Bisection and Fixed-Point Method

Maria Priscillya Pasaribu4103312018

Bilingual Mathematics Education

Bisection Method

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 principles:•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

Bisection Method

Figure 1 At least one root exists between the two points if the function is real, continuous, and changes sign

If , then there may be more than one root between and

Algorithm for The Bisection Method

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Step 1

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.

x

f(x)

xu x

Figure 1

x

f(x)

xu x

xm

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Step 2

Estimate the root, xm of the equation f (x) = 0 as the mid point between x and xu as

xx

m = xu

2

Figure 5 Estimate of xm

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Step 3

Now check the following

a) If , then the root lies between x and xm;

then x = x ; xu = xm.

b) If , then the root lies between xm and xu;

then x = xm; xu = xu.

c) If ; then the root is xm. Stop the

algorithm if this is true.

0ml xfxf

0ml xfxf

0ml xfxf

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Step 4

xx

m = xu

2

100

newm

oldm

new

a x

xxm

root of estimatecurrent newmx

root of estimate previousoldmx

Find the new estimate of the root

Find the absolute relative approximate error

where

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Step 5

Is ?

Yes

No

Go to Step 2 using new upper and lower

guesses.

Stop the algorithm

Compare the absolute relative approximate error with the pre-specified error tolerance .

as

sa

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.

Example 1

Find the root of X3 – 9X2 + 18X – 6 = 0 has a unique root in [2, 2.5] with accuracy 10-3.

Graph

Step 1: Assume that x= 2 and xu= 2,5 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 from the equation X3 – 9X2 + 18X – 6 = 0

Step 2

Step 3

Step 4: Iteration 1

Cont.

Root of f(x)=0 as function of number of iterations for bisection method

Iteration xl xu xm f(xl) f(xm) Error (xu- xm)

0 2 2,5 2,25 2 0,328 0,251 2,25 2,5 2,375 0,328 -2,86 0,1252 2,25 2,375 2,3125 0,328 -0,137451 0,06253 2,25 2,3125 2,28125 0,328 0,09744 0,031254 2,28125 2,3125 2,296875 0,09744 -0,019489 0,0156255 2,28125 2,296875 2,2890625 0,09744 0,039107 0,00781256 2,2890625 2,296875 2,29296875 0,039107 0,00984 0,003906257 2,29296875 2,296875 2,29492188 0,00984  -0,004816  0,001953128 2,29296875 2,29492188 2,29394532 0,00984  0,00251476 0,0009779 2,29394532 2,29492188 2,2944336 0,00251476 -0,0011502 0,000488

The iteration is stopped because the error is approximate to 10-3.

So, the estimated root is 2,294436.

Fixed-Point Method

Fixed-point method is one of the opened methods that is finding approximate solutions of the equation f(x)=0

Algorithm of Fixed-point Method

• Given an equation f(x)=0

• Rewrite the equation f(x)=0 in the form of x=g(x)

• Let the initial guess be x0 and consider the recursive process

• xn+1=g(xn), n= 0, 1, 2, ...

Example

Find the root of X3 – 9X2 + 18X – 6 = 0 has a unique root in [2, 2.5] with accuracy 10-3.

Solution

With x0 = 2.25, this is the result of the fixed-point 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.