Solve a system of 2x2 linear equations using the Jacobi iterative method.
Matrix A and Vector b
Equation 1: a₁₁x + a₁₂y = b₁
Equation 2: a₂₁x + a₂₂y = b₂
Convergence Steps
(1.000, 1.000)Solution Vector
Final Solution (x, y)
Step
Approx X
Approx Y
0
0.000
0.000
1
1.4000
1.3750
2
0.8500
0.8500
3
1.0600
1.0562
4
0.9775
0.9775
5
1.0090
1.0084
6
0.9966
0.9966
7
1.0013
1.0013
8
0.9995
0.9995
9
1.0002
1.0002
10
0.9999
0.9999
Convergence10 Iterations
Dominant?Row1 Yes
About this calculator
What is the Jacobi Method?
The Jacobi method is an iterative algorithm for determining the solutions of a system of linear equations in numerical linear algebra. It is used primarily for large, sparse matrices where direct methods like Gaussian elimination are inefficient. The method requires the matrix to be 'diagonally dominant' to guarantee convergence.
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Pro Tips
Diagonally Dominant: The magnitude of the diagonal element must be greater than the sum of the magnitudes of other elements in that row.
Synchronous: In each step, every variable is updated using the values from the PREVIOUS iteration.
Parallel: Jacobi is highly parallelizable because the update for one variable doesn't depend on the current step's update of another.
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Fun Facts
"Carl Gustav Jacob Jacobi developed this method in the mid-19th century."
"The Jacobi method is the basis for more advanced solvers like SOR (Successive Over-Relaxation)."
"It's widely used in computational physics to solve heat and wave equations."
