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.
“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.
The Gauss-Seidel method calculator utilizes the algorithm to achieve fast calculations. So just follow the below points:
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;
The determination of the triangular matrix by the above formula tells us that the matrix is equal to the product of principle diagonal elements.
“The matrix has non-zero entries on the down-ward diagonal and above the main diagonal is zero”.
$$ \begin{array}{ccc} 9 & 0 & 0 & 0 \\ 7 & 3 & 0 & 0 \\ 2 & 4 & 7 & 0 \\ 1 & -1 & -2 & 6 \end{array} $$
“The matrix in which all the elements are below the diagonal matrix is zero and above have non-zero”.
$$ \begin{array}{ccc} 1 & -1 & -2 & 6 \\ 2 & 4 & 7 & 0 \\ 7 & 3 & 0 & 0 \\ 9 & 0 & 0 & 0 \end{array} $$
4_x1 + 3_x2 = 7
6_x1 + 9_x2 = 7
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
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:
Output:
Our gauss seidel method calculator will give you the following answers:
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.
From the Source Wikipedia: Gauss-Seidel method, Convergence.
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