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Multi Equation Solver

In this we use the Gauss Jordan method of solving a matrix to determine the exact solution of a set of multi variable equations, provided that the system of equations aren't inconsistent.

Gauss Jordan

Row Echelon Form & The Reduced Row Echelon Form

In this we make the reduced row echelon form of the matrix.
The properties of a Reduced Row Echelon Matrix are as follows :

  • If a row doesn't consist of zeroes only, the first non zero number must be 1, and it's called the leading one.
  • If a row only consists of zeroes, it must be moved to the bottom.
  • For each non zero row, the leading one must be to the right, and below the other leading one's.
  • In a column with a leading one, the other elements must be 0.

The only difference between the row echelon and a reduced row echelon form is the 4th point of the above mentioned properties.

Why don't we consider any of the rows to be completely filled with zeroes in this project?

In this project however, we don't consider any of the rows to be completely filled with zeroes, as it would mean that the system of equations is inconsistent (since the rank of the matrix would be less than the number of variables).

Rank of a Matrix

The rank of a matrix is the number of non zero rows in the reduced row echelon form of the matrix.

Usage of rank to determine whether the system of equations is consistent or not

If we consider the equation matrix to be A, and the augmented matrix to be B, the form of the equation is AX=B. Also, we consider n to be the number of variables in the system of equations.

  • If Rank(A) = Rank(A|B) < n -> Infinite Solutions
  • If Rank(A) = Rank(A|B) = n -> Unique Solution
  • If Rank(A) ≠ Rank(A|B) -> No Solution

Cramers Rule Method for inconsistent systems

If the system of equations is inconsistent, we can use the Cramers Rule method to determine the solution of the system of equations.

It states that if the determinant of the coefficient matrix is zero, then the system of equations is inconsistent.

Usage

In this project, we use Cramers rule to determine if the set of equations is consistent of not. If it is consistent, we use the Gauss Jordan method to determine the solution of the system of equations.

We could have used just the cramers rule method, but I used the Gauss Jordan method as I just learnt it and wanted to implement it. Also, I heard that it's faster than the Cramers rule method.

How to use

  • The first line of the input is the number of variables or the number of equations. We take this value as 'n'.
  • The next 'n' lines are the coefficients of the variables in the equations. The last number in each line is the constant.
  • For example, for the equation, 2x + 3y = 5, the input would be 2 3 5.

Example

Input

g++ main.cpp -o main
./main
3
1 2 3 6
4 5 6 15
1 1 1 3

Number of variables : 3
1x + 2y + 3z = 6 : 1 2 3 6
4x + 5y + 6z = 15 : 4 5 6 15
1x + 1y + 1z = 3 : 1 1 1 3

Output

1 0 0 1
0 1 0 1
0 0 1 1

The output is the reduced row echelon form of the augmented matrix.

1x + 0y + 0z = 1 : 1 0 0 1 -> x = 1
0x + 1y + 0z = 1 : 0 1 0 1 -> y = 1
0x + 0y + 1z = 1 : 0 0 1 1 -> z = 1

The solution of the system of equations is x = 1, y = 1, z = 1.

References