Gauss Elimination Method Algorithm and Flowchart


Gauss Elimination method can be adopted to find the solution of linear simultaneous equations arising in engineering problems. In the method, equations are solved by elimination procedure of the unknowns successively.

The method overall reduces the system of linear simultaneous equations to an upper triangular matrix. Then backward substitution is used to derive the unknowns. This is the key concept in writing an algorithm or program, or drawing a flowchart for Gauss Elimination.

Partial pivoting or complete pivoting can be adopted in Gauss Elimination method. So, this method is considered superior to the Gauss Jordan method.

In the Gauss Elimination method algorithm and flowchart given below, the elimination process is carried out until only one unknown remains in the last equation. It is straightforward to program, and partial pivoting can be used to control rounding errors.

Gauss Elimination Algorithm:

  1. Start
  2. Declare the variables and read the order of the matrix n.
  3. Take the coefficients of the linear equation as:
    Do for k=1 to n
    Do for j=1 to n+1
    Read a[k][j]
    End for j
    End for k
  4. Do for k=1 to n-1
    Do for i=k+1 to n
    Do for j=k+1 to n+1
    a[i][j] = a[i][j] – a[i][k] /a[k][k] * a[k][j]
    End for j
    End for i
    End for k
  5. Compute x[n] = a[n][n+1]/a[n][n]
  6. Do for k=n-1 to 1
    sum = 0
    Do for j=k+1 to n
    sum = sum + a[k][j] * x[j]
    End for j
    x[k] = 1/a[k][k] * (a[k][n+1] – sum)
    End for k
  7. Display the result x[k]
  8. Stop

Gauss Elimination Flowchart:

Here is a basic layout of Gauss Elimination flowchart which includes input, forward elimination, back substitution and output.

Flowchart for Gauss Elimination MethodThe one given below shows pivoting and elimination procedure.

Gauss Elimination Flowchart with Pivoting
Here is an attachment showing how the forward elimination and back substitution take place.

Forward Elimination and Back Substitution
Also see,
Gauss Elimination C Program
Gauss Elimination MATLAB Program

While solving linear simultaneous equations, analytical methods often fail when the problems are complicated. These algorithm and flowchart can be referred to write source code for Gauss Elimination Method in any high level programming language.

It is complete, and though it is somewhat long for coding, it is clearly differentiated for the elimination phase and the back substitution phase. If you have any questions, discuss them in comments.


  1. Programming can be made easier with flowcharts. Specially suda code can be directly converted to flowcharts. By the way you have used correct shape and the program is very clear


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