Let us turn to the general problem (1.1):
If 
, we can eliminate the unknown 
from each
of the succeeding equations. A typical step is to subtract from
equation i the multiple 
of the first
equation. The results will be denoted with a superscript 2 . The step is
carried out by first forming
and then forming
and
The multiple of the first equation is chosen to make 
, that
is, to eliminate the unknown 
from equation i . Doing this for
each 
we arrive at the system
Notice that if we start out with A stored in a C or FORTRAN array, we
can save a considerable amount of storage by overwriting the
with the 
as they are created. Also,
we can save the multipliers 
in the space formerly occupied
by the 
entries of
A and just remember that all the elements below the diagonal in the first
column are really zero after elimination. Later we shall see why it
is useful to save the multipliers. Similarly the original vector b can
be overwritten with the 
as they are formed.
Now we set the first equation aside and eliminate 
from equations
in the same way. If 
, then for
we first form
and then
and
This results in
As before, we set the first two equations aside and eliminate 
from
equations 
. This can be done as long as 
.
The elements 
are called
pivot elements or simply pivots.
Clearly, the process can be continued as long as no pivot vanishes.
Assuming this to be the case, we finally arrive at
In the computer the elements of the original matrix A are successively
overwritten by the 
as they are formed, and the
multipliers 
are saved in the places corresponding to the
variables eliminated.
The process of reducing the system of equations (1.1) to the form
(1.11) is called (forward) elimination. The result is a
system with a kind of coefficient matrix called upper triangular.
An upper triangular matrix 
is one for which
It is easy to solve a system of equations (1.11) with an upper
triangular matrix by a process known as back substitution. If
, we solve the last equation for 
Using the known value of 
, we then solve equation n - 1 for
, and so forth. A typical step is to solve equation k ,
for 
using the previously computed 
:
The only way this process can break down (in principle) is if a pivot element is zero. The objective of this lesson is to exercise the array constructs in the language. Therefore, the other important considerations will be avoided.
The algorithm for elimination is quite compact:
