Gauss Seidel Iteration Calculator

The Gauss-Seidel method is an iterative technique for solving a system of linear equations Ax = b. For a system with 'n' equations and 'n' unknowns (x₁, x₂, ..., x_n), the update for each unknown x_i at iteration (k+1) is derived by solving the i-th equation for x_i, using the most recently computed values for the other unknowns.

For a system:
a₁₁x₁ + a₁₂x₂ + ... + a_1nx_n = b₁
a₂₁x₁ + a₂₂x₂ + ... + a_2nx_n = b₂
...
a_n1x₁ + a_n2x₂ + ... + a_nnx_n = b_n

The iterative formula for x_i at step (k+1) is:
x_i^(k+1) = (1/a_ii) [b_i - ∑_j<i (a_ijx_j^(k+1)) - ∑_j>i (a_ijx_j^(k))]

This means that when calculating x_i, any x_j (where j < i) that have already been updated in the current iteration (k+1) are used, while x_j (where j > i) still use their values from the previous iteration (k).