteed. For systems that are not diagonally dominant, direct methods such as Gaussian elimination or LU decomposition are more reliable.