Linear Equations by Gaussian Elimination

### Program Information

Name: Linear Equations by Gaussian Elimination
Domain: Numerical Program
Functionality: A program which solves a system of linear equations, $Ax=b$ by using Gaussian elimination where A is the square matrix of coeffcients and $b$ is a column vector
Input: The original input pair is $(A,b)$;
Output: $x=(x_1,x_2,\dots ,x_n)$ is a solution of the system ;

#### Reference

 Metamorphic Testing:A New Approach for Generating Next Test Cases

### MR Information

#### MR1------

Description:
Property: $Ax=b$ and $A'x'=b' \Rightarrow x=x'$ where $A'$ is obtained from $A$ by swapping row $i$ and $j$ and $b'$ is obtained from $b$ by swapping corresponding entries;
Source input: $(A,b)$
Source output: $x$
Follow-up input: $(A',b')$
Follow-up output: $x'$
Input relation: $(A,b) \Rightarrow (A',b')$ where $A'$ is obtained from $A$ by swapping row $i$ and $j$ and $b'$ is obtained from $b$ by swapping corresponding entries;
Output relation: $x=x'$
Pattern:

#### MR2------

Description:
Property: $Ax=b$ and $A''x''=b \Rightarrow x''=(x_1,\dots ,x_{i-1},x_j,x_{i+1},\dots ,x_{j-1},x_i,x_{j+1},\dots ,x_n)$ where $A''$ is obtained from $A$ by swapping col $i$ and $j$ with $i < j$ ; And $x''_i=x_j,x_j''=x_i$ and $x''_k=x_k$ for $k \neq i,j$
Source input: $(A,b)$
Source output: $x$
Follow-up input: $(A'',b)$
Follow-up output: $x''$
Input relation: $(A,b) \Rightarrow (A'',b)$ where $A''$ is obtained from $A$ by swapping col $i$ and $j$ with $i < j$
Output relation: $x''=(x_1,\dots ,x_{i-1},x_j,x_{i+1},\dots ,x_{j-1},x_i,x_{j+1},\dots ,x_n)$ where $x''_i=x_j,x_j''=x_i$ and $x''_k=x_k$ for $k \neq i,j$
Pattern:
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