Solve systems of linear equations quickly with our free online Gauss-Jacobi method calculator. This numerical analysis tool provides accurate, step-by-step iterations for strictly diagonally dominant matrices. Ideal for students, engineers, and researchers, it verifies matrix convergence criteria and computes precise approximations for complex linear systems. Input your coefficients below to visualize the iterative process instantly.
Formula:
Understanding the Gauss-Jacobi Iteration Formula
For a system of linear equations represented as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector:
A = Matrix A, x = Vector x, b = Vector b
The Gauss-Jacobi method iteratively updates the values of x, y, and z using the following formulas:
- For the first variable (x):
- For the second variable (y):
- For the third variable (z):
xk+1 = (1/a11) * (b1 - a12yk - a13zk)
yk+1 = (1/a22) * (b2 - a21xk - a23zk)
zk+1 = (1/a33) * (b3 - a31xk - a32yk)
Where 'k' denotes the current iteration, and 'k+1' denotes the next iteration. The process continues until the difference between successive iterations falls below a specified tolerance or the maximum number of iterations is reached.
Note: For effective convergence, it is often required that the matrix A is diagonally dominant (i.e., for each row, the absolute value of the diagonal element is greater than the sum of the absolute values of the other elements in that row).