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Gauss Seidel Method Calculator

Gauss Seidel Method Calculator

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No of equations:

x1 +

x2 +

x3 =

x1 +

x2 +

x3 =

x1 +

x2 +

x3 =

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Use this online Gauss Seidel method calculator that allows you to resolve a system of linear simultaneous equations. You can also compute the values regarding to gauss seidel method problems by using our online power method calculator in a fraction of seconds.

What is the Gauss Seidel method?

“The method in which the first given system of linear equation is placed in diagonally dominant form is termed as Gauss-Seidel method”

Gauss Seidel iteration method is also known as the Liebmann method or the method of successive displacement which is an iterative method used to solve a system of linear equations.

Formula:

The formula to find the Gauss Seidel Method is given as:

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

Where

  • L* = Lower Triangular Matrix
  • U = Upper Triangular Matrix

Lower Triangular Matrix:

“If all the entries above the main diagonal are zero is termed as a lower triangular matrix”

A = \left[\begin{array}{ccc} 2 & 0 & 0 \\ 1 & 5 & 0 \\ 1 & -1 & -2 \end{array}\right]

Upper Triangular Matrix:

“Similarly if all the entries below the main diagonal are zero is known as upper triangular matrix”

A = \left[\begin{array}{ccc} 2 & -1 & 3 \\ 0 & 5 & 2\\ 0 & 0 & -2 \end{array}\right]

Gauss Seidel Iterative Method Algorithm:

Let’s discuss the Gauss Seidel Iterative Method Algorithm regarding the coefficient of variables. Following are the steps to calculate it easily.

  • First, start the process
  • After that, you need to arrange the given system of linear equations in diagonally dominant form.
  • Once read to check the error (e).
  • Convert the first equation in terms of the first variable, the second equation in terms of the second variable, and so on.
  • Once you convert the variables then set initial guesses for x_0, y_0, z_0, and so on.
  • Substitute the value of y_0, z_0 … from step 5 in the first equation fetched from step 4 to estimate the new value of x1_. Use x_1, z_0, u_0 …. in the second equation obtained from step 4 to compute the new value of y1. Similarly, use x_1, y_1, u_0… to find new z_1, and so on.
  • If| x0 – x1| > e and | y0 – y1| > e and | z0 – z1| > e
    and so on then go to step 9.
  • Set x_0=x_1, y_0=y_1, z0=z1, and so on, and go to step 6.
  • Print the value of x_1, y_1, z_1, and so on.
  • Finally, stop the process and obtain your results.

Besides, our online gauss seidel method calculator also supports Gauss Seidel Iterative Method Algorithm and you can calculate it in a couple of seconds.

Gauss Seidel Method Example:

1x_1 + 2x_2 = 7
8x_1 + 9x_2 = 7
Solution:

X = A^-1b

$$ \begin{bmatrix}783061.99 \\ -696054.33 \\\end{bmatrix} $$

x_1 = 783061.99
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

$$ \begin{bmatrix}1 & 0 \\ -0.89 & 0.11 \\\end{bmatrix} \times \begin{bmatrix} 7 \\ 7 \\ 7 \\\end{bmatrix} = \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} $$

Gauss Seidel Algorithm

$$ \times^{(0)}= \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} $$

$$ \times^{(1)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 17.889 \\ -15.123 \\\end{bmatrix} $$

$$ \times^{(2)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 17.889 \\ -15.123 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix}37.247 \\ -32.331 \\\end{bmatrix} $$

$$ \times^{(3)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 37.247 \\ -32.331 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 71.661 \\ -62.921 \\\end{bmatrix} $$

$$ \times^{(4)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix}
71.661 \\ -62.921 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix}
= \begin{bmatrix} 132.842 \\ -117.304 \\\end{bmatrix} $$

$$ \times^{(5)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 132.842 \\ -117.304 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 241.608 \\ -213.985 \\\end{bmatrix} $$

$$ \times^{(6)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 241.608 \\ -213.985 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 434.97 \\ -385.862 \\\end{bmatrix} $$

$$ \times^{(7)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix}
434.97 \\ -385.862 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 778.725 \\ -691.422 \\\end{bmatrix} $$

$$ \times^{(8)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 778.725 \\ -691.422 \\\end{bmatrix} + \begin{bmatrix} 7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 1389.844 \\ -1234.639 \\\end{bmatrix} $$

$$ \times^{(9)}= \begin{bmatrix} 0 & -2 \\ 0 &1.78 \\\end{bmatrix} \times \begin{bmatrix} 1389.844 \\ -1234.639 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 2476.278 \\ -2200.358 \\\end{bmatrix} $$

$$ \times^{(10)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix}
2476.278 \\ -2200.358 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 4407.716 \\ -3917.192 \\\end{bmatrix} $$

$$ \times^{(11)}= \begin{bmatrix} 0 & -2 \\ 0 &1.78 \\\end{bmatrix} \times \begin{bmatrix} 4407.716 \\ -3917.192 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 7841.384 \\ -6969.341 \\\end{bmatrix} $$

$$ \times^{(12)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 7841.384 \\ -6969.341 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 13945.683 \\ -12395.385 \\\end{bmatrix} $$

$$ \times^{(13)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 13945.683 \\ -12395.385 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 24797.769 \\ -22041.684 \\\end{bmatrix} $$

$$ \times^{(14)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 24797.769 \\ -22041.684 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 44090.367 \\ -39190.66 \\\end{bmatrix} $$

$$ \times^{(15)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix}
44090.367 \\ -39190.66 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 78388.319 \\ -69677.728 \\\end{bmatrix} $$

$$ \times^{(16)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \time \begin{bmatrix} 78388.319 \\ -69677.728 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 139362.457 \\ -123876.962 \\\end{bmatrix} $$

$$ \times^{(17)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 139362.457 \\-123876.962 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 247760.923 \\ -220231.154 \\\end{bmatrix} $$

$$ \times^{(18)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 247760.923 \\ -220231.154 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix 440469.308 \\ -391527.496 \\\end{bmatrix} $$

$$ \times^{(19)}= \begin{bmatrix} 0 & -2 \\ 0 & 1.78 \\\end{bmatrix} \times \begin{bmatrix} 440469.308 \\ -391527.496 \\\end{bmatrix} + \begin{bmatrix}7 \\ -5.44 \\\end{bmatrix} = \begin{bmatrix} 783061.991 \\ -696054.326 \\\end{bmatrix} $$

However, you can use our gaussian elimination with the partial pivoting calculator to calculate the values of Guass Seidel method in a fraction of seconds.

How Does Gauss Seidel Method Calculator Work?

This online power method calculator lets you perform calculations by simply entering the following inputs:
Inputs:

  • First, enter the number of equations (2 or 3)
  • After that, enter coefficient values for the equations
  • Simply, click on the “Calculate” button

Outputs:

Gauss Seidel method calculator calculates the following results:

  • Inverse of L-1*
  • Calculation of T
  • Calculation of C
  • Apply Gauss Seidel Algorithm

You can also calculate the resolving systems of equations with the help of the gaussian elimination calculator.

FAQ’s:

Is the Gauss Jacobi method an iterative method?

Yes, Gauss Jacobi or Jacobi method is typically an iterative method that is used for solving equations of the diagonally dominant system of linear equations. And, you can calculate the values of the Gauss Siedal method with respect to the iterative method by using this gauss seidel method calculator

What is the difference between Jacobi and Gauss-Seidel methods?

The difference between Jacobi and Gauss-Seidel methods is that in the Jacobi method the variable values are not modified until the next iteration. While in the Gauss Seidel method the variable values are modified as soon as the new value is considered. You can calculate the values regarding the Gauss Seidel method by using our gauss seidel method calculator

Which iterative method is more efficient between Jacobi method or the Gauss-Seidel method?

The gauss-Seidel method is more efficient as compared to the Jacobi method since the Gauss-Seidel method requires less number of iterations to combine the actual solution with a certain degree of accuracy. We provide you with an online gauss seidel method calculator to make calculations regarding gauss seidel method problems swiftly.

What is the disadvantage of the Jacobi method?

The disadvantage of the Jacobi method includes that after the modified value of a variable is estimated in the present iteration, it is not used up to the next iteration. In simple words, the value of all the variables which are used in the current iteration is from the previous iteration, hence increasing the number of iterations to reach the exact solution.

What is the difference between Gauss elimination and Gauss-Seidel method?

Gauss-elimination is the direct method while Gauss-seidel is the iterative method. These two methods are different from each other and are commonly used for different purposes.

When does the Gauss Seidel method applicable?

The gauss seidel method is applicable if it follows strictly diagonally dominant or symmetric definite matrices.

Conclusion:

Gauss-Seidel Method is commonly used to find the linear system Equations. This method is given and named by German Scientists Carl Friedrich Gauss and Philipp Ludwig Siedel. Generally, the gauss seidel method is applicable if iteration to solve n linear equations with unknown variables. This method is very simple and calculates the values with the help of our online Gauss Seidel method calculator with a couple of steps.

References:

From the source of Wikipedia: Gauss–Seidel method, Algorithm, Examples
From the source of sciencedirect.com: Iterative Methods of Solution, Solution to a System of Linear Algebraic Equations