In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a linear system of equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel, and is similar to the Jacobi method. Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823.^[1] A publication was not delivered before 1874 by Seidel.

The Gauss–Seidel method is an iterative technique for solving a square system of n linear equations with unknown x:

It is defined by the iteration

In more detail, write out A, x and b in their components:

Then the decomposition of A into its lower triangular component and its strictly upper triangular component is given by:

The system of linear equations may be rewritten as:

The Gauss–Seidel method now solves the left hand side of this expression for x, using previous value for x on the right hand side. Analytically, this may be written as:

The procedure is generally continued until the changes made by an iteration are below some tolerance, such as a sufficiently small residual.

The element-wise formula for the Gauss–Seidel method is extremely similar to that of the Jacobi method.

The computation of x(k+1) uses the elements of x(k+1) that have already been computed, and only the elements of x(k) that have not been computed in the k+1 iteration. This means that, unlike the Jacobi method, only one storage vector is required as elements can be overwritten as they are computed, which can be advantageous for very large problems.

However, unlike the Jacobi method, the computations for each element cannot be done in parallel. Furthermore, the values at each iteration are dependent on the order of the original equations.

The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either:

The Gauss–Seidel method sometimes converges even if these conditions are not satisfied.

Since elements can be overwritten as they are computed in this algorithm, only one storage vector is needed, and vector indexing is omitted. The algorithm goes as follows:

We want to use the equation

in the form

where:

Now we can find:

We suppose:

We can then calculate:

As expected, the algorithm converges to the exact solution:

In fact, the matrix A is strictly diagonally dominant (but not positive definite).

We want to use the equation

in the form

where:

Now we can find:

We suppose:

We can then calculate:

If we test for convergence we'll find that the algorithm diverges. In fact, the matrix A is neither diagonally dominant nor positive definite. Then, convergence to the exact solution

is not guaranteed and, in this case, will not occur.

To make it clear consider an example.

Suppose we choose (0, 0, 0, 0) as the initial approximation, then the first approximate solution is given by

Using the approximations obtained, the iterative procedure is repeated until the desired accuracy has been reached. The following are the approximated solutions after four iterations.

The exact solution of the system is (1, 2, −1, 1).

The following numerical procedure simply iterates to produce the solution vector.

Produces the output: