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CPE310:NumericalAnalysisforEngineersChapter1:SolvingNonlinearEquations
AhmedTamrawi
Copyrightnotice:carehasbeentakentouseonlythosewebimagesdeemedbytheinstructortobeinthepublicdomain.Ifyouseeacopyrightedimageonanyslideandarethecopyrightowner,pleasecontacttheinstructor.Itwillberemoved.
Animportantprobleminappliedmathematicsisto:"solveđ đ„ = 0"wheređ đ„ isafunctionof đ„
Thevaluesofđ„ thatmakeđ đ„ = 0arecalledtherootsoftheequation orzerosofđ
Theproblemisknownasrootfinding orzerofinding
Weareconcernedaboutsolvingsinglenonlinearequation inoneunknown,wheređ: â â â
Solutionisscalar đ„ forwhichđ đ„ = 0
â1-DNonlinearEquationâ
Example:đ đ„ = 3đ„ + sin đ„ â đ1 forwhichđ„ = 0.3604 isoneapproximatesolution
Theselecturesdescribesomeofthemanymethodsforsolving đ đ„ = 0 bynumericalprocedures wheređ đ„ is singlenonlinearequation inoneunknown
IntervalHalving(Bisection) SecantMethod
False-PositionMethod
NewtonâsMethod Fixed-PointIterationMethod
LinearInterpolationMethods
IntervalHalving(Bisection)Method
IntervalHalving(Bisection)Method
Ancientbuteffectivemethodforfindingazeroofđ(đ„)
Itbeginswithtwovalues forđ„ thatbracket aroot
Itdeterminesthattheydobracketarootbecauseđ(đ„) changessigns at
thesetwox-values
ifđ(đ„) iscontinuous,theremustbe atleastonerootbetweenthevalues
Aplotofđ(đ„) isusefultoknowwheretostart
REPEATSET đ„7 =
(1891:);
IF đ đ„7 đ đ„< < 0 THENSET đ„; = đ„7
ELSESET đ„< = đ„7
UNTIL (|đ„< â đ„;| <tol) OR đ đ„7 = 0
âą đ„< andđ„; suchthatđ đ„< đ đ„; < 0âą tol:thespecifiedtolerancevalueINPUT
IntervalHalving(Bisection)Algorithm
âą Thefinalvalueofđ„7 approximatestheroot,anditisinerrorbynotmorethan|18?1:|;
âą Thealgorithmmayproduceafalseroot ifđ(đ„) isdiscontinuouson[đ„<, đ„;]
NOTESâMaximumErrorâ
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„< = 0, đ„; = 1 suchthatđ 0 đ 1 < 0tol = 1đž â 3 OR (0.001)
đ„< đ„;
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ)MaxError
ActualError
AbsoluteError
1 0.00000 1.00000 -1.00000 1.12320
|đ„< â đ„;|2 đ„CDEFGH â đ„7
|đ„CDEFGH â đ„7|
REPEAT
SET !" = (&'(&))+IF , !" , !- < 0 THEN
SET !+ = !"ELSE
SET !- = !"UNTIL (|!- â !+| <tol) OR , !" = 0
âą !- and!+ suchthat, !- , !+ < 0âą tol:thespecified tolerancevalueINPUT
IntervalHalving(Bisection)Algorithm
Actualđ„ = .36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„< = 0, đ„; = 1 suchthatđ 0 đ 1 < 0tol = 1đž â 3 OR (0.001)
đ„< đ„;
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ)MaxError
ActualError
AbsoluteError
1 0.00000 1.00000 0.50000 -1.00000 1.12320 0.33070 0.50000 -0.13958 0.13958
2 0.00000 0.50000 -1.00000 0.33070
REPEAT
SET !" = (&'(&))+IF , !" , !- < 0 THEN
SET !+ = !"ELSE
SET !- = !"UNTIL (|!- â !+| <tol) OR , !" = 0
âą !- and!+ suchthat, !- , !+ < 0âą tol:thespecified tolerancevalueINPUT
IntervalHalving(Bisection)Algorithm
Actualđ„ = .36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„< = 0, đ„; = 1 suchthatđ 0 đ 1 < 0tol = 1đž â 3 OR (0.001)
đ„< đ„;
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ)MaxError
ActualError
AbsoluteError
1 0.00000 1.00000 0.50000 -1.00000 1.12320 0.33070 0.50000 -0.13958 0.13958
2 0.00000 0.50000 0.25000 -1.00000 0.33070 -0.28662 0.25000 0.11042 0.11042
3 0.25000 0.50000 -0.28662 0.33070
REPEAT
SET !" = (&'(&))+IF , !" , !- < 0 THEN
SET !+ = !"ELSE
SET !- = !"UNTIL (|!- â !+| <tol) OR , !" = 0
âą !- and!+ suchthat, !- , !+ < 0âą tol:thespecified tolerancevalueINPUT
IntervalHalving(Bisection)Algorithm
Actualđ„ = .36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„< = 0, đ„; = 1 suchthatđ 0 đ 1 < 0tol = 1đž â 3 OR (0.001)
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ)MaxError
ActualError
AbsoluteError
1 0.00000 1.00000 0.50000 -1.00000 1.12320 0.33070 0.50000 -0.13958 0.13958
2 0.00000 0.50000 0.25000 -1.00000 0.33070 -0.28662 0.25000 0.11042 0.11042
3 0.25000 0.50000 0.37500 -0.28662 0.33070 0.03628 0.12500 -0.01458 0.01458
4 0.25000 0.37500 0.31250 -0.28662 0.03628 -0.12190 0.06250 0.04792 0.04792
5 0.31250 0.37500 0.34375 -0.12190 0.03628 -0.04196 0.03125 0.01667 0.01667
6 0.34375 0.37500 0.35938 -0.04196 0.03628 -0.00262 0.01563 0.00105 0.00105
7 0.35938 0.37500 0.36719 -0.00262 0.03628 0.01689 0.00781 -0.00677 0.00677
8 0.35938 0.36719 0.36328 -0.00262 0.01689 0.00715 0.00391 -0.00286 0.00286
9 0.35938 0.36328 0.36133 -0.00262 0.00715 0.00227 0.00195 -0.00092 0.00092
10 0.35938 0.36133 0.36035 -0.00262 0.00227 -0.00018 0.00098 0.00007 0.00007
11 0.36035 0.36133 0.36084 -0.00018 0.00227 0.00105 0.00049 -0.00042 0.00042
0.36035 â 0.36133 < 0.001 âč 0.00098 < 0.001 ToleranceMet
đ„< đ„;
REPEAT
SET !" = (&'(&))+IF , !" , !- < 0 THEN
SET !+ = !"ELSE
SET !- = !"UNTIL (|!- â !+| <tol) OR , !" = 0
âą !- and!+ suchthat, !- , !+ < 0âą tol:thespecified tolerancevalueINPUT
IntervalHalving(Bisection)Algorithm
Needediterationstoachieveaspecifiedaccuracyisknowninadvance
Errorafterđ iterations< | 1:?18;\
|
Simpleandguaranteed toworkif:1. đ(đ„) iscontinuous in[đ„<, đ„;]; and2. Thevaluesđ„< andđ„; actuallybracketaroot
Plottingthefunctionhelpsdefiningthebracket
Goodforinitialguessforotherrootfindingalgorithms
REPEAT
SET !" = (&'(&))+IF , !" , !- < 0 THEN
SET !+ = !"ELSE
SET !- = !"UNTIL (|!- â !+| <tol) OR , !" = 0
âą !- and!+ suchthat, !- , !+ < 0âą tol:thespecified tolerancevalueINPUT
IntervalHalving(Bisection)Algorithm
SlowConvergence comparedtoothertechniqueswewillseenext
Othermethodsrequirelessnumberofiterations
Iter !" !# !$ %(!") %(!#) %(!$) MaxError
ActualError
AbsoluteError
1 0.00000 1.00000 0.50000 -1.00000 1.12320 0.33070 0.50000 -0.13958 0.13958
2 0.00000 0.50000 0.25000 -1.00000 0.33070 -0.28662 0.25000 0.11042 0.11042
3 0.25000 0.50000 0.37500 -0.28662 0.33070 0.03628 0.12500 -0.01458 0.01458
4 0.25000 0.37500 0.31250 -0.28662 0.03628 -0.12190 0.06250 0.04792 0.04792
5 0.31250 0.37500 0.34375 -0.12190 0.03628 -0.04196 0.03125 0.01667 0.01667
6 0.34375 0.37500 0.35938 -0.04196 0.03628 -0.00262 0.01563 0.00105 0.00105
7 0.35938 0.37500 0.36719 -0.00262 0.03628 0.01689 0.00781 -0.00677 0.00677
8 0.35938 0.36719 0.36328 -0.00262 0.01689 0.00715 0.00391 -0.00286 0.00286
9 0.35938 0.36328 0.36133 -0.00262 0.00715 0.00227 0.00195 -0.00092 0.00092
10 0.35938 0.36133 0.36035 -0.00262 0.00227 -0.00018 0.00098 0.00007 0.00007
11 0.36035 0.36133 0.36084 -0.00018 0.00227 0.00105 0.00049 -0.00042 0.00042
Someearlieriterationmayresultonacloserapproximationthanlaterones
đ„<đ„;
đ„7đ„âČ7
đ„CDEFGH
đ„âČ7 isclosertođ„CDEFGH thanđ„7
CloserFurther
CloserFurther
Whentherearemultipleroots,intervalhalvingmaynotbeapplicable,becausethefunctionmaynotchangesign atpointsoneithersideoftheroots.
WemaybeabletofindtherootsbyworkingwithđâČ(đ„), whichwillbezeroatamultipleroot.
đ đ„ = đ„; â 2đ„ + 1 = 0,inbracket[0, 2]Example:
http://mathfaculty.fullerton.edu/mathews/a2001/Animations/RootFinding/BisectionMethod/BisectionMethod.html
BisectionMethod- Animation
LinearInterpolationMethodsMostfunctionscanbeapproximatedbyastraightlineoverasmallinterval
SecantMethod
SecantMethod
In geometry,a secant ofa curve isalinethatintersectstwo points onthecurve
Itbeginswithtwovalues forđ„ thatarenear the root
Thetwovaluesmaybeonthesamesidefromtherootoronoppositeside
ifđ(đ„) islinear,thesecantintersectsthex-axisattherootexactly
Aplotofđ(đ„) isusefultoknowwheretostart
đ đ„ = 3đ1 â 4 cos đ„ = 0
đ„CDEFGH
đ đ„
đ„_đ„<đ„; đ„CDEFGH
đ đ„<
đ đ„_
đ„_đ„<đ„; đ„CDEFGH
đ đ„<
đ đ„_
đ đ„_ â đ(đ„<)
đ„_ â đ„<
đ„_ â đ„;
đ„_đ„<đ„; đ„CDEFGH
đ đ„<
đ đ„_
đ đ„_ â đ(đ„<)
đ„_ â đ„<
đ„_ â đ„;
(đ„_ â đ„;)đ(đ„_)
=(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
đ„; = đ„_ â đ đ„_(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
(đ„_ â đ„;)đ(đ„_)
=(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
đ„_đ„<đ„; đ„CDEFGH
đ đ„<
đ đ„_
đ đ„_ â đ(đ„<)
đ„_ â đ„<
đ„_ â đ„;
IF đ đ„_ < đ đ„< THENSWAP đ„_withđ„<
REPEATSET đ„; = đ„_ â đ(đ„_)
1b?18c 1b ?c 18
SET đ„_ = đ„<SET đ„< = đ„;
UNTIL (|đ(đ„;)| <tol)
âą đ„_ andđ„< thatarenear therootâą tol:thespecifiedtolerancevalueINPUT
SecantMethod
âą Anotherstoppingâterminationâcriteriaiswhenthepairofpointsbeingusedaresufficientlyclosetogether:âUNTIL (|đ„_ â đ„<| <tol)â
âą Thealgorithmmayfail ifđ(đ„) isnotcontinuous.
NOTES
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 7 OR (0.0000001)
đ„< đ„_
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 1.000000 0.000000 1.12320 -1.00000
đ„;
IF ! "# < ! "% THENSWAP "#with"%
REPEATSET "+ = "# â !("#) 01203
4 01 24 03SET "# = "%SET "% = "+
UNTIL (|!("+)| <tol)
âą "# and"% thatarenear therootâą tol:thespecified tolerancevalueINPUT
SecantMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 7 OR (0.0000001)
đ„_ đ„<
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 1.00000 0.00000 0.4709896 1.12320 -1.000000 0.2651588 -0.110567
2 0.00000 0.4709896 -1.000000 0.2651588
đ„;
IF ! "# < ! "% THENSWAP "#with"%
REPEATSET "+ = "# â !("#) 01203
4 01 24 03SET "# = "%SET "% = "+
UNTIL (|!("+)| <tol)
âą "# and"% thatarenear therootâą tol:thespecified tolerancevalueINPUT
SecantMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 7 OR (0.0000001)
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 1.00000 0.00000 0.4709896 1.12320 -1.000000 0.2651588 -0.110567
2 0.00000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.4709896 0.3722771 0.2651588 0.0295336
đ„< đ„_
đ„;
IF ! "# < ! "% THENSWAP "#with"%
REPEATSET "+ = "# â !("#) 01203
4 01 24 03SET "# = "%SET "% = "+
UNTIL (|!("+)| <tol)
âą "# and"% thatarenear therootâą tol:thespecified tolerancevalueINPUT
SecantMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 7 OR (0.0000001)đ„;
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 1.00000 0.00000 0.4709896 1.12320 -1.000000 0.2651588 -0.110567
2 0.00000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.4709896 0.3722771 0.3599043 0.2651588 0.0295336 -0.001294 0.000517
4 0.3722771 0.3599043 0.3604239 0.0295336 -0.001294 0.0000552 -0.000002
5 0.3599043 0.3604239 0.3604217 -0.001294 0.0000552 0.00000003 0.000000002
đ„_
đ„<
IF ! "# < ! "% THENSWAP "#with"%
REPEATSET "+ = "# â !("#) 01203
4 01 24 03SET "# = "%SET "% = "+
UNTIL (|!("+)| <tol)
âą "# and"% thatarenear therootâą tol:thespecified tolerancevalueINPUT
SecantMethod
ToleranceMetđ đ„; < 1E â 7
Ifthefunctionisfarfromlinearneartheroot,thesuccessiveiteratescanflyoffto
pointsfarfromtheroot
đ„_ đ„< đ„;đ„fggE
Thebadchoiceofđ„_ andđ„< orduplicatingapreviousresult,mayresultonanendlessloop,
thus,neverreachingananswer
đ đ„ = đ„đ?1 andđ„_ = 2, đ„< = 3
http://mathfaculty.fullerton.edu/mathews/a2001/Animations/RootFinding/SecantMethod/SecantMethod.html
SecantMethod- Animation
LinearInterpolationMethodsMostfunctionscanbeapproximatedbyastraightlineoverasmallinterval
False-PositionMethodâRegulaFalsiâMethod
False-PositionMethod
ProblemofSecantMethodIfthefunctionisfarfromlinearneartheroot,thesuccessiveiteratescanflyoff
topointsfarfromtheroot
False-Positionbeginswithtwovaluesforđ„ thatbracket aroot
Ensurethattherootisbracketedbetweenthetwostartingvaluesandremainsbetweenthesuccessivepairs.
đ đ„ = đ„7 + 4đ„; â 10 = 0
Similartobisectionexceptthenextiterateistakenattheintersectionofalinebetweenthepairofx-valuesandthex-axisratherthanatthemidpoint
đ„_đ„<
đ„hijklmino
đ„CDEFGH
đ„pqrjk?snjimino
FasterthanBisectionbutmorecomplicated
đ„; = đ„_ â đ đ„_(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
(đ„_ â đ„;)đ(đ„_)
=(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
REPEATSET đ„; = đ„_ â đ(đ„_)
1b?18c 1b ?c 18
IF đ(đ„;) isofoppositesigntođ(đ„_) THEN SET đ„< = đ„;
ELSESET đ„_ = đ„<
UNTIL (|đ(đ„;)| <tol)
âą đ„_ andđ„< thatbracketaroot suchthatđ(đ„_)andđ(đ„<) areofoppositesign
âą tol:thespecifiedtolerancevalueINPUT
False-Position(regula falsi)Method
âą Anotherstoppingâterminationâcriteriaiswhenthepairofpointsbeingusedaresufficientlyclosetogether:âUNTIL (|đ„_ â đ„<| <tol)â
âą Thealgorithmmayfail ifđ(đ„) isnotcontinuous.
NOTES
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0
đ„_ đ„<
đ„;Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 -1.000000 1.12320
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
tol = 1đž â 4 OR (0.0001)
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0
đ„_ đ„<
đ„;
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 0.4709896 -1.000000 1.12320 0.2651588 -0.110567
2 0.000000 0.4709896 -1.000000 0.2651588
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
tol = 1đž â 4 OR (0.0001)
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 0.4709896 -1.000000 1.12320 0.2651588 -0.110567
2 0.000000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.000000 0.3722771 -1.000000 0.0295336
đ„_ đ„<
đ„;
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
tol = 1đž â 4 OR (0.0001)
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 0.4709896 -1.000000 1.12320 0.2651588 -0.110567
2 0.000000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.000000 0.3722771 0.361598 -1.000000 0.0295336 0.002941 -0.0011760
đ„;
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
tol = 1đž â 4 OR (0.0001)
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 4 OR (0.0001)
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 0.4709896 -1.000000 1.12320 0.2651588 -0.110567
2 0.000000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.000000 0.3722771 0.36159774 -1.000000 0.0295336 0.002941 -0.0011760
4 0.000000 0.36159774 0.36053740 -1.000000 0.002941 0.00028944 -0.0001157
5 0.000000 0.36053740 0.36043307 -1.000000 0.00028944 0.00002845 0.00001373
đ„;
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 4 OR (0.0001)
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 0.4709896 -1.000000 1.12320 0.2651588 -0.110567
2 0.000000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.000000 0.3722771 0.36159774 -1.000000 0.0295336 0.002941 -0.0011760
4 0.000000 0.36159774 0.36053740 -1.000000 0.002941 0.00028944 -0.0001157
5 0.000000 0.36053740 0.36043307 -1.000000 0.00028944 0.00002845 0.00001373
đ„;
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
ToleranceMetđ đ„; < 1E â 4
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
đ„_ = 1đ„< = 0tol = 1đž â 5 OR (0.00001)
Iter đđ đđ đđ đ(đđ) đ(đđ) đ(đđ) ActualError
1 0.000000 1.000000 0.4709896 -1.000000 1.12320 0.2651588 -0.110567
2 0.000000 0.4709896 0.3722771 -1.000000 0.2651588 0.0295336 -0.011849
3 0.000000 0.3722771 0.36159774 -1.000000 0.0295336 0.002941 -0.0011760
4 0.000000 0.36159774 0.36053740 -1.000000 0.002941 0.00028944 -0.0001157
5 0.000000 0.36053740 0.36043307 -1.000000 0.00028944 0.00002845 0.00001373
đ„;
FalsePositionconvergestotherootfromonlyoneside,slowingitdown,especiallyifthatendoftheintervalisfartherfromtheroot.Thereisawaytoavoidthisresult,calledmodifiedlinearinterpolation
REPEATSET !" = !$ â &(!$) *+,*-
. *+ ,. *-IF &(!") isofoppositesignto&(!$) THEN
SET !/ = !"ELSE
SET !$ = !/UNTIL (|&(!")| <tol)
âą !$ and!/ thatbracketaroot suchthat&(!$)and&(!/)areofopposite sign
âą tol:thespecified tolerancevalueINPUT
False-Position(regula falsi)Method
http://mathfaculty.fullerton.edu/mathews/a2001/Animations/RootFinding/RegulaFalsi/RegulaFalsi.html
False-PositionMethod- Animation
NewtonâsMethodLinearapproximationofthefunctionusingtangentlineoversmallinterval
NewtonâsMethod
Oneofthemostwidelyusedmethodsofsolvingequations
Basedonalinearapproximationofthefunctionusingatangent tothecurve
Startingfromasingleinitialestimate,đ„_ thatisnottoofarfromaroot
đ đ„ = 3đ1 â 4 cos đ„
Movealongthetangenttoitsintersectionwiththex-axis,andtake
thatasthenextapproximation
Continueuntileitherthesuccessivex-valuesaresufficientlycloseorthevalueofthefunctionissufficientlynearzero
đ„_đ„CDEFGH
đ„<
đ
đ„< = đ„_ âđ(đ„_)đâČ đ„_
tan Ξ = đv đ„_ =đ(đ„_)đ„_ â đ„<
đ„_đ„CDEFGH
đ„<
đ
SlopeofTangent atđ đ„_ isequaltothederivativeofcurveatđ„_
đ„o9< = đ„o âđ(đ„o)đâČ đ„o
COMPUTE đ đ„_ , đâČ(đ„_)SET đ„< = đ„_IF đ đ„_ â 0 AND đâČ đ„_ â 0 THEN
REPEATSET đ„_ = đ„<SET đ„< = đ„_ â
c(1b)cv(1b)
UNTIL ( đ„< â đ„_ <tol1)OR(|đ(đ„<)| <tol2)
âą đ„_ reasonablyclose totherootâą tol1:thespecifiedtolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecifiedtolerancevalueforhowcloseđ(đ„<) tozero.
INPUT
NewtonâsMethod
âą Themethodmayconvergetoarootdifferentfromtheexpectedoneordivergeifthestartingvalueisnotcloseenoughtotheroot.
NOTES
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)
đ„_
tol2 = 1đž â 5 OR (0.00001)đ„<
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)
đ„_
tol2 = 1đž â 5 OR (0.00001)
đ đ„ = 3đ„ + sin đ„ â đ1 đâČ đ„ = 3 + cos đ„ â đ1
đ„<
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)tol2 = 1đž â 5 OR (0.00001)
đ„_ = 0.0, đ„< = 0.0
đ đ„ = 3đ„ + sin đ„ â đ1 đâČ đ„ = 3 + cos đ„ â đ1
đ(đ„_) = â1.0, đâČ(đ„_) = 3.0
đ„_
đ„<
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)tol2 = 1đž â 5 OR (0.00001)
đ„_ = 0.0, đ„< = 0.0đ đ„ = 3đ„ + sin đ„ â đ1
đâČ đ„ = 3 + cos đ„ â đ1 đ(đ„_) = â1.0, đâČ(đ„_) = 3.0
Iteration1:đ„_ = 0.0, đ„< = đ„_ âc 1bcx 1b
= 0.0 â ?<._7._
= 0.33333
0.33333 â 0.0 < 0.00001đđ â0.068418 < 0.00001
đ„_
đ„<
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)tol2 = 1đž â 5 OR (0.00001)
đ„_ = 0.0, đ„< = 0.0đ đ„ = 3đ„ + sin đ„ â đ1
đâČ đ„ = 3 + cos đ„ â đ1 đ(đ„_) = â1.0, đâČ(đ„_) = 3.0
Iteration1:đ„_ = 0.0, đ„< = đ„_ âc 1bcx 1b
= 0.0 â ?<._7._
= 0.33333
Iteration2:đ„< = 0.33333, đ„; = đ„< âc 18cx 18
= 0.33333 â ?_._{|}<|;.~}ïżœ7}
= 0.36017
0.36017 â 0.33333 < 0.00001đđ â0.0006279 < 0.00001
đ„; đ„<
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)tol2 = 1đž â 5 OR (0.00001)
đ„_ = 0.0, đ„< = 0.0đ đ„ = 3đ„ + sin đ„ â đ1
đâČ đ„ = 3 + cos đ„ â đ1 đ(đ„_) = â1.0, đâČ(đ„_) = 3.0
Iteration1:đ„_ = 0.0, đ„< = đ„_ âc 1bcx 1b
= 0.0 â ?<._7._
= 0.33333
Iteration2:đ„< = 0.33333, đ„; = đ„< âc 18cx 18
= 0.33333 â ?_._{|}<|;.~}ïżœ7}
= 0.36017
Iteration3:đ„; = 0.36017, đ„7 = đ„; âc 1:cx 1:
= 0.36017 â ?_.___{;ïżœïżœ;.~_;;{
= 0.3604217
0.3604217 â 0.36017 < 0.00001đđ â0.00000005 < 0.00001
đ„7 đ„;
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = 0.36042170296032440136932951583028
đ đ„ = 3đ„ + sin đ„ â đ1 = 0
Webeginwithđ„_ = 0.0tol1 = 1đž â 5 OR (0.00001)tol2 = 1đž â 5 OR (0.00001)
đ„_ = 0.0, đ„< = 0.0đ đ„ = 3đ„ + sin đ„ â đ1
đâČ đ„ = 3 + cos đ„ â đ1 đ(đ„_) = â1.0, đâČ(đ„_) = 3.0
Iteration1:đ„_ = 0.0, đ„< = đ„_ âc 1bcx 1b
= 0.0 â ?<._7._
= 0.33333
Iteration2:đ„< = 0.33333, đ„; = đ„< âc 18cx 18
= 0.33333 â ?_._{|}<|;.~}ïżœ7}
= 0.36017
Iteration3:đ„; = 0.36017, đ„7 = đ„; âc 1:cx 1:
= 0.36017 â ?_.___{;ïżœïżœ;.~_;;{
= 0.3604217
0.3604217 â 0.36017 < 0.00001đđ â0.00000005 < 0.00001
ActualError=đ„ïżœlmïżœqr â đ„7 = 0.000000002960324AbsoluteError=|đ„ïżœlmïżœqr â đ„7| = 0.000000002960324
đ„7 đ„;
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
Newtonâsalgorithmiswidelyusedbecause,atleastinthenearneighborhoodofaroot,itismorerapidlyconvergentthananyofthe
methodsdiscussedsofar
Thenumberofdecimalplacesofaccuracynearlydoublesateachiteration
Thereistheneedfortwofunctionevaluationsateachstepđ đ„ andđâČ(đ„) andwemustobtainthederivativefunctionatthestart
FindingđâČ(đ„)maybedifficult.Computeralgebrasystemscanbearealhelp
InsomecasesNewtonâsmethodwillnotconverge
Thebadchoiceofđ„_ mayneverleadustoreachananswer;wearestuckinaninfiniteloop
Reachinglocalminimumorlocalmaximum ofthefunction,theanswerwillflyofftoinfinity
đv(đ„_) = 0
đ„o9< = đ„o âđ(đ„o)đâČ đ„o
đ„~ đ„7đ„fggEđ„_ đ„; đ„} đ„<
đ„{
RelatingNewtonâstoOtherMethodsLinearInterpolationMethods
đ„; = đ„_ âđ đ„_
đ đ„_ â đ(đ„<)(đ„_ â đ„<)
đ„; = đ„_ âđ(đ„_)đâČ đ„_
NewtonâsMethod
đv đ„_ = limïżœâ_
đ đ„_ + â â đ(đ„_)â
đ„< = đ„_ + â
đv đ„_ = lim(18?1b)â_
đ đ„< â đ(đ„_)đ„< â đ„_
đ„; = đ„_ âđ(đ„_)
lim(18?1b)â_
đ đ„< â đ(đ„_)đ„< â đ„_
Thedenominatorofthefractionisanapproximation ofthederivativeatđ„_
đ„; = đ„_ â đ đ„_(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
SecantmethodlookslikeNewtonâsanditsgoodtousewhenthederivativeisnoteasytoachieve
SecantMethod NewtonâsMethod
đ„; = đ„_ âđ(đ„_)đâČ đ„_
đ„; = đ„_ â đ đ„_(đ„_ â đ„<)
đ đ„_ â đ(đ„<)
Newtonâsmethodworkswithcomplexrootsifwegiveitacomplexvalueforthestartingvalueđ„_
đ„CDEFGH = â2.9259, 0.46293 ± 1.22254đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5 = 0
Webeginwithđ„_ = 1 + đtol1 = 1đž â 4 OR (0.0001)tol2 = 1đž â 4 OR (0.0001)
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„_ = 1 + đ, đ„< = 1 + đ
đ(đ„_) = 2 + 5đ, đâČ(đ„_) = 3 + 10đ
đ„CDEFGH = â2.9259, 0.46293 ± 1.22254đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5 = 0
Webeginwithđ„_ = 1 + đtol1 = 1đž â 4 OR (0.0001)tol2 = 1đž â 4 OR (0.0001)
đ đ„ = đ„7 + 2đ„; â đ„ + 5
đâČ đ„ = 3đ„; + 4đ„ â 1
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = â2.9259, 0.46293 ± 1.22254đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5 = 0
Webeginwithđ„_ = 1 + đtol1 = 1đž â 4 OR (0.0001)tol2 = 1đž â 4 OR (0.0001)
đ„_ = 1 + đ, đ„< = 1 + đ
đ(đ„_) = 2 + 5đ, đâČ(đ„_) = 3 + 10đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5
đâČ đ„ = 3đ„; + 4đ„ â 1
Iteration1:đ„_ = 1 + đ, đ„< = đ„_ âc 1bcx 1b
= 1 + đ â ;9~i79<_i
= 0.486238 + 1.04587đ
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = â2.9259, 0.46293 ± 1.22254đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5 = 0
Webeginwithđ„_ = 1 + đtol1 = 1đž â 4 OR (0.0001)tol2 = 1đž â 4 OR (0.0001)
Iteration2:đ„< = 0.486238 + 1.04587đ, đ„; = đ„< âc 18cx 18
=
= 0.486238 + 1.04587đ â0.519354 â 0.402202đâ1.62730 + 7.23473đ
= 0.448139 + 1.23665đ
đ„_ = 1 + đ, đ„< = 1 + đ
đ(đ„_) = 2 + 5đ, đâČ(đ„_) = 3 + 10đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5
đâČ đ„ = 3đ„; + 4đ„ â 1
Iteration1:đ„_ = 1 + đ, đ„< = đ„_ âc 1bcx 1b
= 1 + đ â ;9~i79<_i
= 0.486238 + 1.04587đ
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
đ„CDEFGH = â2.9259, 0.46293 ± 1.22254đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5 = 0
Webeginwithđ„_ = 1 + đtol1 = 1đž â 4 OR (0.0001)tol2 = 1đž â 4 OR (0.0001)
Iteration2:đ„< = 0.486238 + 1.04587đ, đ„; = đ„< âc 18cx 18
=
= 0.486238 + 1.04587đ â0.519354 â 0.402202đâ1.62730 + 7.23473đ
= 0.448139 + 1.23665đ
Iteration3:đ„; = 0.448139 + 1.23665đ, đ„7 = đ„; âc 1:cx 1:
=
= 0.448139 + 1.23665đ ââ0.0711105 â 0.166035đâ3.19287 + 8.27175đ
= 0.462720 + 1.22242đ
đ„_ = 1 + đ, đ„< = 1 + đ
đ(đ„_) = 2 + 5đ, đâČ(đ„_) = 3 + 10đ
đ đ„ = đ„7 + 2đ„; â đ„ + 5
đâČ đ„ = 3đ„; + 4đ„ â 1
Iteration1:đ„_ = 1 + đ, đ„< = đ„_ âc 1bcx 1b
= 1 + đ â ;9~i79<_i
= 0.486238 + 1.04587đ
COMPUTE ! "# ,!âČ("#)SET ") = "#IF ! "# â 0 AND !âČ "# â 0 THEN
REPEATSET "# = ")SET ") = "# â .(/0)
.1(/0)UNTIL ( ") â "# <tol1)OR(|!("))| <tol2)
âą "# reasonablyclose totherootâą tol1:thespecified tolerancevaluefordifference
betweensuccessivex-values.âą tol2:thespecified tolerancevalueforhowclose!(")) tozero.
INPUT
NewtonâsMethod
http://mathfaculty.fullerton.edu/mathews/a2001/Animations/RootFinding/NewtonMethod/NewtonMethod.html
NewtonâsMethod- Animation
Fixed-PointIterationMethod
Fixed-PointIterationMethod
A veryusefulwaytogetarootofagivenđ(đ„)
Werearrangeđ(đ„) intoanequivalentformđ„ = đ(đ„)
Ifđ isarootofđ(đ„),thenđ đ = đ â đ đ = 0 âč đ = đ(đ)
đ đ„ = đ„; â 2đ„
đ isafixed-pointforfunctionđ
đ„ = đ đ„ = 2đ„ïżœ
đ„o9< = đ đ„o đ = 0,1,2,3, âŠ
Forgivenequationđ(đ„) = 0,theremaybemanyequivalentfixed-pointproblemsđ„ = đ(đ„) withdifferentchoiceofđ(đ„)
đ đ„ = đ„; â 2đ„ â 3 = 0
đ< đ„ = 2đ„ + 3ïżœ đ; đ„ =3
đ„ â 2 đ7 đ„ =đ„; â 32
!" # = #% â 32
REARRANGE đ đ„ INTO đ„ = đ(đ„)SET đ„; = đ„<REPEAT
SET đ„< = đ„;SET đ„; = đ(đ„<)UNTIL ( đ„< â đ„_ <tol)
âą đ„< reasonablyclose totherootâą tol:thespecifiedtolerancevalueINPUT
Fixed-PointIterationMethod(đ„ = đ(đ„)Method)
âą Themethodmayconvergetoarootdifferentfromtheexpectedoneordiverge.âą Differentrearrangementswillconvergeatdifferentrates.
NOTES
đ„CDEFGH = â1, 3đ đ„ = đ„; â 2đ„ â 3 = 0
Webeginwithđ„_ = 4tol = 1đž â 4 OR (0.0001)
REARRANGE ! " INTO " = $(")SET "' = "(REPEAT
SET "( = "'SET "' = $("()UNTIL ( "( â "* <tol)
âą "( reasonablyclose totherootâą tol:thespecified tolerancevalueINPUT
Fixed-PointIterationMethod(" = $(")Method)
đ< đ„ = 2đ„ + 3ïżœ
đ„_ = 4,đ„< = 2 4 + 3ïżœ = 11ïżœ = 3.31663đ„; = 2 3.31663 + 3ïżœ = 9.63325ïżœ = 3.10375đ„7 = 2 3.10375 + 3ïżœ = 9.20750ïżœ = 3.03439đ„} = 2 3.31663 + 3ïżœ = 9.06877ïżœ = 3.01144đ„~ = 2 3.01144 + 3ïżœ = 9.02288ïżœ = 3.00381đ„{ = 2 3.00381 + 3ïżœ = 9.00762ïżœ = 3.00127đ„ïżœ = 2 3.00127 + 3ïżœ = 9.00254ïżœ = 3.00042đ„| = 2 3.00042 + 3ïżœ = 9.00084ïżœ = 3.00013đ„ïżœ = 2 3.00013 + 3ïżœ = 9.00026ïżœ = 3.00004
đ„_ đ„o9< = đ(đ„o)
đ„CDEFGH = â1, 3đ đ„ = đ„; â 2đ„ â 3 = 0
Webeginwithđ„_ = 4tol = 1đž â 4 OR (0.0001)
REARRANGE ! " INTO " = $(")SET "' = "(REPEAT
SET "( = "'SET "' = $("()UNTIL ( "( â "* <tol)
âą "( reasonablyclose totherootâą tol:thespecified tolerancevalueINPUT
Fixed-PointIterationMethod(" = $(")Method)
đ„_ = 4,đ„< = 1.5đ„; = â6đ„7 = â0.375đ„} = â1.263158đ„~ = â0.919355đ„{ = â1.02762đ„ïżœ = â0.990876đ„| = â1.00305
đ; đ„ =3
đ„ â 2
đ„_
đ„CDEFGH = â1, 3đ đ„ = đ„; â 2đ„ â 3 = 0
Webeginwithđ„_ = 4tol = 1đž â 4 OR (0.0001)
REARRANGE ! " INTO " = $(")SET "' = "(REPEAT
SET "( = "'SET "' = $("()UNTIL ( "( â "* <tol)
âą "( reasonablyclose totherootâą tol:thespecified tolerancevalueINPUT
Fixed-PointIterationMethod(" = $(")Method)
đ„_ = 4,đ„< = 6.5đ„; = 19.625đ„7 = 191.070
đ7 đ„ =đ„; â 32
Diverges
đ„_
đ7 đ„ =đ„; â 32
Startonthex-axisattheinitialđ„_, goverticallytothecurveđ(đ„),thenhorizontallytotheline đŠ = đ„, thenverticallytothecurve,andagainhorizontallytotheline.
Repeatthisprocessuntilthepointsonthecurveconverge toafixedpointorelsediverge
Howitworks?
!" # = #% â 32
Thedifferentbehaviorsdependonwhethertheslopeofthecurve isgreater,less,orofoppositesigntotheslopeoftheline
Howitworks?
DivergesConvergesConverges
!"
0<SlopeofCurve<1
monotonicconvergence
|SlopeofCurve|â„ 1-1<SlopeofCurve<0
oscillatoryconvergence
!"
#$ ! = !& â 32
!"
http://mathfaculty.fullerton.edu/mathews/a2001/Animations/RootFinding/FixedPoint/FixedPoint.html
Fixed-PointIteration- Animation