Numerical Methods Bracketing method

7/24/2019 Numerical Methods Bracketing method

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FEU Institute of TechnologyElectrical & Electronics Engineering (EEE)

Roots: Bracketing Methods

Ronald M. Pascual, Ph.D.

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Roots Problems in Engineering

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Where:v = downward vertical velocity (m/s),t = time (s),g = the acceleration due to gravity (=9.81 m/s2),cd = a lumped drag coefficient (kg/m),m = the jumpers mass (kg)

Recall the bungee jumper model:

Problem: Given all other variables and parameters,how to solve for mass, m??

Fact: You cannot manipulate this equation to explicitly solve for m!


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Two Major Classes of Root-

Finding Methods

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1.) Bracketing methods

based on two initial guesses that

bracket the root

2.) Open methods can involve one or more initial guesses,

but there is no need for them to bracket the

root


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Incremental Search

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Value of m thatmakes thefunction equal tozero?


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FEU Institute of TechnologyElectrical & Electronics Engineering (EEE) 5

Incremental Search

Illustration of a number of

general ways that a root mayoccur in an interval prescribedby a lower boundxl andan upper boundxu . Parts (a)and (c) indicate that if bothf (xl ) and f (xu) have the

same sign, either there willbe no roots or there will be aneven number of rootswithin the interval. Parts (b)and (d) indicate that if thefunction has different signs at

the end points, there willbe an odd number of roots inthe interval.


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FEU Institute of TechnologyElectrical & Electronics Engineering (EEE) 6

Incremental Search

In general, if f (x) is real and continuous in theinterval fromxl to xu and f (xl ) and f (xu) haveopposite signs, that is,

f (xl ) f (xu) < 0

then there is at least one real root betweenxl andxu.


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Bisection Method

Incremental search method in which the intervalis always divided in half. That is,

xr = (xl + xu) / 2

If a function changes sign over an interval, the

function value at the midpoint is evaluated

The location of the root is then determined as

lying within the subinterval where the sign

change occurs The process is repeated until the root is known

to the required precision.

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Bisection Method

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Problem Use bisection method to solve for mass, m, in the

bungee jumper mathematical model. Assumev=36, t=4, g=9.81 and cd=0.25. Use initial

guesses of xl=50 and xu=200. Tabulate the

values for xl, xu, xr, and the approximate percent

relative error for each iteration. Note also the

signs of f(xl), f(xu) and f(xr). Use a stopping

criterion of s = 0.5%.

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Solution

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FALSE POSITION

(also called the linear interpolationmethod)

locates the root by joining f(xl ) and f(xu)

with a line The intersection of this line with thex-axis

represents an improved estimate of the

root

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FALSE POSITION

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Problem

Use false position method to solve thesame (previous) problem approached with

bisection method.

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Parameters: v=36, t=4, g=9.81, cd=0.25.Initial guesses: xl=50 and xu=200

Stopping criterion: s = 0.5%.


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Solution

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xl xu xr |Ea|%50.0000 200.0000 176.2773


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Solution

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|Ea|% = |(162.3828-176.2773)/162.3828| x 100% = 8.5566%

xl xu xr |Ea|%50.0000 200.0000 176.277350.0000 176.2773 162.3828 8.5566


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Solution

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xl xu xr |Ea|%50.0000 200.0000 176.2773

50.0000 176.2773 162.3828 8.556650.0000 162.3828 154.2446 5.276250.0000 154.2446 149.4777 3.1890

50.0000 149.4777 146.6856 1.903550.0000 146.6856 145.0501 1.1275

50.0000 145.0501 144.0922 0.6648

50.0000 144.0922 143.5311 0.3909


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References

Applied Numerical Methods w/MATLAB

for Engineers & Scientist (3rd Edition).

Chapra. 2012.

Numerical Methods for Engineers (6th

Edition). Chapra and Canale. 2010.

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Numerical Methods Bracketing method

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