10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS
As a numerical technique, Gaussian elimination is rather unusual because it is
direct
. That
is, a solution is obtained after a single application of Gaussian elimination. Once a “solu-
tion” has been obtained, Gaussian elimination offers no method of refinement. The lack of
refinements can be a problem because, as the previous section shows, Gaussian elimination
is sensitive to rounding error.
Numerical techniques more commonly involve an iterative method. For example, in
calculus you probably studied Newton’s iterative method for approximating the zeros of a
differentiable function. In this section you will look at two iterative methods for approxi-
mating the solution of a system of
n
linear equations in
n
variables.
The Jacobi Method
The first iterative technique is called the
Jacobi method,
after Carl Gustav Jacob Jacobi
(1804–1851). This method makes two assumptions: (1) that the system given by
has a unique solution and (2) that the coefficient matrix
A
has no zeros on its main diago-
nal. If any of the diagonal entries
are zero, then rows or columns must be
interchanged to obtain a coefficient matrix that has nonzero entries on the main diagonal.
To begin the Jacobi method, solve the first equation for
the second equation for
and so on, as follows.
Then make an
initial approximation
of the solution,
Initial approximation
and substitute these values of
into the right-hand side of the rewritten equations to obtain
the
first approximation.
After this procedure has been completed, one
iteration
has been
x
i
(
x
1
,
x
2
,
x
3
, . . . ,
x
n
),
x
n
5
1
a
nn
(
b
n
2
a
n
1
x
1
2
a
n
2
x
2
2
. . .
2
a
n
,
n
2
1
x
n
2
1
d
.
.
.
x
2
5
1
a
22
(
b
2
2
a
21
x
1
2
a
23
x
3
2
. . .
2
a
2
n
x
n
)
x
1
5
1
a
11
(
b
1
2
a
12
x
2
2
a
13
x
3
2
. . .
2
a
1
n
x
n
)
x
2
,
x
1
,
a
11
,
a
22
,
. . . ,
a
nn
a
n
1
x
1
1
a
n
2
x
2
1
. . .
1
a
nn
x
n
5
b
n
.
.
.
.
.
.
.
.
.
.
.
.
a
21
x
1
1
a
22
x
2
1
. . .
1
a
2
n
x
n
5
b
2
a
11
x
1
1
a
12
x
2
1
. . .
1
a
1
n
x
n
5
b
1
578
CHAPTER 10
NUMERICAL METHODS