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2.7 Solution of systems of linear equations
Our goal in this section is to develop methods for solving a system of
n
linear equations with
n
unknowns. We will use these methods later in the course since solving ODEs/PDEs often requires
solving systems of equations.
Recall notations. Consider for example the following system of 3 equations with 3 unknowns
⎪
⎩
⎪
⎨
⎧
−
=
+
+
=
+
−
=
+
+
3
5
3
0
4
2
7
3
2
z
y
x
z
y
x
z
y
x
or
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
3
0
7
5
3
1
4
1
2
3
2
1
3
2
1
x
x
x
where x
1
=x, x
2
=y, and x
3
=z
Shorthand notation:
b
x
A
r
r
=
Equation (*)
A

n
x
n
coefficient matrix
x
r
 vector of
n
unknowns, i.e., (x
1
, x
2
,…, x
n
)
b
r
 known RHS vector, i.e., (b
1
, b
2
,…, b
n
)
Why do we need to “complicate” things by using vectors and matrices? For some mathematical models
in engineering,
n
can get very large (e.g., 1000s) and it is much easier to set up numerical solutions of
such problems when the equations are written in the form given by Eq. (*).
There are two main approaches to finding
x
r
:
(i)
Direct methods:
Gaussian Elimination is the most common (Math 115)

perform organised raw operations to eliminate unknowns, which is similar to the solution by
hand for small
n
;

the method is easy to code to handle
n
of any size, but the number of operations increases as
n
increases;
(ii)
Indirect (Iterative) methods:
GaussSeidel is the most common

the idea is similar to that of the direct iteration method for a single equation, i.e., x
new
=g(x
old
)

can employ relaxation to help convergence
There are variations of (i) and (ii) but ideas are similar. We will now consider some examples of (i) and
(ii).
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