Gaussian elimination
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m (Gauss elimination moved to Gaussian elimination) |
(towards a uniform notation for linear systems : A*Phi = B) |
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== Gauss Elimination == | == Gauss Elimination == | ||
- | We consider the system of linear equations '''<math> | + | We consider the system of linear equations '''<math> A\cdot\phi = B </math>''' or <br> |
:<math> | :<math> | ||
\left[ | \left[ | ||
Line 12: | Line 12: | ||
\left[ | \left[ | ||
\begin{matrix} | \begin{matrix} | ||
- | { | + | {\phi_1 } \\ |
- | { | + | {\phi_2 } \\ |
. \\ | . \\ | ||
- | { | + | {\phi_n } \\ |
\end{matrix} | \end{matrix} | ||
\right] | \right] | ||
Line 52: | Line 52: | ||
\left[ | \left[ | ||
\begin{matrix} | \begin{matrix} | ||
- | { | + | {\phi_1 } \\ |
- | { | + | {\phi_2 } \\ |
. \\ | . \\ | ||
- | { | + | {\phi_n } \\ |
\end{matrix} | \end{matrix} | ||
\right] | \right] | ||
Line 86: | Line 86: | ||
By using the formula: <br> | By using the formula: <br> | ||
:<math> | :<math> | ||
- | + | \phi_i = {1 \over {a_{ii}^' }}\left( {b_i^' - \sum\limits_{j = i + 1}^n {a_{ij}^' \phi_j } } \right) | |
</math> <br> | </math> <br> | ||
Solve the equation of the k<sup>th</sup> row for x<sup>k</sup>, then substitute back into the equation of the (k-1)<sup>st</sup> row to obtain a solution for (k-1)<sup>st</sup> raw, and so on till k = 1. | Solve the equation of the k<sup>th</sup> row for x<sup>k</sup>, then substitute back into the equation of the (k-1)<sup>st</sup> row to obtain a solution for (k-1)<sup>st</sup> raw, and so on till k = 1. |
Revision as of 20:37, 15 December 2005
Gauss Elimination
We consider the system of linear equations or
To perform Gaussian elimination starting with the above given system of equations we compose the augmented matrix equation in the form:
After performing elementary raw operations the augmented matrix is put into the upper triangular form:
By using the formula:
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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