Page 224 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 9 SYSTEMS OF SIMULTANEOUS EOUATIONS 20 1

simultaneous replacement. Improvement in one of the variables does not have an
effect until the next cycle of iteration. For this reason it does not converge as
rapidly as the Gauss-Seidel method, to be described in the following section.
To illustrate, consider a system of order 3,

allxl + al2x2 + a13x3 = cl
a21xI a22x2 + a2@3 = c2
a31x1 + a32x2 + a33x3 = c3
These equations can be rearranged to give

x1 = c1 - a12X2 - a13x3
a1 1

x2 = ‘2 - a21X1 - a23x3
a22

c3 - a31xl - a32x2
x3 =
a33
Begin with initial estimates for XI, x2 and x3; in the following example, initial
estimates of zero were used. Then solve for each unknown value; thus
c1-0-0
x1 =
a1 1

c2 -0-0
x2 =
a22

c3 -0-0
x3 =
a33

In the second iteration,

x1 = ‘1 - a12x2 - a13x3
a1 1
and so on.

The Jacobi method is shown implemented on a spreadsheet. Figure 9-9
shows the table of coefficients and constants.