Maria Priscillya Pasaribu
Bilingual Mathematics Education
Bisection method is one of the closed methods
(bracketing method) to determine the root of a
nonlinear equation f(x) = 0, with the following main
•Using two initial values to confine one or more
roots of non-linear equations.
•Root value is estimated by the midpoint between
two existing initial values
Choose x and xu as two guesses for the root such that
f(x) f(xu) < 0, or in other words, f(x) changes sign
between x and xu. This was demonstrated in Figure 1.
Estimate the root, xm of the equation f (x) = 0 as the mid
point between x and xu as
x + xu
Figure 5 Estimate of xm
Now check the following
a) If f ( xl ) f ( xm ) < ,0 then the root lies between x and xm;
then x = x ; xu = xm.
f ( xl ) f ( xm ) > 0
, then the root lies between x m and xu;
then x = xm; xu = xu.
f ( xl ) f ( xm ) = 0
; then the root is x m. Stop the algorithm
if this is true.
Find the new estimate of the root
x + xu
Find the absolute relative approximate error
x new − xm
xm = previous estimate of root
xm = current estimate of root
Compare the absolute relative approximate error ∈a with
the pre-specified error tolerance ∈s .
Go to Step 2 using new
upper and lower
Stop the algorithm
Is ∈a >∈s?
Note one should also check whether the number of
iterations is more than the maximum number of iterations
allowed. If so, one needs to terminate the algorithm and
notify the user about it.
Find the root of X3 – 9X2 + 18X – 6 = 0 has
a unique root in [2, 2.5] with accuracy 10 -3.
The iteration is stopped because the
error is approximate to 10-3.
So, the estimated root is 2,294436.
Fixed-point method is one of the opened
methods that is finding approximate
solutions of the equation f(x)=0
Algorithm of Fixed-point
• Given an equation f(x)=0
• Rewrite the equation f(x)=0 in the form of
• Let the initial guess be x0 and consider
the recursive process
• xn+1=g(xn), n= 0, 1, 2, ...
Find the root of X3 – 9X2 + 18X – 6 = 0 has a
unique root in [2, 2.5] with accuracy 10 -3.
With x0 = 2.25, this is the result of the fixedpoint method for all five choices of g.
The root of the equation we got is
2,2944336, as was noted in example of
Bisection Method. Comparing the results to
the Bisection method given in that example,
it can be seen that the same result at least
have been obtained for choice d.