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Gaussian elimination

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Gauss Elimination

We consider the system of linear equations  Ax = b or

 
\left[ 
\begin{matrix}
   {a_{11} } & {a_{12} } & {...} & {a_{1n} }  \\ 
   {a_{21} } & {a_{22} } & . & {a_{21} }  \\ 
   . & . & . & .  \\ 
   {a_{n1} } & {a_{n1} } & . & {a_{nn} }  \\ 
\end{matrix}
\right]
\left[ 
\begin{matrix}
   {x_1 }  \\ 
   {x_2 }  \\ 
   .  \\
   {x_n }  \\
\end{matrix}
\right]
=
\left[ 
\begin{matrix}
   {b_1 }  \\ 
   {b_2 }  \\ 
   .  \\
   {b_n }  \\
\end{matrix}
\right]

To perform Gaussian elimination starting with the above given system of equations we compose the augmented matrix equation in the form:


\left[ 
\begin{matrix}
   {a_{11} } & {a_{12} } & {...} & {a_{1n} }  \\ 
   {a_{21} } & {a_{22} } & . & {a_{21} }  \\ 
   . & . & . & .  \\ 
   {a_{n1} } & {a_{n1} } & . & {a_{nn} }  \\ 
\end{matrix}

\left| 
\begin{matrix}
   {b_1 }  \\ 
   {b_2 }  \\ 
   .  \\ 
   {b_n }  \\ 
\end{matrix}

\right.

\right]
\left[ 
\begin{matrix}
   {x_1 }  \\ 
   {x_2 }  \\ 
   .  \\ 
   {x_n }  \\ 
\end{matrix}
\right]

After performing elementary raw operations the augmented matrix is put into the upper triangular form:


\left[ 
\begin{matrix}
   {a_{11}^' } & {a_{12}^' } & {...} & {a_{1n}^' }  \\ 
   0 & {a_{22}^' } & . & {a_{2n}^' }  \\ 
   . & . & . & .  \\ 
   0 & 0 & . & {a_{nn}^' }  \\ 
\end{matrix}

\left| 
\begin{matrix}
    {b_1^' }  \\ 
   {b_1^' }  \\ 
   .  \\
   {b_1^' }  \\ 

\end{matrix}

\right.
\right]

By using the formula:


x_i  = {1 \over {a_{ii}^' }}\left( {b_i^'  - \sum\limits_{j = i + 1}^n {a_{ij}^' x_j } } \right)

Solve the equation of the kth row for xk, then substitute back into the equation of the (k-1)st row to obtain a solution for (k-1)st raw, and so on till k = 1.

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