Gauss–Seidel method

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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 system of linear 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 strictly diagonally dominant,[1] or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823.[2] A publication was not delivered before 1874 by Seidel.[3]


Let be a square system of n linear equations, where:

When and are known, and is unknown, we can use the Gauss–Seidel method to approximate . The vector denotes our initial guess for (often for ). We denote as the k-th approximation or iteration of , and is the next (or k+1) iteration of .

Matrix-based formula[edit]

The solution is obtained iteratively via

where the matrix is decomposed into a lower triangular component , and a strictly upper triangular component such that .[4] More specifically, the decomposition of into and is given by:

Why the matrix-based formula works[edit]

The system of linear equations may be rewritten as:

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

Element-based formula[edit]

However, by taking advantage of the triangular form of , the elements of can be computed sequentially for each row using forward substitution:[5]

Notice that the formula uses two summations per iteration which can be expressed as one summation that uses the most recently calculated iteration of . 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 similar to that of the Jacobi method.

The computation of uses the elements of that have already been computed, and only the elements of that have not been computed in the (k+1)-th 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 are generally much harder to implement in parallel, since they can have a very long critical path, and are thus most feasible for sparse matrices. Furthermore, the values at each iteration are dependent on the order of the original equations.

Gauss-Seidel is the same as successive over-relaxation with .


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.

Golub and Van Loan give a theorem for an algorithm that splits into two parts. Suppose is nonsingular. Let be the spectral radius of . Then the iterates defined by converge to for any starting vector if is nonsingular and .[8]


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:

algorithm Gauss–Seidel method is
    inputs: A, b
    output: φ

    Choose an initial guess φ to the solution
    repeat until convergence
        for i from 1 until n do
            σ ← 0
            for j from 1 until n do
                if ji then
                    σσ + aijφj
                end if
            end (j-loop)
            φi ← (biσ) / aii
        end (i-loop)
        check if convergence is reached
    end (repeat)


An example for the matrix version[edit]

A linear system shown as is given by:

We want to use the equation

in the form

We must decompose into the sum of a lower triangular component and a strict upper triangular component :

The inverse of is:

Now we can find:

Now we have and and we can use them to obtain the vectors iteratively.

First of all, we have to choose : we can only guess. The better the guess, the quicker the algorithm will perform.

We choose a starting point:

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).

Another example for the matrix version[edit]

Another linear system shown as is given by:

We want to use the equation

in the form

We must decompose into the sum of a lower triangular component and a strict upper triangular component :

The inverse of is:

Now we can find:

Now we have and and we can use them to obtain the vectors iteratively.

First of all, we have to choose : we can only guess. The better the guess, the quicker will perform the algorithm.

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.

An example for the equation version[edit]

Suppose given k equations where xn are vectors of these equations and starting point x0. From the first equation solve for x1 in terms of For the next equations substitute the previous values of xs.

To make it clear consider an example.

Solving for and gives:

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.

0.6 2.32727 −0.987273 0.878864
1.03018 2.03694 −1.01446 0.984341
1.00659 2.00356 −1.00253 0.998351
1.00086 2.0003 −1.00031 0.99985

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

An example using Python and NumPy[edit]

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

import numpy as np


# initialize the matrix
A = np.array(
        [10.0, -1.0, 2.0, 0.0],
        [-1.0, 11.0, -1.0, 3.0],
        [2.0, -1.0, 10.0, -1.0],
        [0.0, 3.0, -1.0, 8.0],
# initialize the RHS vector
b = np.array([6.0, 25.0, -11.0, 15.0])

print("System of equations:")
for i in range(A.shape[0]):
    row = [f"{A[i,j]:3g}*x{j+1}" for j in range(A.shape[1])]
    print("[{0}] = [{1:3g}]".format(" + ".join(row), b[i]))

x = np.zeros_like(b, np.float_)
for it_count in range(1, ITERATION_LIMIT):
    x_new = np.zeros_like(x, dtype=np.float_)
    print(f"Iteration {it_count}: {x}")
    for i in range(A.shape[0]):
        s1 =[i, :i], x_new[:i])
        s2 =[i, i + 1 :], x[i + 1 :])
        x_new[i] = (b[i] - s1 - s2) / A[i, i]
    if np.allclose(x, x_new, rtol=1e-8):
    x = x_new

print(f"Solution: {x}")
error =, x) - b
print(f"Error: {error}")

Produces the output:

System of equations:
[ 10*x1 +  -1*x2 +   2*x3 +   0*x4] = [  6]
[ -1*x1 +  11*x2 +  -1*x3 +   3*x4] = [ 25]
[  2*x1 +  -1*x2 +  10*x3 +  -1*x4] = [-11]
[  0*x1 +   3*x2 +  -1*x3 +   8*x4] = [ 15]
Iteration 1: [ 0.  0.  0.  0.]
Iteration 2: [ 0.6         2.32727273 -0.98727273  0.87886364]
Iteration 3: [ 1.03018182  2.03693802 -1.0144562   0.98434122]
Iteration 4: [ 1.00658504  2.00355502 -1.00252738  0.99835095]
Iteration 5: [ 1.00086098  2.00029825 -1.00030728  0.99984975]
Iteration 6: [ 1.00009128  2.00002134 -1.00003115  0.9999881 ]
Iteration 7: [ 1.00000836  2.00000117 -1.00000275  0.99999922]
Iteration 8: [ 1.00000067  2.00000002 -1.00000021  0.99999996]
Iteration 9: [ 1.00000004  1.99999999 -1.00000001  1.        ]
Iteration 10: [ 1.  2. -1.  1.]
Solution: [ 1.  2. -1.  1.]
Error: [  2.06480930e-08  -1.25551054e-08   3.61417563e-11   0.00000000e+00]

Program to solve arbitrary no. of equations using Matlab[edit]

The following code uses the formula

function x = gauss_seidel(A, b, x, iters)
    for i = 1:iters
        for j = 1:size(A,1)
            x(j) = (b(j) - sum(A(j,:)'.*x) + A(j,j)*x(j)) / A(j,j);

See also[edit]


  1. ^ Sauer, Timothy (2006). Numerical Analysis (2nd ed.). Pearson Education, Inc. p. 109. ISBN 978-0-321-78367-7.
  2. ^ Gauss 1903, p. 279; direct link.
  3. ^ Seidel, Ludwig (1874). "Über ein Verfahren, die Gleichungen, auf welche die Methode der kleinsten Quadrate führt, sowie lineäre Gleichungen überhaupt, durch successive Annäherung aufzulösen" [On a process for solving by successive approximation the equations to which the method of least squares leads as well as linear equations generally]. Abhandlungen der Mathematisch-Physikalischen Klasse der Königlich Bayerischen Akademie der Wissenschaften (in German). 11 (3): 81–108.
  4. ^ Golub & Van Loan 1996, p. 511.
  5. ^ Golub & Van Loan 1996, eqn (10.1.3)
  6. ^ Golub & Van Loan 1996, Thm 10.1.2.
  7. ^ Bagnara, Roberto (March 1995). "A Unified Proof for the Convergence of Jacobi and Gauss-Seidel Methods". SIAM Review. 37 (1): 93–97. doi:10.1137/1037008. JSTOR 2132758.
  8. ^ Golub & Van Loan 1996, Thm 10.1.2


This article incorporates text from the article Gauss-Seidel_method on CFD-Wiki that is under the GFDL license.

External links[edit]