Page 226 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 9 SYSTEMS OF SIMULTANEOUS EOUATIONS 203

Cells BIO, C10 and DIO contain, respectively, the formulas
=( $E$4-$C$4*Cg-$D$4*Dg)/$B$4
=($E$5-$8$5*89-$D$S*D9)/$C$5

=($E$6-$B$6*Bg-$C$6*Cg)/$D$6
When these formulas are filled down into successive rows, as shown in
Figure 9-10, the values of the variables XI, x2 and x3 converge. Convergence to a
suitable level is observed visually. In this particular example, twenty-three
iteration cycles were required to get below the lo6 percent error level (here, the
percentage error in the variable xl is shown).

The Gauss-Seidel Method
Implemented on a Worksheet
In the Gauss-Seidel method, an improved value of one of the variables is
used in the iteration cycle as soon as it has been calculated. The Gauss-Seidel
method is sometimes called the method of successive replacement.
To illustrate, consider the same system of order 3 that was used previously to
illustrate the Jacobi method. Again, begin with initial estimates of zero for xI, x2
and x3. Now solve for each unknown value in turn, using the latest values of the
variables as they are calculated; thus

c1 -0-0
x1 =
a1 1
c2 - aZlxl - 0
x2 =
a22

x3 = c3 - a31X1 - a32x2
a3 3
In the second iteration,

XI = cl - a12x2 - a13x3
a1 1
and so on.
Using the same constants and coefficients that were used in the preceding
example (Figure 9-10), the spreadsheet formulas in Figure 9-1 1 can be modified
to implement the Gauss-Seidel method, in which the value of a variable is used
as soon as it is calculated. The formulas in cells B14, C14 and D14 are,
respectively,