Analisis Numerico METODO STEFFENSEN

7/23/2019 Analisis Numerico METODO STEFFENSEN

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Conminucionfotacionlixiviacionminapirometalurgiae

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Integrantes:-BERNAL HEREDIA, Hctor Jesus

-CARRANZA LEVANO, Renato-JARA SOLORZANO, Ednson

-!IRANDA !ARCOS , J"on#

!etodo Nu$ercoS%E&&ENSEN

'( de Sete$)re de* (+'


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Analisis Numerico

!todo Ne$erco S%E&&ENSEN ro0esor: Lc12*ad#s 21!e*gare3o

Estre$ado#ro

CONTN!"O#

'1 Conce4to %eorco de* !todo S%E&&ENSEN----------------------------+((1 asos a segur 4ara *a reso*uc5n de un 4ro)*e$a-------------------+61 A*gort$o de* !etodo Ste7ensen------------------------------------------+891 ro)*e$as de* !etodo Ste7ensen-----------------------------------------+/

$%TO"O " &T''N&N

Jo"an &reder Ste7ensen descu)r5 un a*gort$o ;ue 4er$teca*cu*ar *as ra # no re;uere, co$o e* caso de* $todo de ne?ton, *ae=a*uac5n de der=ada a*guna1

So*o necesta un 4unto nca* en e* 4roceso de terac5n1 En cadaterac5n,e* n@$ero de d


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Analisis Numerico


Estre$ado#ro

a4ro$acones suces=as co$ena$os a a4*car e* $todo de Aten1 E*$todo de Ste7ensen consste en to$ar este =a*or co$o nue=o 4unto de4artda 4ara ca*cu*ar tres nue=as teracones de* $todo dea4ro$acones suces=as # a4*car de nue=o e* $todo Aten1E* $todo se es;ue$ata de *a 0or$a sguente:

En *a terac5n (n+1 ) se conoce xn # se ca*cu*a xn+1 as


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Analisis Numerico


Estre$ado#ro

Esta condc5n "ace *a 0unc5n adecuada co$o una correcc5n 4araencontrar *a so*uc5n, aun;ue no es necesara 4ara ;ue este $todotra)a3e ecente$ente1

ara a*gunas 0uncones, e* $todo de Ste7ensen 4uede 0unconar nc*usos esta condc5n no es cu$4*da, 4ero en ta* caso, e* =a*or de 4artdade)e ser $u# cercano a *a so*uc5n , # *a con=ergenca a *a so*uc5n

4odr


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Estre$ado#ro

M

La 0unc5n es e* =a*or ;ue ree$4*aa *a 0unc5n 4endente de , # es "a**ado a

4artr de* segundo =a*or de *a coordenada # *a

coordenada au*ar , con *a ac*arac5n ;ue

Es @nca$ente 4ara 4ro45sto de encontrar 4ara este 4unto au*ar, ;ue e*

=a*or nca* dado de *a 0unc5n de)e estar adecuada$ente cerca de su 4ro4aso*uc5n, # 4or esa ra5n cu$4*r e* re;usto de ;ue:

ara todas *as de$.s 4artes de* c.*cu*o, e* $todo de Ste7ensen s5*o re;uere;ue *a 0unc5n sea contnua # tener una so*uc5n cercana a* =a*or nca*1Entonces 4ode$os resu$r *a 05r$u*a a:

xn+1=xn

( f(xn))2

f(xn+ f(xn ))f(xn) > n+, ',M

(A&O& A &)*!+ (A+A ,A +&O,*C!-N " *N (+O.,$A#

- %ene$os una 0unc5n cua*;uera:

f(x )=anxn+an1x

n1+........+a2x2+a1x

1+a0

- Crea$os otra 0unc5n des4e3ando *a =ara)*e de $a#or grado:


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Analisis Numerico


Estre$ado#ro

g (x )=n

an1x

n1 ..a2x2a1x

1a0an

- Consderando e* 4unto nca*

p0=x0

xn

p1=g(p0 )

p

g(1)p2=

- &na*$ente, tenendo *os 4untos p0 , p1 , p2 1 Estos datos *oree$4*aa$os en *a 0or$u*a genera* de Ste7ensen:

xn+1=p2 (p2p1)

2

p22p

1+p

0

- Hace$os *a ta)*a corres4ondente:

n p0 p1 p2 xn+1 /

0

1

2

A,)O+!T$O " &T''N&N


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Estre$ado#ro


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Estre$ado#ro

(+O.,$A 1

Dada *a 0unc5n f(x )=xex

ca*cu*e *a ra


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Analisis Numerico


Estre$ado#ro

x1=0.582226097

SEGN!A ITERACI N

n=1

p0=0.582226097

p1=g(x0)=g (0.582226097 )=ex=e0.582226097=0.558653365

p2=g(p1)=g (0.558653365 )=ex=e0.558653365=0.571978792

Entonces en:

xn+1=p2 (p2p1)

2

p22p

1+p

0

x2=p2 (p2p1 )

2

p22p1+p0

0.571978792 (0.5719787920.558653365 )2

0.5719787922(0.558653365 )+0.582226097

x2=0.0567166438

%ERCERA I%ERACIPN

n=2

p0=0.0567166438

p1=g(x0)=g (0.0567166438 )=ex=e0.0567166438=0.567130162

p2=g(p1)=g (0.567130162 )=ex=e0.567130162=0.567150736

Entonces en:


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Estre$ado#ro

xn+1=p2 (p2p1)

2

p22p

1+p

0

x3=p2 (p2p1 )

2

p22p1+p0

0.56150736 (0.561507360.567130162 )2

0.561507362(0.567130162)+0.0567166438

x3=0.567143290

CQAR%A I%ERACIPN

n=3

p0=0.567143290

p1=g(x0)=g (0.567143290 )=ex=e0.567143290=0.567143291

p2=g(p1)=g (0.567143291 )=ex=e0.567143291=0.567143290

Entonces en:

xn+1=p2

(p2p1)2

p22p1+p0

x4=p2 (p2p1 )

2

p22p1+p0

0.567143290 (0.5671432900.567143291 )2

0.5671432902(0.567143291)+0.567143290

x4=0.567143290


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Estre$ado#ro

EN LA %ABLA:

n p0 p1 p2 xn+1 /

0 ' 0.367879441 0.692200628 0.582226097 ----

1 0.582226097 0.558653365 0.571978792 0.567166438 (188

3 0.0567166438 0.567130162 0.567150736 0.567143290 91+'+-6

4 0.567143290 0.567143291 0.567143290 0.567143290 +

Entonces *a ra


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Estre$ado#ro

g(x)= 9

x2+cos (x )ln (x )=x

r$era terac5n x0=1

"0=g(x0)

"0=

9

x2+cos (x )ln (x )=5.843008847

#0=g("0)

#0=

9

"2+cos (" ) ln (" )=0.270431205

x1=x0 ("0x0 )

2

#02"0+x0=3.251888045

segunda terac5n x1=3.251888045

"0=g(x0)

"1=

9

x2+cos (x )ln (x )=1.071222430

#0=g("0)

#1=

9

"2+cos (" )ln (" )=5.777494830

x2

=x1

("1x1 )

2

#12" 1+x1=2.561406656


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Estre$ado#ro

tercera terac5n x2=2.561406656

"0=g(x0)

"2= 9

x2+cos (x )ln (x )

=1.881315431

#0=g("0)

#2= 9

"2+cos (" )ln (" )

=3.459112339

x3=x2 (" 2x2 )

2

#22" 2+x2=2.356558565

cuarta terac5n x3=2.356558565

"3=g(x3)

"3= 9

x2+cos (x )ln (x )

=2.256316685

#0=g("0)

#3= 9

"2+cos (" )ln (" )

=2.469708081

x3=x0 ("0x0 )

2

#02"0+x0=2.324519763

;unta terac5n x4=2.324519763


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Estre$ado#ro

"4=g (x4)

"4= 9

x2+cos (x ) ln (x )

=2.322268869

#0=g("0)

#4= 9

"2+cos (" ) ln (" )=2.326980850

x5=x0 ("0x0 )

2

#02"0+x0=2.323792115

ERR$R=|(2.3237921152.324519763 )2.323792115 |% 100=0.031312958


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Estre$ado#ro

"0=g (x0 )

"0=

9

((1 )2+9)

"0=0 9

#0=g ("0 )

#0= 9

((0,9)2+9)

#0=0 9174311927

x=x0 ("0x0 )

2

#02" 0+x0

x=1 (0,9(1))2

0,91743119272(0,9)+(1)

x=0 . 91484375

2da. Aproximacina"iendo #ue x0=0.91484375

"0=g (x0 )

"0=

9

((0.91484375 )2+9)

"0=0 914918748

#0=g ("0 )


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Estre$ado#ro

#0=

9

((0,914918748 )2+9)

#0=0 914905985

x=x0 ("0x0 )

2

#02" 0+x0

x=1 (0,914918748(0.91484375))2

0,9149059852(0,914918748)+(0.91484375)

x=0 '914907841

$ecordando

x= 9

(x2+9 )

%or lo tanto conocemox0=1

"0=g (x0 )

#0=g ("0 )

x=x0 ("0x0 )

2

#02" 0+x0

i x "=g (x ) #=g (" ) &a

0 &1 & 0,' & 0,'1311'2 &

1 &0,'1*3+ &0,'1'1'* &0,'1'0+'*+ &'.31

2 &0,'1'0*1 &0.'1'0*1 &0,'1'0*1 0.00+2

3 &0,'1'0*1


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Analisis Numerico


Estre$ado#ro

Comparacin con otros mtodos

Funcin: f(x )=x3x1

(+O.,$A 5

Dada *a sguente 0unc5n:

Ha**ar *a ra


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Analisis Numerico


Estre$ado#ro

e* =a*or x0=5 , # co$4are e* n@$ero de teracones necesaras

4ara "a**ar *a ra


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Analisis Numerico


Estre$ado#ro

n x (x) (x +(x)) g(x) %ER

+ 9 818 '(+1+9 'UU19 '(T

'61898U'

1'+U8UU( 1+966

'8+1U6 ''T

(

61(U'

9U

1('U/9

+/ /+U1/86/6

'691//6

'+T

661UU96/8/

91(69/8 +/1'/

'''1+/+U T

961U9U89/U

61U669(9 69/1('U9++'

U1/8/U UT

61U+UU6

61'9/68('' ((91/69(

/+19'++99 /T

8

61/8''

6

(19/9(UU

8 '616998'

61/++99

U8 8T

/61/''(686

'1U(9/( /91'+88'6(U

61U(U6( 9T

U6188U8/

'1(('((9 618U+8/'

(U1(+6U/6 6T

618(8/'U(

+1898(( '91689/'99

'18U+8/8' (T

'+61+6/U'

+1((8'8 919'6U'U(

'91+U''+ 'T

''618/'6

+1+8//U6(U +1U66'6++/

''1('++ +T

'(6186/UU

+1++96'U +1++699'89

'+19++6U +T

'66186'8(89

'1'+UE-+ +1+++('/(/

'+19U9/6 +T

'96186'8+U(

618U88E-'+ 91(6''E-+

'+19U/9/( +T

'6186'8+U(

'19(''E-'9 '186'E-'6 '+ +T

'86186'8+U(

-(1'6'8E-'9 -(1(/6/9E-'6

18888888/ +T

'/6186'8+U(

'19(''E-'9 '1/+6E-'6 '' +T

'U

6186'8+

U(

/1'+9E-

' /1U'/E-'9 '+ +T' 6186'8+

U(-(1'6'8E-'9

-(1(/6/9E-'6 18888888/

+T

(+ 6186'8+U(

'19(''E-'9

'1/+6E-'6 '' +T


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Analisis Numerico


Estre$ado#ro

Co$4arando con Mtodo Newton Raphson:

n x (x) (x) %ER

+ 9 818 (+1' '(T

'618/'89'/

'1(68U'6

'(1886+(/ 6T

(61/(96/

+1+9/8++(

'+18U'+8' +T

66186(9''

+1+++U9(/

'+1++(U' +T

9

6186'8+

U6

81(6/8E-

+U

'+19U/(

/( +T

6186'8+U(

/1'+96E-'

'+19U/(/' +T

86186'8+U(

/1'+96E-'

'+19U/(/' +T

/6186'8+U(

-(1'6'86E-'9

'+19U/(/' +T

U

6186'8+

U(

'19('+E-

'9

'+19U/(

/' +T

6186'8+U(

/1'+96E-'

'+19U/(/' +T

'+6186'8+U(

-(1'6'86E-'9

'+19U/(/' +T

''6186'8+U(

'19('+E-'9

'+19U/(/' +T

'(6186'8+U(

/1'+96E-'

'+19U/(/' +T

'6 6186'8+ -(1'6'86E- '+19U/( +T

xn+1=x f(x)f '(x)


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Analisis Numerico


Estre$ado#ro

U( '9 /'

'96186'8+U(

'19('+E-'9

'+19U/(/' +T

'6186'8+U(

/1'+96E-'

'+19U/(/' +T

'86186'8+U(

-(1'6'86E-'9

'+19U/(/' +T

'/

6186'8+

U(

'19('+E-

'9

'+19U/(

/' +T

'U6186'8+U(

/1'+96E-'

'+19U/(/' +T

'6186'8+U(

-(1'6'86E-'9

'+19U/(/' +T

(+6186'8+U(

'19('+E-'9

'+19U/(/' +T

En *a @*t$a ta)*a, nota$os c*ara$ente ;ue no se cu$4*e *acondc5n:

De gua* $anera se **ega a *a ra

Analisis Numerico METODO STEFFENSEN

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