082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.13 Iterative techniques for the solution of simultaneous equations 313
available and these are known as iterative methods. They generate a sequence of approximate
solutions which may converge to the required solution, and are particularly
advantageous when large systems of equations are to be solved by computer. We shall
studytwo such techniques here: Jacobi’s method and Gauss--Seidel iteration.
Example8.49 Solve the equations
2x+y =4
x−3y = −5
usingJacobi’s iterative method.
Solution We first rewrite the equations as
2x =−y+4
−3y =−x−5
and then as
x = − 1 2 y +2
(8.10)
y = 1 3 x + 5 3
Jacobi’smethodinvolves‘guessing’asolutionandsubstitutingtheguessinther.h.s.of
the equations in(8.10).Suppose we guessx = 0,y = 0.Substitution then gives
x = 2
y = 5 3
We now use these values as estimates of the solution and resubstitute into the r.h.s. of
Equation (8.10).This time wefind
x = − 1 2
( 5
3)
+2 = 1.1667 (tofour decimal places)
y = 1 3 (2) + 5 3 = 2.3333
(tofour decimal places)
The whole process is repeated in the hope that each successive application oriteration
will give an answer close to the required solution, that is successive iterates will converge.
In order to keep track of the calculations, we label the initial guessx (0) ,y (0) , the
resultof the first iterationx (1) ,y (1) and so on. Generally, we find
x (n+1) = − 1 2 y(n) +2
y (n+1) = 1 3 x(n) + 5 3