lesson 23: newton's method
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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)
. . . . . .
Announcements
◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323◮ MidtermII:4/11inclass(§4.3through§4.8)
. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
. . . . . .
Lasttime: Economics
◮ terms: totalcost, averagecost, marginalcost, revenue,marginalrevenue, profit, marginalprofit
◮ Atthepointofminimalaveragecost, averagecostisequaltomarginalcost
◮ Atthepointofmaximumprofit, marginalrevenueisequaltomarginalcost
. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
. . . . . .
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
. . . . . .
BSRE inaction
Trytofind√2.
Iteration Guess0 1.00000000001 1.5000000002 1.4166666673 1.4142156864 1.4142135625 1.414213562
. . . . . .
BSRE inaction
Trytofind√2.
Iteration Guess0 1.00000000001 1.5000000002 1.4166666673 1.4142156864 1.4142135625 1.414213562
. . . . . .
◮ Numericalmethodsforsolvingequationsareusefulinthe“realworld.”
◮ Newton’smethodisageneralizablemethodfordoingso.
. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
. . . . . .
TheProblem
Givenafunction f, find x∗ suchthat f(x∗) = 0.
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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.
. . . . . .
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.
. . . . . .
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.
. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
. . . . . .
Squareroots
ExampleUseNewton’smethodtoextract
√2.
SolutionThisisthesameastheBSRE!
. . . . . .
Squareroots
ExampleUseNewton’smethodtoextract
√2.
SolutionThisisthesameastheBSRE!
. . . . . .
A cubic
ExampleFindthezeroesof
y = x3 − 3x2 + 2x + 0.3
Use VANDER toexperiment.
. . . . . .
A cubic
ExampleFindthezeroesof
y = x3 − 3x2 + 2x + 0.3
Use VANDER toexperiment.
. . . . . .
Rootsofequations
ExampleUseNewton’smethodtofindanumericalsolutiontotheequation
cos(x) = x
SolutionRewritetheequationsothat cos x− x = 0, andapplyNewton’sMethodtothefunction f(x) = cos x− x.
. . . . . .
Rootsofequations
ExampleUseNewton’smethodtofindanumericalsolutiontotheequation
cos(x) = x
SolutionRewritetheequationsothat cos x− x = 0, andapplyNewton’sMethodtothefunction f(x) = cos x− x.
. . . . . .
Applications
◮ Themethodofbisectioncanfindrootswithconvergencelike 2−n
◮ Newton’smethodcanfindrootswithconvergencelike 2−2n
. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
. . . . . .
(lackof)convergence
ExampleUseNewton’smethodtofindthezeroof
f(x) =
{√x x ≥ 0
−√x x ≤ 0
SinceNf(x) = −x
wejustcyclearoundtheroot.
. . . . . .
(lackof)convergence
ExampleUseNewton’smethodtofindthezeroof
f(x) =
{√x x ≥ 0
−√x x ≤ 0
SinceNf(x) = −x
wejustcyclearoundtheroot.
. . . . . .
UseNewton’smethodtofindthezeroof
Example
f(x) = x1/3
SinceNf(x) = −2x
wedivergefromtheroot!
. . . . . .
UseNewton’smethodtofindthezeroof
Example
f(x) = x1/3
SinceNf(x) = −2x
wedivergefromtheroot!
. . . . . .
ExampleFindthezero(es)of
y = x3 − 3x2 + 2x + 0.4
A localminimumvalueclosetozerowillscrewupconvergence.
. . . . . .
ExampleFindthezero(es)of
y = x3 − 3x2 + 2x + 0.4
A localminimumvalueclosetozerowillscrewupconvergence.
. . . . . .
ExampleExperimentwiththefunction
f(x) = x3 − 3x2 + 2x + 0.3
Weseethatwedon’talwaysconvergetothenearestroot.
. . . . . .
ExampleExperimentwiththefunction
f(x) = x3 − 3x2 + 2x + 0.3
Weseethatwedon’talwaysconvergetothenearestroot.
. . . . . .
Outline
Lasttime
Introduction
Newton’sMethodGraphicallySymbolically
ApplicationsZeroesoffunctionsRootsofequationsConvergence
Flaws(lackof)convergenceconvergencetowhat?
Wow
. . . . . .
◮ Wecanrepeatthismethodwith complex numbers◮ Thebasinsofattractionhavebeautifulstructure.◮ Trythe NewtonBasinGeneration applet
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