Easily solve systems of linear equations using the Gauss Jacobi iteration method. This calculator provides approximate solutions for 3x3 systems, allowing you to specify initial guesses, tolerance, and maximum iterations. Ideal for numerical analysis students and professionals seeking iterative solutions.
Formula:
For a system of linear equations represented as Ax = b, the Gauss-Jacobi iterative method finds an approximate solution by rearranging each equation to solve for one variable. For a system with n variables (e.g., x1, x2, ..., xn), the iteration formula for each variable xi at iteration k+1 is:
xi(k+1) = (1 / aii) * (bi - ∑j≠i aij xj(k))
Where:
- xi(k+1): The new approximation for the i-th variable.
- xj(k): The approximation for the j-th variable from the previous iteration.
- aii: The diagonal coefficient for the i-th variable in the i-th equation. (Must be non-zero).
- aij: The coefficient for the j-th variable in the i-th equation (where j ≠ i).
- bi: The constant term in the i-th equation.
- ∑j≠i: Summation over all j except i.
Convergence is often guaranteed if the matrix A is strictly diagonally dominant, meaning |aii| > ∑j≠i |aij| for all i.