What is the Gauss-Seidel method?

Gauss-Seidel method is a mathematical method used to solve the linear equations of the given system. Its name is based on Carl Friedrich Gauss and Philipp Ludwig von Seidel, known as great German Mathematicians. Gauss-Seidel method is defined as the iterative technique that helps us solve a number of linear equations. Gauss-Seidel method is similar to the Jacobi method. Gauss-Seidel method is also termed as Liebmann method and the successive displacement method. An iterative method is referred to the process of solving equations by applying successive linear values (like 1, 2, 3, 4. . .).

What is the Jacobi method?

Jacobi Method is also used to solve numerical linear algebraic equations. Jacobi method is defined as the iterative algorithms that help to know the solutions of the system, which are dominant diagonally.

The general form of the Jacobi method, along with the application of iteration in the given linear equation in terms of the unknown, is as follows:

$x^{k+1}=D^{-1}\left(b-\left(L+U\right)\right)x^k$

Here, $k \mathrm{and} k+1$ show the further iterations, and x shows the unknown variable, which is to be found.

What is the functional process to solve the linear equation by the Gauss-Seidel method?

Gauss-Seidel method for solving a number of linear equations works on the boundary conditions given according to the problem. Gauss-Seidel method works on the continuous given series of the numbers for the implementation in place of x (or the other variables). The Gauss-Seidel method is a successive displacement method because it works in a chain form like every unknown term determined by the previous terms.

Forms of equations and the iteration methods

Gauss-Seidel method is the iteration-based method that solves a number of linear equations having unknown variables.

Basic form- $Ax=b$

Here, x represents the unknown variable of the equation.

This general equation is defined by the general iteration form

$L_{*}{x}^{\left(k+1\right)}=b-U{x}^k$                       ...... (i)

Here, k represents the approximation (i.e. 1,2,3,4,5,...) and k+1 shows the proceeding value.

It shows the iteration of the unknown variable x.

If equation (1) is rearranged in the form of x, i.e., unknown variable, then it will be

$x^{\left(k+1\right)}=L_{*}^{-1}b-U{x}^k$

In general form, without using the iteration, it will be

$x=L_{*}^{-1}b-Ux$

Here, all the proceeding iterations are removed (i.e., k or k+1).

The important part is that A must be diagonally dominant because the Gauss-Seidel method is dependent on the matrix A or the values of matrix A.

Types of problems solved by the Gauss-Seidel method

Gauss-Seidel method is used to solve a matrix-based problem, equation-based problems, and most importantly, in programming languages like Python, NumPy, etc.

General example to show the Gauss-Seidel method values:

Taking three variables into consideration, $X_1, X_2, X_3$.

Assume three values of them which show the precision of the values.

$$\begin{gathered} X_1=2.1102, 2.1001, 2.1000 \\ {X}_2=3.21004, 3.21000, 3.20005 \\ {X}_3=5.01234, 5.01204, 5.01200 \end{gathered}$$

The above three assumed values and variables clearly show that the values go on more precise after applying iterations in the linear equations to find the unknown variables.

It is like a directly proportional principle between a number of iterations and the precise value. More the number of iteration steps more the precise value.

Example of an equation-based problem solved by Gauss-Seidel method.

$$\begin{gathered} x_1-2x_2=5 \\ 3x_1+x_2=6 \end{gathered}$$

These are two equations. Now balance them to find $x_1 \mathrm{and} x_2$.

$$\begin{gathered} x_1=5+2x_2 \\ {x}_2=6-3x_1 \end{gathered}$$

By using the Gauss-Seidel condition, use the boundary conditions or iterative method.

Put zero as a boundary condition. Then,

$$\begin{gathered} x_1=5+2\left(0\right) \\ {x}_1=5 \\ {x}_2=6-3x_1 \\ {x}_2=6 \end{gathered}$$

Here, the value of $x_1 \mathrm{and} x_2$ are on the boundary condition.

By the use of an iterative method, apply the successive values (1, 2, 3, 4 . . .) to find more roots which became more precise.

Gauss-Seidel method gives the values at the end of successive values are the approximated solutions. Usually, after four iteration terms, the values will be more precise.

Difference Between Gauss-Seidel Method and Jacobi method

The Jacobi and Gauss-Seidel methods are almost the same and used to solve the equations, but the key difference between them is shown in the iteration method procedure.  In the Jacobi method, after implementing the first boundary condition, it is mandatory to use the answered values for the next step of iteration or the values we got by applying boundary conditions. On the other hand, in the Gauss-Seidel method, the values applied in the iteration process are predefined or successive. Clearly, we can say that the Gauss-Seidel method is the advanced and more accurate Jacobi method. Jacobi method is hard compared to the Gauss-Seidel method because it implements in parallel, so it possesses a lengthy process that is critical to understand and has much confusion.

Similar example by both the methods:

Jacobi method

$$\begin{gathered} {\left(u_2\right)}_{step 1}=\frac{5u_3}{10}=0.5u_3=0 \\ {\left(u_3\right)}_{step 1}=\frac{5u_2+5u_4}{10}=0.5\left(u_2+u_4\right)=0 \\ {\left(u_4\right)}_{step 1}=\frac{15+5u_3}{10}=3+u_3=3 \end{gathered}$$

Gauss Seidel method

$$\begin{gathered} {\left(u_2\right)}_{step 1}=\frac{5u_3}{10}=0.5u_3=0 \\ {\left(u_3\right)}_{step 1}=\frac{5{\left(u_2\right)}_{step 1}+5u_4}{10}=0.5{\left(u_2\right)}_{step 1}+0.5u_4=0 \\ {\left(u_4\right)}_{step 1}=\frac{15+5{\left(u_3\right)}_{step 1}}{10}=3+{\left(u_3\right)}_{step 1}=3 \end{gathered}$$

Limitation of the Gauss-Seidel method

The only limitation of the Gauss-Seidel method is that it is not possible to be a convergence of every matrix because it only happens when the matrix is diagonally dominant or symmetric and positive definite.

Uses of the Gauss-Seidel method

Applications of the Gauss-Seidel method are the following:

• Easy to apply
• More precise
• More reliable
• Iteration is fast
• In comparison to the NR method, it carries less space.

Common Mistakes

While using the Gauss-Seidel method, it is important to know the difference between the Jacobi method and Gauss-Seidel Method. Both methods are almost similar, so it is easy to get confused between them.

Context and Applications

This is mainly used to solve equations to find unknown variables using iteration. It is an important method to be known which is used in the following context:

• Bachelor of Technology
• Bachelor of Science in Physics
• Master of Technology
• Master of Science in Physics

Related Concepts

• Jacobi method
• Iteration process
• NR method

Practice Problems

Q1. Which method is similar to the Gauss-Seidel method?

1. Jacobi method
2. L-transformation
3. R-K method
4. None of the above

Correct option- (a)

Explanation- The gauss-Seidel method and Jacobi method follow an almost similar procedure.

Q2. What is/are the other name/s of the Gauss-Seidel method?

1. Successive displacement method
2. Liebmann method
3. Both (a) and (b)
4. None of the above

Correct answer- (c)

Explanation- The successive displacement method and Liebmann method are the other names of the Gauss-Seidel method.

Q3. Gauss-Seidel method works on _________.

1. given conditions
2. boundary conditions
3. self made conditions
4. All of the above

Correct answer- (b)

Explanation- Gauss-Seidel method proceeds on the boundary conditions.

Q4. What is the limitation of the Gauss-Seidel method?

1. Diagonally dominant matrix
2. Boundary conditions
3. Not applicable for matrix-based problem
4. None of the above

Correct answer-(a)

Explanation- The gauss-Seidel method is only applicable to that matrix-based problem that is diagonally dominant.

Q5. Which one is the advanced method between Jacobi and Gauss-Seidel method?

1. Both are same
2. Jacobi
3. Gauss-Seidel method
4. No relation between them

Correct answer-(c)

Explanation- The gauss-Seidel method is the upgrade of the Jacobi method, and it is more precise and accurate than the Jacobi method.