- Therefore, adequate methods should be used in linear equation system solutions such as without calculating determinant or inverse matrix.
- Two of the most useful methods are Gaussian elimination and L-U decomposition.
- Elimination Methods. While it may be satisfactory for hand computations with small systems, it is inadequate for a large system (numbers may getting bigger!).
- The method that is called Gaussian elimination avoids this by subtracting
times the first equation from the
equation to make the transformed numbers in the first column equal to zero.
- We do similarly for the rest of the columns.
- Observe that zeros may be created in the diagonal positions even if they are not present in the original matrix of coefficients.
- As a result, the variable
is eliminated in the other equations and the new system of equations becomes:
- Notice that in this new system of equations at first stage, the coefficient of the
term in the
row and the constant
are as follows:
- Next, let
(If not, pivoting) in this new system of equations, then multiply the
equation by (
) and subtract it from the
equation.
- Repeat this procedure up to equation
as
times to obtain the upper-triangular form:
- As seen, the number of unknowns decreases by one in the
and other equations as
stage, decreases once again in the
and other equations as
stage and so on. In
stage, a single unknown is obtained.
- Then, the
coefficient of the
equation as
stage is as follows:
 |
(6.1) |
 |
(6.2) |
- After reaching to the upper-triangular form, the solution is almost readily obtained.
- From the last equation in the upper-triangular form:
- All other unknowns are obtained consequently by backward-substitution. The general expression would be:
![$\displaystyle {\small\boxed{x_k=\frac{1}{a_{kk}^{(k-1)}}\left[ b_{k}^{(k-1)} - \sum_{j=k+1}^{n} a_{kj}^{(k-1)} x_j\right]}~~(k=n-1,\ldots,1)}$](img653.svg) |
(6.3) |
- Repeat the example of the previous section,
- The method we have just illustrated is called Gaussian elimination.
- In this example, no pivoting was required to make the largest coefficients be on the diagonal.
- Back-substitution, gives us
Example py-file: Show steps in Gaussian elimination and back substitution without pivoting.
myGEshow.py
Figure 6.1:
Steps in Gaussian elimination and back substitution without pivoting.
|
- if we had stored the ratio of coefficients in place of zero (we show these in parentheses), our final form would have been
- The original matrix can be written as the product:
- This procedure is called a
-decomposition of
. (
)
- We have
- When there are m row interchanges