Gauss coeff. external field?

Question

Can any body tell me how to calculate Gauss coeff. for external field contribution....Plzzzz....or Please let me know where I can get the procedure.....!!!

Thanks in advance....

Answer 1

I'll try my best to answer...
Suppose the goal is to find and describe the solution(s), if any, of the following system of linear equations:

The algorithm is as follows: eliminate x from all equations below L1, and then eliminate y from all equations below L2. This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for.

In our example, we eliminate x from L2 by adding to L2, and then we eliminate x from L3 by adding L1 to L3. Formally:



The result is:

Now we eliminate y from L3 by adding − 4L2 to L3:

The result is:

This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete.

The second part, back-substitution, consists of solving for the unknowns in reverse order. Thus, we can easily see that

Then, z can be substituted into L2, which can then be solved easily to obtain

Next, z and y can be substituted into L1, which can be solved to obtain
Thus, the system is solved.

This algorithm works for any system of linear equations. It is possible that the system cannot be reduced to triangular form, yet still have at least one valid solution: for example, if y had not occurred in L2 and L3 after our first step above, the algorithm would have been unable to reduce the system to triangular form. However, it would still have reduced the system to echelon form. In this case, the system does not have a unique solution, as it contains at least one free variable. The solution set can then be expressed parametrically (that is, in terms of the free variables, so that if values for the free variables are chosen, a solution will be generated).

In practice, one does not usually deal with the actual systems in terms of equations but instead makes use of the augmented matrix (which is also suitable for computer manipulations). This, then, is the Gaussian Elimination algorithm applied to the augmented matrix of the system above, beginning with:

which, at the end of the first part of the algorithm looks like this:

That is, it is in row echelon form.

At the end of the algorithm, we are left with

That is, it is in reduced row echelon form, or row canonical form.