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by Lale Yurttas, Texas A&M University Chapter 2 1
Copyright © The McGraw-Hill Companies, Inc. Permission required
for reproduction or display.
Chapter 2
ROOTS OF EQUATION:
BRACKETING METHODS
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Roots of Equations •Why?
•But
by Lale Yurttas, Texas A&M University Chapter 2 2
a
acbb
xcbxax 2
4
0
2
2
?0sin
?02345
x x x
x f exdxcxbxax
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Nonlinear Equation
Solvers
Bracketing Graphical Open Methods
Bisection
False Position
(Regula-Falsi)
Newton Raphson
Secant
by Lale Yurttas, Texas A&M University Chapter 2 3
ALL ITERATIVE
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BRACKETING METHODS (Or, two point methods for finding
roots)
• Two initial guesses for theroot are required. Theseguesses
must “bracket” or beon either side of the root.
== > Fig. 5.1
• If one root of a real andcontinuous function, f(x)=0,is
bounded by values x=xl, x
=xu thenf(xl) . f(xu)
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Part 1:
GRAPHICAL METHODS
by Lale Yurttas, Texas A&M University Chapter 2 5
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GRAPHICAL METHODS
•Simple method – obtain an estimate of root ofequation.
•Used to provide visual insight into thetechnique.
•Make a plot of function & observe the x-axiscross → rough
approximation of the root.
•
Limited practical value – not precise.
by Lale Yurttas, Texas A&M University Chapter 2 6
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• Fig. (a) and (c) → f(xl) & f(xu) have same signs
– noroots / even number of roots within the interval.
• Fig. (b) and (d) → function have different signs
– oddnumber of roots in the interval
by Lale Yurttas, Texas A&M University Chapter 2 7
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f(x) = sin 10x + cos 3x
by Lale Yurttas, Texas A&M University Chapter 2 8
Figure 5.4a
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Example 1
Use the graphical approach to determine the drag coefficient c
neededfor a parachutist of mass m=68.1 kg to have a velocity of 40
m/s afterfree-falling for time t=10 s. (Note: The acceleration due
to gravity is 9.8m/s2)
by Lale Yurttas, Texas A&M University Chapter 2 9
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Part 2:
THE BISECTION METHOD
by Lale Yurttas, Texas A&M University Chapter 2 10
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4. Compare es with ea . If ea
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Evaluation of Method
Pros
• Easy
• Always find root
•
Number of iterations required toattain an absolute error can
becomputed a priori.
Cons
• Slow
• Know a and b that bound root
•
Multiple roots• No account is taken of f(xl) and
f(xu), if f(xl) is closer to zero, it islikely that root is
closer to xl .
by Lale Yurttas, Texas A&M University Chapter 2 14
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Example 2
Use bisection method to solve the same problem
approachedgraphically in Example 1.
by Lale Yurttas, Texas A&M University Chapter 2 17
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THE FALSE-POSITION METHOD
Regula-Falsi)
• Also called linear interpolation method.
• An alternative based on a graphical insight.
• Alternative method to join f(xl) and f(xu) by a straight
line.
•
Intersection line with x-axis improved estimation of
root.• Advantages: Faster & always converges for a single
root.
by Lale Yurttas, Texas A&M University Chapter 2 19
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Procedures
1. Choose xl and xu for the root where functionchanges
sign, check if f(xl).f(xu) < 0.
2. Estimate the value of the root from the
followingformula
and evaluate f(xr).
by Lale Yurttas, Texas A&M University Chapter 2 20
ul
ul l ur
f f
f x f x x
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3. Find the pair• If f(xl). f(xr) < 0, root lies in the lower
interval, then xu= xr and go to step 2.
• If f(xl). f(xr) > 0, root lies in the upper interval, then
xl=xr, go to step 2.• If f(xl). f(xr) = 0, then root is xr and
terminate.
4. Compare es with ea . If ea
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Example 3
Use false-position method to solve the same problem
approachedgraphically in Example 1.
by Lale Yurttas, Texas A&M University Chapter 2 22
