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Chapter 5

Bracketing Method

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• Easy

• But, not easy

• How about these?

a

acbbxcbxax

2

40

22

? 02345 xfexdxcxbxax

Roots of Equations

? 0)3sin()10cos(

? 0sin

xxx

xxx

Graphical Approach

• Make a plot of the

function f(x) and

observe where it

crosses the x-axis,

i.e. f(x) = 0

• Not very practical

but can be used to

obtain rough

estimates for roots

• These estimates can

be used as initial

guesses for numerical

methods that we’ll

study here.

Using MATLAB, plot f(x)=sin(10x)+cos(3x)

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Why Called Bracketing Methods?

They require two initial guesses which “bracket”

either side of the root.

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Example1: the parachutist problem

Given m = 68.1 kg, v = 40 m/s, t = 10 s, g = 9.8 m/s2, find the corresponding c

Graphical Methods

40138667

40116889

1 146843010168 )(

.)(

).(.)()(

.)./()/( cctmc ec

ec

vec

gmcf

)()/( tmce

c

gmv 1

Make a plot of the function and observe where it crosses the x axis

Solution:

c f(c)

4 34.115

8 17.653

12 6.067

16 -2.269

20 -8.401

Root is between 12 and 16

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Estimate Properties of Roots By Graphical Methods

From (a) and (c):

if both f(xl) and f(xu) have the same sign, there must be 0 or even number of roots

From (b) and (d):

if f(xl) and f(xu) have different signs, there must be 1 or odd number of roots

Exceptions:

multiple roots

f(x) = (x-2)2(x-4)

discontinuous f(x)

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Find x such that f(x) = 0:

Step 1: Choose lower xl and upper xu such that f(xl)f(xu) < 0

Step 2: an estimate of the root

Step 3: Make the evaluations to determine in which interval the root lies:

(a) if f(xl)f(xr) < 0, the root lies in the lower interval; set xu = xr and

return to Step 2.

(b) if f(xl)f(xr) > 0, the root lies in the upper interval; set xl = xr and

return to Step 2.

(c) if f(xl)f(xr) = 0, the root = xr, stop.

Bisection Method

2

ulr

xxx

8

s

t

rt

tx

xx

%100

||

||

snewr

oldr

newr

ax

xx

%

||

||100

Error Estimates

But, we don’t know xt !

As an alternative, use:

true relative error εt, approximate relative error εa,

and acceptable error εs

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Example 1

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Example 2

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f(x) = sin x – x3

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Conservative Overestimate by a

a is always larger than t !

a

t

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Bisection Method

Pros

• Easy

• Always finds a root

• Number of iterations

required to attain an

absolute error.

Cons

• Slow

• Need to find initial

guesses for xl and xu

• No account is taken

of the fact that if f(xl)

is closer to zero, it is

likely that root is

closer to xl .

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Incremental Search and Determining Initial Guesses

• For every interval xi – xi+1, apply bisection or false-

position method to find the roots in the interval

• Only roots in x4-x5 will be found