“The method wherein the primary given gadget of linear equation is positioned in diagonally dominant shape is termed as Gauss-Seidel technique”
Gauss Seidel iteration method is likewise called the Liebmann technique or the approach of successive displacement that is an iterative method used to remedy a system of linear equations.
Formula:
The formulation to discover the Gauss Seidel method is given as x^(k+1)= L*^-1(b-Uxk) Where
“If all of the entries above the principle diagonal are zero is termed as a decrease triangular matrix”
$$ A = \left[\begin{array}{ccc} 2 & 0 & 0 \\ 1 & 5 & 0 \\ 1 & -1 & -2 \end{array}\right] $$
“Further if all the entries below the primary diagonal are 0 is known as top triangular matrix”
$$ A = \left[\begin{array}{ccc} 2 & -1 & 3 \\ 0 & 5 & 2\\ 0 & 0 & -2 \end{array}\right] $$
$$ \begin{aligned} 1x_1 + 2x_2 &= 7 \\ 8x_1 + 9x_2 &= 7 \end{aligned} $$
Solution:
$$ X = A^{-1}b \quad \begin{bmatrix} 783061.99 \\ -696054.33 \end{bmatrix}, \quad x_1 = 783061.99, \quad x_2 = -696054.33 $$
Upper triangular component (L):
$$ \begin{bmatrix} 0 & 2 \\ 0 & 0 \end{bmatrix} $$
Lower triangular component (T):
$$ \begin{bmatrix} 1 & 0 \\ 8 & 9 \end{bmatrix} $$
Inverse of \( L^{-1} \):
$$ \begin{bmatrix} 1 & 0 \\ -0.89 & 0.11 \end{bmatrix} $$
Calculation of T:
$$ -\begin{bmatrix} 1 & 0 \\ -0.89 & 0.11 \end{bmatrix} \times \begin{bmatrix} 0 & 2 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \end{bmatrix} $$
Calculation of C:
$$ \begin{bmatrix} 1 & 0 \\ -0.89 & 0.11 \end{bmatrix} \times \begin{bmatrix} 7 \\ 7 \end{bmatrix} = \begin{bmatrix} 7 \\ -5.44 \end{bmatrix} $$
$$ x^{(0)} = \begin{bmatrix} 7 \\ -5.44 \end{bmatrix} $$
Iteration Steps:
$$ x^{(1)} = \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \end{bmatrix} \times \begin{bmatrix} 7 \\ -5.44 \end{bmatrix} + \begin{bmatrix} 7 \\ -5.44 \end{bmatrix} = \begin{bmatrix} 17.889 \\ -15.123 \end{bmatrix} $$
Repeat similar calculations for \( x^{(2)}, x^{(3)}, \dots, x^{(n)} \) until convergence:
Sure, Gauss Jacobi or Jacobi approach is usually an iterative method this is used for fixing equations of the diagonally dominant device of linear equations. And, you could calculate the values of the Gauss Siedal method with recognize to the iterative approach by means of the use of this gauss seidel method calculator
The difference among Jacobi and Gauss-Seidel methods is that within the Jacobi approach the variable values aren't modified until the next new release. even as within the Gauss Seidel approach the variable values are modified as soon as the new price is taken into consideration. you can calculate the values regarding the Gauss Seidel technique via using our gauss seidel technique calculator
