Roots of Equations: Bracketing Methods

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How to Solve?

v(t)' gmcd

tanhgcd

mt

t'4v'36g'9.81cd'0.25m'?

v(t)' gmcd

tanhgcd

mt

t'4v'36g'9.81cd'0.25m'?

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Techniques for Finding Roots

• Graphical Methods• Bracketing Methods• Open Methods

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Graphical Methods

• The best.• The worst.

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Bracketing Methods

• Find roots between a lower bound xl and an upper bound xu such that f(xl)*f(xu)<0.

• Initial guess needed (incremental search).• Always work if roots exist between the two

limits.• Converge slower than open methods.

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Conditions

a) f(xl)*f(xu)>0, no root.b) f(xl)*f(xu)<0, one root.c) f(xl)*f(xu)>0, even number

of roots.d) f(xl)*f(xu)<0, odd number

of roots.

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Conditions (cont.)

a) f(xl)*f(xu)<0, repeated roots.

b) f(xl)*f(xu)<0, no root.

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Incremental Search

• Goal: find xl and xu such that f(xl)*f(xu)<0.• Range of search must be decided. • Large step size may miss some roots.• Small step size increases computation time.

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Example• Cases where roots could be missed because the incremental length

of the search procedure is too large.• Note that the last root on the right is multiple and would be missed

regardless of the increment length.

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Example 5.2

• sin(10x)+cos(3x)• 50 steps:

3.2449, 3.30613.3061, 3.36733.7347, 3.79594.6531, 4.71435.6327, 5.6939

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Example 5.2 (cont.)• sin(10x)+cos(3x)• 100 steps:

3.2424, 3.27273.3636, 3.39393.7273, 3.75764.2121, 4.24244.2424, 4.27274.6970, 4.72735.1515, 5.18185.1818, 5.21215.6667, 5.6970

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Bisection

• Iterative.• Every iteration reduces the bound by half.• Algorithm: repeat the following

xr = ( xl + xu )/2if f(xr)*f(xu)<0

xl = xrelse

xu = xrend

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Example 5.3

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Converge Criteria

• Use relative approximate errora=| ( xr

new - xrold ) / xr

new ) | 100%.• Not suitable when the root is close to 0.• In that case, use approximate error.

a=| xrnew - xr

old |• Number of iteration can be determined if a is

specified.n'1%log2

Δx 0

Ea,d

Δx 0: initial boundEa,d: desired Ea

n'1%log2Δx 0

Ea,d

Δx 0: initial boundEa,d: desired Ea

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False Position

• Iterative.• Similar to bisection but determine the next bound by line

equation.• Algorithm: repeat the following

xr'xu&f(xu)(xl&xu)f(xl)&f(xu)

if f(xr)f(xu)<0xl'xr

elsexu'xr

end

xr'xu&f(xu)(xl&xu)f(xl)&f(xu)

if f(xr)f(xu)<0xl'xr

elsexu'xr

end

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False Position (cont.)

• Usually converge faster than bisection but not always.

• Example: f(x)=x10-1