lesson 23: newton's method
DESCRIPTION
Newton's method allows us to find zeros of functions quickly.TRANSCRIPT
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Section4.8Newton’sMethod
Math1aIntroductiontoCalculus
April4, 2008
Announcements
◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323◮ MidtermII:4/11inclass(§4.3through§4.8)
![Page 2: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/2.jpg)
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Announcements
◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323◮ MidtermII:4/11inclass(§4.3through§4.8)
![Page 3: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/3.jpg)
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Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
![Page 4: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/4.jpg)
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Lasttime: Economics
◮ terms: totalcost, averagecost, marginalcost, revenue,marginalrevenue, profit, marginalprofit
◮ Atthepointofminimalaveragecost, averagecostisequaltomarginalcost
◮ Atthepointofmaximumprofit, marginalrevenueisequaltomarginalcost
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. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
![Page 6: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/6.jpg)
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TheBabylonianSquareRootExtractor
Toestimatethesquarerootofa:
◮ Makeaguess x
◮ If x =√a, x =
ax
◮ Otherwise, oneof x andaxisbiggerthan
√a and
oneissmaller◮ averageof x and
axis
closerto√a than x
◮ rinse, lather, repeat
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BSRE inaction
Trytofind√2.
Iteration Guess0 1.00000000001 1.5000000002 1.4166666673 1.4142156864 1.4142135625 1.414213562
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BSRE inaction
Trytofind√2.
Iteration Guess0 1.00000000001 1.5000000002 1.4166666673 1.4142156864 1.4142135625 1.414213562
![Page 9: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/9.jpg)
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◮ Numericalmethodsforsolvingequationsareusefulinthe“realworld.”
◮ Newton’smethodisageneralizablemethodfordoingso.
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Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
![Page 11: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/11.jpg)
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TheProblem
Givenafunction f, find x∗ suchthat f(x∗) = 0.
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Graphicalillustration
◮ Chooseapoint x0 tostart
◮ If f(x0) ̸= 0, Drawthelinetangenttothegraphof y = f(x) at (x0, f(x0))
◮ Thislineintersectsthex-axisanewpoint, callit x1
◮ Rinse, lather, repeat
..x
.y
.x0 .x1.x2
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. . . . . .
Graphicalillustration
◮ Chooseapoint x0 tostart
◮ If f(x0) ̸= 0, Drawthelinetangenttothegraphof y = f(x) at (x0, f(x0))
◮ Thislineintersectsthex-axisanewpoint, callit x1
◮ Rinse, lather, repeat
..x
.y
.x0
.x1.x2
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. . . . . .
Graphicalillustration
◮ Chooseapoint x0 tostart
◮ If f(x0) ̸= 0, Drawthelinetangenttothegraphof y = f(x) at (x0, f(x0))
◮ Thislineintersectsthex-axisanewpoint, callit x1
◮ Rinse, lather, repeat
..x
.y
.x0
.x1.x2
![Page 15: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/15.jpg)
. . . . . .
Graphicalillustration
◮ Chooseapoint x0 tostart
◮ If f(x0) ̸= 0, Drawthelinetangenttothegraphof y = f(x) at (x0, f(x0))
◮ Thislineintersectsthex-axisanewpoint, callit x1
◮ Rinse, lather, repeat
..x
.y
.x0 .x1
.x2
![Page 16: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/16.jpg)
. . . . . .
Graphicalillustration
◮ Chooseapoint x0 tostart
◮ If f(x0) ̸= 0, Drawthelinetangenttothegraphof y = f(x) at (x0, f(x0))
◮ Thislineintersectsthex-axisanewpoint, callit x1
◮ Rinse, lather, repeat
..x
.y
.x0 .x1
.x2
![Page 17: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/17.jpg)
. . . . . .
Graphicalillustration
◮ Chooseapoint x0 tostart
◮ If f(x0) ̸= 0, Drawthelinetangenttothegraphof y = f(x) at (x0, f(x0))
◮ Thislineintersectsthex-axisanewpoint, callit x1
◮ Rinse, lather, repeat
..x
.y
.x0 .x1.x2
![Page 18: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/18.jpg)
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Symbolicexpression
Bydefinition, thelinebetween (xn, f(xn)), (xn+1, 0) istangenttothegraphof f(x) at xn.
Thus
f(xn) − 0xn − xn+1
= f′(xn)
So
xn+1 = xn −f(xn)f′(xn)
Iterating thismethodgivesussuccessive“guesses”forazerotothefunction.
![Page 19: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/19.jpg)
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Symbolicexpression
Bydefinition, thelinebetween (xn, f(xn)), (xn+1, 0) istangenttothegraphof f(x) at xn. Thus
f(xn) − 0xn − xn+1
= f′(xn)
So
xn+1 = xn −f(xn)f′(xn)
Iterating thismethodgivesussuccessive“guesses”forazerotothefunction.
![Page 20: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/20.jpg)
. . . . . .
Symbolicexpression
Bydefinition, thelinebetween (xn, f(xn)), (xn+1, 0) istangenttothegraphof f(x) at xn. Thus
f(xn) − 0xn − xn+1
= f′(xn)
So
xn+1 = xn −f(xn)f′(xn)
Iterating thismethodgivesussuccessive“guesses”forazerotothefunction.
![Page 21: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/21.jpg)
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Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
![Page 22: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/22.jpg)
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Squareroots
ExampleUseNewton’smethodtoextract
√2.
SolutionThisisthesameastheBSRE!
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Squareroots
ExampleUseNewton’smethodtoextract
√2.
SolutionThisisthesameastheBSRE!
![Page 24: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/24.jpg)
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A cubic
ExampleFindthezeroesof
y = x3 − 3x2 + 2x + 0.3
Use VANDER toexperiment.
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A cubic
ExampleFindthezeroesof
y = x3 − 3x2 + 2x + 0.3
Use VANDER toexperiment.
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Rootsofequations
ExampleUseNewton’smethodtofindanumericalsolutiontotheequation
cos(x) = x
SolutionRewritetheequationsothat cos x− x = 0, andapplyNewton’sMethodtothefunction f(x) = cos x− x.
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Rootsofequations
ExampleUseNewton’smethodtofindanumericalsolutiontotheequation
cos(x) = x
SolutionRewritetheequationsothat cos x− x = 0, andapplyNewton’sMethodtothefunction f(x) = cos x− x.
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Applications
◮ Themethodofbisectioncanfindrootswithconvergencelike 2−n
◮ Newton’smethodcanfindrootswithconvergencelike 2−2n
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Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
![Page 30: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/30.jpg)
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(lackof)convergence
ExampleUseNewton’smethodtofindthezeroof
f(x) =
{√x x ≥ 0
−√x x ≤ 0
SinceNf(x) = −x
wejustcyclearoundtheroot.
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(lackof)convergence
ExampleUseNewton’smethodtofindthezeroof
f(x) =
{√x x ≥ 0
−√x x ≤ 0
SinceNf(x) = −x
wejustcyclearoundtheroot.
![Page 32: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/32.jpg)
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UseNewton’smethodtofindthezeroof
Example
f(x) = x1/3
SinceNf(x) = −2x
wedivergefromtheroot!
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UseNewton’smethodtofindthezeroof
Example
f(x) = x1/3
SinceNf(x) = −2x
wedivergefromtheroot!
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ExampleFindthezero(es)of
y = x3 − 3x2 + 2x + 0.4
A localminimumvalueclosetozerowillscrewupconvergence.
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ExampleFindthezero(es)of
y = x3 − 3x2 + 2x + 0.4
A localminimumvalueclosetozerowillscrewupconvergence.
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ExampleExperimentwiththefunction
f(x) = x3 − 3x2 + 2x + 0.3
Weseethatwedon’talwaysconvergetothenearestroot.
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ExampleExperimentwiththefunction
f(x) = x3 − 3x2 + 2x + 0.3
Weseethatwedon’talwaysconvergetothenearestroot.
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Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
![Page 39: Lesson 23: Newton's Method](https://reader034.vdocuments.us/reader034/viewer/2022052410/555dc4f8d8b42ab56b8b47f7/html5/thumbnails/39.jpg)
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◮ Wecanrepeatthismethodwith complex numbers◮ Thebasinsofattractionhavebeautifulstructure.◮ Trythe NewtonBasinGeneration applet