Gaussian elimination

(Difference between revisions)
 Revision as of 22:40, 16 December 2005 (view source)Jasond (Talk | contribs)← Older edit Revision as of 22:59, 16 December 2005 (view source)Jasond (Talk | contribs) (→Algorithm)Newer edit → Line 99: Line 99: == Algorithm == == Algorithm == - Code or psuedo-code goes here. + Forward elimination phase + :    for k:= 1 step until n-1 do
+ ::    for i:=k+1 step until n do
+ :::    $m = {{A_{ik} } \over {A_{kk} }}$
+ :::    for j:=k+1,3
+ ::::    $A_{ij}=A_{ij}-mA_{kj}$
+ :::    end loop (j)
+ :::    $b_i=b_i-mb_k$
+ ::    end loop (i)
+ :    end loop (k)
+ Back substitution phase + :    for k:=n stepdown until 1 do
+ ::    for i:=k+1 step until n
+ :::    $b_k=b_k-A_{ki}b_{i}$
+ ::    end loop (i)
+ ::    $\phi_{k}=b_{k}/A_{kk}$
+ :      end loop (k) == Important Considerations == == Important Considerations == Gaussian elimination is best used for relatively small, relatively full systems of equations.  If properly used, it will outperform iterative methods for these systems.  However, it is important to keep in mind that the basic algorithm is vulnerable to accuracy issues.  In practice, it is best to employ safeguards against such problems (e.g. pivoting). Gaussian elimination is best used for relatively small, relatively full systems of equations.  If properly used, it will outperform iterative methods for these systems.  However, it is important to keep in mind that the basic algorithm is vulnerable to accuracy issues.  In practice, it is best to employ safeguards against such problems (e.g. pivoting).

Description

We consider the system of linear equations $A\phi = B$ or

$\left[ \begin{matrix} {a_{11} } & {a_{12} } & {...} & {a_{1n} } \\ {a_{21} } & {a_{22} } & . & {a_{21} } \\ . & . & . & . \\ {a_{n1} } & {a_{n1} } & . & {a_{nn} } \\ \end{matrix} \right] \left[ \begin{matrix} {\phi_1 } \\ {\phi_2 } \\ . \\ {\phi_n } \\ \end{matrix} \right] = \left[ \begin{matrix} {b_1 } \\ {b_2 } \\ . \\ {b_n } \\ \end{matrix} \right]$

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

$\left[ \begin{matrix} {a_{11} } & {a_{12} } & {...} & {a_{1n} } \\ {a_{21} } & {a_{22} } & . & {a_{21} } \\ . & . & . & . \\ {a_{n1} } & {a_{n1} } & . & {a_{nn} } \\ \end{matrix} \left| \begin{matrix} {b_1 } \\ {b_2 } \\ . \\ {b_n } \\ \end{matrix} \right. \right] \left[ \begin{matrix} {\phi_1 } \\ {\phi_2 } \\ . \\ {\phi_n } \\ \end{matrix} \right]$

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

$\left[ \begin{matrix} {a_{11}^' } & {a_{12}^' } & {...} & {a_{1n}^' } \\ 0 & {a_{22}^' } & . & {a_{2n}^' } \\ . & . & . & . \\ 0 & 0 & . & {a_{nn}^' } \\ \end{matrix} \left| \begin{matrix} {b_1^' } \\ {b_2^' } \\ . \\ {b_n^' } \\ \end{matrix} \right. \right]$

The solution to the original system is found via back substitution. The solution to the last equation is

$\phi_n = b_n^'/a_{nn}'.$

This result may now be substituted into the second to last equation, allowing us to solve for $\phi_{n-1}$. Repetition of this substitution process will give us the complete solution vector. The back substitution process may be expressed as

$\phi_i = {1 \over {a_{ii}^' }}\left( {b_i^' - \sum\limits_{j = i + 1}^n {a_{ij}^' \phi_j } } \right),$

where $i=n,n-1,\ldots,1$.

Algorithm

Forward elimination phase

for k:= 1 step until n-1 do
for i:=k+1 step until n do
$m = {{A_{ik} } \over {A_{kk} }}$
for j:=k+1,3
$A_{ij}=A_{ij}-mA_{kj}$
end loop (j)
$b_i=b_i-mb_k$
end loop (i)
end loop (k)

Back substitution phase

for k:=n stepdown until 1 do
for i:=k+1 step until n
$b_k=b_k-A_{ki}b_{i}$
end loop (i)
$\phi_{k}=b_{k}/A_{kk}$
end loop (k)

Important Considerations

Gaussian elimination is best used for relatively small, relatively full systems of equations. If properly used, it will outperform iterative methods for these systems. However, it is important to keep in mind that the basic algorithm is vulnerable to accuracy issues. In practice, it is best to employ safeguards against such problems (e.g. pivoting).