Gauss Seidel Method Calculator

Get the assistance of this Gauss Seidel method calculator to resolve the system of linear equations. Take the help of Gauss seidel method to solve the linear equations with unknown variables. 

What is Gauss Seidel Method?

“The method that is used to solve the n linear equations with the unknown variables. In this, the first given system of a linear equation is placed in a diagonally dominant form”. 

This iteration method is also referred to as the Leibmann method or the method of successive displacement. This method is applicable when it follows the symmetric definite matrices or is diagonally dominant.

Algorithm of Gauss Seidel Iterative Method:

The Gauss-Seidel method calculator utilizes the algorithm to achieve fast calculations. So just follow the below points:

• Start the system and arrange the linear equations in the diagonally dominant form. 
• Change first equation for first variable and the second for the second variable and so on. 
• After converting the variables, set the initial gauss like x_0, y_0, z_0, and so on. 
• Substitute the value of y_0, z_0 … in the 1st equation realized from step 4 to estimate the new value of x1
• Use x_1, z_0, u_0 …. in the 2nd equation realized from step 4 to compute the new value of y1, and similarly, use x_1, y_1, u_0… to find new z_1, and so on.
• Set x_0 = x_1, y_0 = y_1, z0 = z1, and so on, and go to step 6.
• In the end, you will be able to get the final results. 

Gauss Seidel Formula:

This Gauss seidel method allows you to solve linear system equations. The formula that is used in their calculations is as follows:

x^(k+1)= L*^-1(b-Uxk)

Where;

• L* is a lower triangular matrix
• U is an upper triangular matrix

The determination of the triangular matrix by the above formula tells us that the matrix is equal to the product of principle diagonal elements.

Lower Triangular Matrix: 

“The matrix has non-zero entries on the down-ward diagonal and above the main diagonal is zero”.

$$ \begin{array}{ccc} 9 & 0 & 0 & 0 \\ 7 & 3 & 0 & 0 \\ 2 & 4 & 7 & 0 \\ 1 & -1 & -2 & 6 \end{array} $$

Upper Triangular Matrix: 

“The matrix in which all the elements are below the diagonal matrix is zero and above have non-zero”.

$$ \begin{array}{ccc} 1 & -1 & -2 & 6 \\ 2 & 4 & 7 & 0 \\ 7 & 3 & 0 & 0 \\ 9 & 0 & 0 & 0 \end{array} $$

Practical Example:

4_x1 + 3_x2 = 7
6_x1 + 9_x2 = 7

Solution:

Upper Triangular Component L:

$$ \begin{bmatrix}0 & 3 \\ 0 & 0 \\ \end{bmatrix} $$

Lower Triangular Component:

$$ \begin{bmatrix} 4 & 0 \\ 6 & 9 \\ \end{bmatrix} $$

The inverse of L-1*

$$ \begin{bmatrix} 0.25 & 0 \\ -0.17 & 0.11 \\ \end{bmatrix} $$

Calculation of T:

$$ -\begin{bmatrix} 0.25 & 0 \\ -0.17 & 0.11 \\ \end{bmatrix} \times \begin{bmatrix}0 & 3 \\ 0 & 0 \\ \end{bmatrix}= \begin{bmatrix}0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} $$

Calculation of C:

$$ \begin{bmatrix} 0.25 & 0 \\ -0.17 & 0.11 \\ \end{bmatrix} \times \begin{bmatrix} 7 \\ 7 \\ \end{bmatrix} = \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} $$

Gauss Seidel Algorithm:

$$ \times^{(0)}= \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} $$

$$ \times^{(1)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.042 \\ -0.583 \\ \end{bmatrix} $$

$$ \times^{(2)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.042 \\ -0.583 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.188 \\ -0.681 \\ \end{bmatrix} $$

$$ \times^{(3)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.188 \\ -0.681 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.26 \\ -0.729 \\ \end{bmatrix} $$

$$ \times^{(4)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.26 \\ -0.729 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.297 \\ -0.753 \\ \end{bmatrix} $$

$$ \times^{(5)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.297 \\ -0.753 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.315 \\ -0.766 \\ \end{bmatrix} $$

$$ \times^{(6)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.315 \\ -0.766 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.324 \\ -0.772 \\ \end{bmatrix} $$

$$ \times^{(7)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.324 \\ -0.772 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.329 \\ -0.775 \\ \end{bmatrix} $$

$$ \times^{(8)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.329 \\ -0.775 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.331 \\ -0.776 \\ \end{bmatrix} $$

$$ \times^{(9)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.331 \\ -0.776 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.332 \\ -0.777 \\ \end{bmatrix} $$

$$ \times^{(10)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.332 \\ -0.777 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.777 \\ \end{bmatrix} $$

$$ \times^{(11)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.777 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(12)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(13)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(14)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(15)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(16)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(17)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(18)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

$$ \times^{(19)}= \begin{bmatrix} 0 & -0.75 \\ 0 & 0.5 \\ \end{bmatrix} \times \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} + \begin{bmatrix}1.75 \\ -0.39 \\ \end{bmatrix} = \begin{bmatrix} 2.333 \\ -0.778 \\ \end{bmatrix} $$

x=A^-1b

$$ \begin{bmatrix}2.33 \\ -0.78 \\\end{bmatrix} $$

X1 = 2.33

X2 = -0.78

Working of Gauss Seidel Method Calculator:

In order to solve the linear equations the gauss seidel calculator is the best approach and uses the following points at the time of calculations.

Input:

• Put the values of equations in the designated field of the tool 
• Insert the coefficient value for the equation
• Press on the “Calculate” button

Output:

Our gauss seidel method calculator will give you the following answers:

• Inverse of L-1
• Calculation of T and C 
• Lower and upper triangular matrix 
• Implementation of Gauss-Seidel Algorithm
• Step-by-step calculations 

FAQs:

Is Gauss Method Similar To The Jacobi Method?

These are nearly similar, but the difference is equations of the Jacobi method are solved by using the same set of data. On the other hand, the Gauss method is solved by using all the available data during calculations.

References:

From the Source Wikipedia: Gauss-Seidel method, Convergence.