GAUSS-SEIDEL METHOD: EXPLANATION, APPLICATIONS, EXAMPLES - MATHS - 2025

The Gauss-Seidel method is an iterative procedure for finding approximate solutions to a system of linear algebraic equations with arbitrarily chosen precision. The method is applied to square matrices with nonzero elements in their diagonals and convergence is guaranteed if the matrix is ​​diagonally dominant.

It was created by Carl Friedrich Gauss (1777-1855), who gave a private demonstration to one of his students in 1823. It was later formally published by Philipp Ludwig von Seidel (1821-1896) in 1874, hence the name of both mathematicians.

Figure 1. The Gauss-Seidel method converges rapidly to obtain the solution of a system of equations. Source: F. Zapata.

For a complete understanding of the method, it is necessary to know that a matrix is ​​diagonally dominant when the absolute value of the diagonal element of each row is greater than or equal to the sum of the absolute values ​​of the other elements of that same row.

Mathematically it is expressed like this:

Explanation using a simple case

To illustrate what the Gauss-Seidel method consists of, we will take a simple case, in which the values ​​of X and Y can be found in the 2 × 2 system of linear equations shown below:

5X + 2Y = 1

X - 4Y = 0

Steps to follow

1- First, you have to determine if the convergence is safe. It is immediately observed that, in effect, it is a diagonally dominant system, since in the first row the first coefficient has a higher absolute value than the others in the first row:

-5 -> - 2-

Likewise, the second coefficient in the second row is also diagonally dominant:

--4 -> - 1-

2- The variables X and Y are cleared:

X = (1 - 2Y) / 5

Y = X / 4

3- An arbitrary initial value is placed, called "seed": Xo = 1, I = 2.

4-The iteration begins: to obtain the first approximation X1, Y1, the seed is substituted in the first equation of step 2 and the result in the second equation of step 2:

X1 = (1 - 2 I) / 5 = (1 - 2 × 2) / 5 = -3/5

Y1 = X1 / 4 = (-3/5) / 4 = -3/20

5- We proceed in a similar way to obtain the second approximation of the solution of the system of equations:

X2 = (1 - 2 Y1) / 5 = (1 - 2x (-3/20)) / 5 = 13/50

Y2 = X2 / 4 = (13/50) / 4 = 13/200

6- Third iteration:

X3 = (1 - 2 Y2) / 5 = (1 - 2 (13/200)) / 5 = 87/500

Y3 = X3 / 4 = (87/500) / 4 = 87/2000

7- Fourth iteration, as the final iteration of this illustrative case:

X4 = (1 - 2 Y3) / 5 = (1 - 2 (87/2000)) / 5 = 913/5000

Y4 = X4 / 4 = (913/5000) / 4 = 913/20000

These values ​​agree quite well with the solution found by other resolution methods. The reader can quickly check it with the help of an online math program.

Analysis of the method

As can be seen, in the Gauss-Seidel method, the approximate values ​​obtained for the previous variable in that same step must be substituted in the following variable. This differentiates it from other iterative methods such as Jacobi's, in which each step requires the approximations of the previous stage.

The Gauss-Seidel method is not a parallel procedure, while the Gauss-Jordan method is. It is also the reason that the Gauss-Seidel method has a faster convergence - in fewer steps - than the Jordan method.

As for the diagonally dominant matrix condition, this is not always satisfied. However, in most cases simply swapping the rows from the original system is sufficient for the condition to be met. Furthermore, the method almost always converges, even when the diagonal dominance condition is not met.

The previous result, obtained by four iterations of the Gauss-Seidel method, can be written in decimal form:

X4 = 0.1826

Y4 = 0.04565

The exact solution to the proposed system of equations is:

X = 2/11 = 0.1818

Y = 1/22 = 0.04545.

So with just 4 iterations you get a result with one thousandth of precision (0.001).

Figure 1 illustrates how successive iterations rapidly converge to the exact solution.

Applications

The Gauss-Seidel method is not limited to 2 × 2 system of linear equations only. The previous procedure can be generalized to solve a linear system of n equations with n unknowns, which is represented in a matrix like this:

A X = b

Where A is an nxn matrix, while X is the vector n components of the n variables to be calculated; and b is a vector containing the values ​​of the independent terms.

To generalize the sequence of iterations applied in the illustrative case to an nxn system, from which the variable Xi wants to be calculated, the following formula will be applied:

In this equation:

- k is the index for the value obtained in iteration k.

-k + 1 indicates the new value in the following.

The final number of iterations is determined when the value obtained in iteration k + 1 differs from the one obtained immediately before, by an amount ε which is precisely the desired precision.

Examples of the Gauss-Seidel method

- Example 1

Write a general algorithm that allows to calculate the vector of approximate solutions X of a linear system of equations nxn, given the matrix of coefficients A, the vector of independent terms b, the number of iterations (i ter) and the initial value or "seed "of the vector X.

Solution

The algorithm consists of two “To” cycles, one for the number of iterations and the other for the number of variables. It would be as follows:

For k ∊

For i ∊

X: = (1 / A) * (b - ∑ _{j = 1} ⁿ (A * X) + A * X)

- Example 2

Check the operation of the previous algorithm through its application in the free and free-to-use mathematical software SMath Studio, available for Windows and Android. Take as an example the case of the 2 × 2 matrix that helped us to illustrate the Gauss-Seidel method.

Solution

Figure 2. Solution of the system of equations of the 2 x 2 example, using the SMath Studio software. Source: F. Zapata.

- Example 3

Apply the Gauss-Seidel algorithm for the following 3 × 3 system of equations, which has been previously ordered in such a way that the coefficients of the diagonal are dominant (that is, of greater absolute value than the absolute values ​​of the coefficients of the same row):

9 X1 + 2 X2 - X3 = -2

7 X1 + 8 X2 + 5 X3 = 3

3 X1 + 4 X2 - 10 X3 = 6

Use the null vector as a seed and consider five iterations. Comment on the result.

Solution

Figure 3. Solution of the system of equations of solved example 3, using SMath Studio. Source: F. Zapata.

For the same system with 10 iterations instead of 5 the following results are obtained: X1 = -0.485; X2 = 1.0123; X3 = -0.3406

This tells us that five iterations are enough to obtain three decimal places of precision and that the method quickly converges to the solution.

- Example 4

Using the Gauss-Seidel algorithm given above, find the solution to the 4 × 4 system of equations given below:

10 x1 - 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

To start the method, make use of this seed:

x1 = 0, x2 = 0, x3 = 0 and x4 = 0

Consider 10 iterations and estimate the error of the result, comparing with iteration number 11.

Solution

Figure 4. Solution of the system of equations of solved example 4, using SMath Studio. Source: F. Zapata.

When comparing with the next iteration (number 11), the result is identical. The largest differences between the two iterations are on the order of 2 × 10 ^-8, which means that the displayed solution has a precision of at least seven decimal places.

References

1. Iterative solution methods. Gauss-Seidel. Recovered from: cimat.mx
2. Numerical methods. Gauss-Seidel. Recovered from: test.cua.uam.mx
3. Numerical: Gauss-Seidel method. Recovered from: aprendeenlinea.udea.edu.co
4. Wikipedia. Gauss-Seidel method. Recovered from: en. wikipedia.com
5. Wikipedia. Gauss-Seidel method. Recovered from: es.wikipedia.com