Case Study #1Case Study #1Finding Roots of EquationsFinding Roots of Equations

~~CE402 Numerical Methods for CE402 Numerical Methods for

EngineersEngineers

Dr. Fritz FiedlerDr. Fritz Fiedler

~~

Andy AbramsAndy Abrams

David CrosbyDavid Crosby

Zack MunstermannZack Munstermann

IntroductionIntroduction

Why find roots of equations?Why find roots of equations? Equations are used to model physical Equations are used to model physical

systemssystems Knowing the roots of these equations Knowing the roots of these equations

helps us understand the physical system helps us understand the physical system How do we find the roots?How do we find the roots?

AnalyticallyAnalytically NumericallyNumerically

GraphingGraphing Graphing the equation over a useful Graphing the equation over a useful

range can identify several points of range can identify several points of interestinterest Rough estimate of the rootsRough estimate of the roots ContinuityContinuity Local minima and maximaLocal minima and maxima

Graphing is simple, yet roughGraphing is simple, yet rough It is often used to define functions for It is often used to define functions for

initial guesses in other methods.initial guesses in other methods.

Bisection MethodBisection Method The bisection method is a “bracketing The bisection method is a “bracketing

method”method” Initial guesses must surround the root.Initial guesses must surround the root. Function must be continuous near the rootFunction must be continuous near the root

Bracketing values Xu and Xl are chosenBracketing values Xu and Xl are chosen New root estimated by Xr = (Xl + Xu)/2New root estimated by Xr = (Xl + Xu)/2

If f(Xr)f(Xl) < 0 then Xr = Xu…continue…If f(Xr)f(Xl) < 0 then Xr = Xu…continue… If f(Xr)f(Xl) > 0 then Xr = Xl…continue…If f(Xr)f(Xl) > 0 then Xr = Xl…continue…

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Bisection Method: continuedBisection Method: continued

Bracketing values Xu and Xl are chosenBracketing values Xu and Xl are chosen New root estimated by Xr = (Xl + Xu)/2New root estimated by Xr = (Xl + Xu)/2

If f(Xr)f(Xl) < 0 then Xr = Xu…continue…If f(Xr)f(Xl) < 0 then Xr = Xu…continue… If f(Xr)f(Xl) > 0 then Xr = Xl…continue…If f(Xr)f(Xl) > 0 then Xr = Xl…continue… If f(Xr)f(Xl) = 0 then Xr = root…stop.If f(Xr)f(Xl) = 0 then Xr = root…stop.

Limitations of the bisection methodLimitations of the bisection method Relatively inefficientRelatively inefficient Must know equation wellMust know equation well

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Newton Raphson MethodNewton Raphson Method The Newton-Raphson method is an “open The Newton-Raphson method is an “open

method”method” Need only one initial guessNeed only one initial guess Must know derivative of functionMust know derivative of function

Initial guess of root Xi is chosenInitial guess of root Xi is chosen New root estimated by Xr = Xi – New root estimated by Xr = Xi –

f(Xi)/f’(Xi)f(Xi)/f’(Xi) If error of Xr is acceptable, stop.If error of Xr is acceptable, stop. If error is too large, Xi = Xr…continue…If error is too large, Xi = Xr…continue…

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Newton-Raphson Method: continuedNewton-Raphson Method: continued

Constraints on N.R.Constraints on N.R. Good initial guess (convergence)Good initial guess (convergence) Derivative must be easily evaluatedDerivative must be easily evaluated Derivative must be continuous and non-Derivative must be continuous and non-

zero near rootzero near root Problems from inflection points and local Problems from inflection points and local

extremes near rootextremes near root

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ExampleExample To compare these methods we will consider To compare these methods we will consider

water flowing in a rectangular open channelwater flowing in a rectangular open channel Flow governed by Manning’s equationFlow governed by Manning’s equation

Where: Where: QQ = volumetric flow rate (m3/s) = volumetric flow rate (m3/s) BB = channel width (m) = channel width (m)

H H = height of water in channel (m) = height of water in channel (m) nn = Manning’s coefficient of roughness = Manning’s coefficient of roughness RR = hydraulic radius (m) = hydraulic radius (m) SS = channel slope = channel slope

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BHQ 2/13/2 SR

n

BHQ

Example: continuedExample: continued In the design of this channel, we wish In the design of this channel, we wish

to find depth of flow and flow velocity to find depth of flow and flow velocity for a range of channel widthsfor a range of channel widths

V = Q/(BH)V = Q/(BH)

First we graphed the equation for a given First we graphed the equation for a given set of parametersset of parameters

Used bisection and N.R. to solve for depth Used bisection and N.R. to solve for depth of flow for a range of channel widthsof flow for a range of channel widths

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Example: continuedExample: continued

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Manning's Equation

-30

-25

-20

-15

-10

-5

0

5

10

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

depth of Flow (m)

f(H

)Graphing:Graphing:

S = 0.0002S = 0.0002B = 20 mB = 20 mn = 0.03n = 0.03Q = 100 m3/sQ = 100 m3/s

Range:Range:4< H < 54< H < 5

Root in this case (by inspection) is approximately 4.82 mRoot in this case (by inspection) is approximately 4.82 m

Example: continuedExample: continued

Bisection method:Bisection method:

S = 0.0002S = 0.0002n = 0.03n = 0.03Q = 100 m3/sQ = 100 m3/s

Range:Range:5m < B < 35m5m < B < 35m

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n

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Channel Width vs. Depth

0

5

10

15

20

25

30

0 10 20 30 40

Width (m)D

ep

th (

m)

Channel Width vs. Velocity

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

Width (m)

Ve

locit

y (

m/s

)

Example: continuedExample: continued

Newton-Raphson:Newton-Raphson:

S = 0.0002S = 0.0002n = 0.03n = 0.03Q = 100 m3/sQ = 100 m3/s

Range:Range:5m < B < 35m5m < B < 35m

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Channel Width vs. Depth

0

5

10

15

20

25

30

0 10 20 30 40

Width (m)D

ep

th (

m)

Channel Width vs. Velocity

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40

Width (m)

Ve

locit

y (

m/s

)

ConclusionsConclusions

Results of the two methods being Results of the two methods being relatively the same shows that they relatively the same shows that they are convergent on the real solutionare convergent on the real solution

Differences between the twoDifferences between the two N.R. much more efficient (4:1 iterations)N.R. much more efficient (4:1 iterations) N.R. more accurate in this caseN.R. more accurate in this case N.R. easier to programN.R. easier to program

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Questions?Questions?

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