Índice:
.............................. ..................... ..................... ..................... ...................... ......................... ..............2 2 1-. INTRODUCCIÓN:.................... 2.- MODELO MATEMÁTICO:...........................................................................3
............................... ..................... ..................... ....................... .............7 7 3.-ALGORITMO COMPUTACIONAL: ..................... ............................... ..................... .............................................. ................................... 9 4-. APLICACIÓN MANUAL: ..................... .............................. ..................... ...................... ......................9 ...........9 4.1.- APLICACIÓN COMPUTACIONAL .................... .............................. ..................... ..................... ..................... ...................... ....................... ............10 10 5-. CONCLUSIONES: .................... ............................... ..................... ............................................... ..................................... 10 6-. RECOMENDACIONES RECOMENDACIONES.................... 7-. BIBLIOGRAFIA: BIBLIOGRAFIA:......................................................................................10
1
METODO DE INTEGRACIÓN DE GAUSS LEGENDRE
1-. INTRODUCCIÓN: En n!"#$#$ n%&'(#)*+ " #n,()#/n n%&'(#) )*n$,#,%0 %n &"# & "*(#,&*$ ( )")%"( " "*( n%&'(#)* %n #n,(" #n# 0+ *( ,n$#/n+ " ,'(&#n* $ %$ )$ ( $)(##( "*(#,&*$ n%&'(#)*$ ( ($*"( )%)#*n$ #(n)#"$. E" ,'(&#n* )%(,%( n%&'(#) &n%* (#* )%(,%(8 $ &!$ * &n*$ $#n/n#&* #n,()#/n n%&'(#)+ $)#"&n, $# $ "#) #n,("$ %n #&n$#/n $( 9% ( " )$* *$ * &!$ #&n$#*n$ #n,(" &",#"8 ,&#'n $ %,#"#;n. E" (*"& !$#)* )*n$#(* *( " #n,()#/n n%&'(#) $ )")%"( %n $*"%)#/n (*#& " #n,(" #n#:
E$, (*"& ,&#'n % $( n%n)#* )*&* %n (*"& "*( #n#)#" ( %n )%)#/n #(n)#" *(#n(#+ )*&* $#%:
En)*n,(( 08 $ 9%#"n, )")%"( " #n,(". L*$ &',**$ $((*""*$ ( )%)#*n$ #(n)#"$ *(#n(#$+ )*&* " &',** R%n-<%,,+ %n $( "#)*$ " (*"& (*(&%"*. En $, (,=)%"* $ #$)%,n &',**$ $((*""*$ $)=#)&n, ( " (*"& *(&%"* )*&* %n #n,(" #n#.
1.1.- RA>ONES PARA LA INTEGRACIÓN NUM?RICA @0 (#$ (;*n$ ( ""( )* " #n,()#/n n%&'(#). L (#n)#" % $( " #&*$##"# ("#;( " #n,()#/n *(& n"=,#). E$ )#(+ #n,("$ 9% (9%(#(=n %n (n )*n*)#&#n,* 0 &n* &,&!,#) n; %n $( ($%",$ %n &n( &!$ $n)#"" &#n, &',**$ n%&'(#)*$. In)"%$* #$,n %n)#*n$ #n,("$ (* )%0 (#&#,# n* % $( )")%"+ $#n* " #n,()#/n n%&'(#) #," #&*(,n)#. L $*"%)#/n n"=,#) %n #n,(" n*$ ((*(= %n $*"%)#/n ),+ &#n,($ 9% " $*"%)#/n n%&'(#) n*$ (= %n $*"%)#/n (*#&. E" ((*( " (*#&)#/n+ 9% n " &',** 9% $ %,#"#) 0 9%' ,n #n* $+ % ""( $( ,n 9%* 9% $ *$#" *,n( %n ($%",* #'n,#)* " $*"%)#/n n"=,#) n "$ (#&($ )#($ )#&"$. 2
M',**$ ( #n,("$ %n##&n$#*n"$. L*$ &',**$ #n,()#/n n%&'(#) %n $( $)(#,*$ n("&n, )*&* )*&#n)#/n "%)#*n$ " #n,(n* ( *,n( %n (*#&)#/n " #n,(". Un (, #&*(,n, " n!"#$#$ )%"9%#( &',** #n,()#/n n%&'(#) $ $,%#( " )*&*(,&#n,* " ((*( (*#&)#/n )*&* %n %n)#/n " n&(* "%)#*n$ " #n,(n*. Un &',** 9% (*%) %n 9%* ((*( ( %n 9%* n&(* "%)#*n$ $ n*(&"&n, )*n$#(* $%(#*(. R%)#n* " n&(* "%)#*n$ " #n,(n* $ (%) " n&(* *()#*n$ (#,&',#)$ #n*"%)($+ 0 *( ,n,* $ (%) " ((*( (*n* ,*,". T&#'n+ ) "%)#/n )%$, ,#&*+ 0 " #n,(n* % $( (#,((#&n, )*&"#)*. D ,**$ &**$+ %n &** #n,()#/n *( %(; (%, % )($ $#&(+ %n &** &%0 $#&"#$,+ "%n* " #n,(n* )*n #n)(&n,*$ &%0 9%*$. M',**$ $*$ n %n)#*n$ #n,(*")#/n. @0 %n ,n$ &#"# &',**$ 9% $ $n n (*#&( " %n)#/n #n,(( 8 *( *,(* %n)#/n 8 " )%" $ )*n*) " #n,(" ),. L %n)#/n 9% $%$,#,%0 " *(##n" $ n)%n,( *(& 9% n %n )#(,* n&(* %n,*$ ,n " &#$&* "*( 9% " *(##n". C*&* "*$ %n,*$ ,(&*$ *(&n (, $#&( $, )*n%n,* %n,*$+ " n% %n)#/n $ ""& %n #n,(*")#/n " %n)#/n *(##n". C%n* "*$ %n,*$ ,(&*$ n* $ %,#"#;n ( n)*n,(( " %n)#/n 9% $%$,#,%0 " *(##n" n,*n)$ $ #) ,(*")#/n. T=#)&n, $,$ %n)#*n$ $*n *"#n*&#*$. F/(&%"$ N,*n-C*,$. L #n,(*")#/n )*n *"#n*&#*$ "% n %n,*$ #%"&n, $(*$ n +H "$ /(&%"$ N,*n-C*,$+ "$ 9% " (" " (),!n%"*+ " " ,()#* 0 " S#&$*n $*n &"*$. S# $ $)*n "*$ n**$ $, J n K 1 $(! " /(&%" N,*n-C*,$ )(( 0 $# $ $)*n J n 1 $(! " /(&%" N,*n-C*,$ #(,. R" " (),!n%"*. E" &',** &!$ $#&" $, ,#* $ )( " %n)#/n #n,(*"*( $( %n %n)#/n )*n$,n, %n *"#n*&#* *(n )(*8 9% $ ,('$ " %n,* +88. E$, &',** $ ""& " (" " (),!n%"*:
L$ )%(,%($ G%$$ - Ln( $ " n*&( %n )"$ ,')n#)$ 9% "#) ," $,(,# ( *,n( %n (*#&)#/n &!$ ()#$ " #n,(".
2.- MODELO MATEMÁTICO: An,$ n"#;( " ($#/n n(" "$ )%(,%($ %$$+ (#$(&*$ " (" " ,()#* ( " #n,()#/n. L (" " ,()#* $ $ n *,n( " !( * " "=n (), 9% %n "*$ "*($ " %n)#/n+ n "*$ ,(&*$ "*$ #n,("*$ #n,()#/n #. 8. L /(&%" ( )")%"( " !( $:
3
D*n a 0 b $*n "*$ "=&#,$ #n,()#/n 0 b a J " n)* " #n,("* #n,()#/n. A*(+ $%*n&*$ 9% $ "#&#n " ($,(#))#/n "*$ %n,*$ #*$ 0 $ ,%#( " "#(, "%( " !( * %n "=n (), 9% %n#( *$ %n,*$ )%"$9%#( " )%( #. 8.
A*( #n "#)&*$ " $,(,# " )%(,%( %$$#n+
*n )1J)2J "$ ) $*n $)*n*)#$ #(n)# " (" " ,()#* 9% %,#"#; %n,*$ ,(&*$ #*$ a 0 b+ "*$ (%&n,*$ " %n)#/n x 0 x1 n* $,!n #*$ n "*$ ,(&*$+ $#n* 9% $*n #n)/n#,$+ n,*n)$ ,n&*$ 4 #n)/n#,$.
E$ *$#" *,n( *$ $$ )*n#)#*n$ " $%*n( 9% " )%)#/n "n, %$, )*n ),#,% " #n,(" %n )*n$,n, 0 %n %n)#/n "#n". D$%'$+ ( ,n( "$ *,($ *$ )*n#)#*n$+ $/"* $ &"#(! $, (;*n&#n,* " $%*n( 9% ,&#'n 4
%$, " #n,(" %n %n)#/n (/"#) y J x28 0 %n )#) y J x38. A" )("*+ $ ,(&#nn "$ )%,(* #n)/n#,$ 0 &!$ $ *,#n %n /(&%" #n,()#/n
"#n" *$ %n,*$ 9% $ ), ( )#)$. L$ )%,(* )%)#*n$ 9% (! 9% ($*"( $*n: L*$ "=&#,$ #n,()#/n $*n 1 0 1 0 $#&',(#)*$ )*n ($),* J + *( "* 9% )&*$ 2 J 1 0 (9%(#&*$ 9% "*$ %n,*$ $,'n $#,%*$ n *(& $#&',(#). D (#&( 0 $%n )%)#/n *,n&*$ c = c =1 0
1
C*n $,*$ "*($+ I )%(, )%)#/n $ $,#$) %,*&!,#)&n,. L ,()( )%)#/n $
E" ($%",* (*%) " $#%#n, *(&%" (*#&)#/n " #n,(":
P*"#n*&#*$ Ln(: P0 ( x ) =1 P1 ( x ) = x 2
P2 ( x ) = x −
1 3
3
P3 ( x ) = x − x 3
5
6
P4 ( x )= x − x + 4
7
2
3 35
5
Raices
Coefcientes
E$,* )*&", " $*"%)#/n " (*"& (*#&)#/n #n,("$ #n#$ ( %n)#*n$ n " #n,("* -1+1H+ *( #n+ $, $*"%)#/n $ $%#)#n, ( )%"9%#( #n,("* )((* *(9% " $n)#"" (")#/n "#n": t =
2 x − a−b
b− a
D*n $ " )**(n *(##n" n 0 , $ " )**(n n*(&"#; n -1J ,J1. L ,(n$*(&)#/n , n $: x =
( b− a ) t + b + a 2
b
1
a
−1
∫ f ( x ) dx =∫ f
(
)
( b− a ) t + b + a b −a ( ) dt 2
2
U$n* "$ (=)$ Ln( 0 "*$ )*#)#n,$ )1+ )2+ )3+.+ *$ n " ,"+ *,n&*$ " $#%#n, *(&%" (*#&)#/n 6
b
∫ f ( x ) dx ≈ (
b− a 2
a
n
)∑ c f (
( b− a ) x i +b + a
1
2
i= 1
)
3.-ALGORITMO COMPUTACIONAL: D,*$: F%n)#/n8+ #n,("*$ + H R$%",*$: C")%"* " #n,(" (*#& %$n* " #n,()#/n %$$-"n( PASO1. L( " %n)#/n PASO2. L( "*$ #n,("*$. PASO 3. @J- PASO4. In#)#"#;&*$ %n $%&*( )J8 PASO5. P( #J1+ $, 2 PASO6. S# #J1 #8J1Q √ 3 #8J1 C$* )*n,((#*: #8J-1Q √ 3 #8J1 PASO7. A)%&%"&*$ " $%& c = c + w ( i )∗f
(
( b −a ) x (i )+ b + a 2
)
PASO. C")%"&*$ " #n,(" (*#& #n,JQ2) PASO . G(#)( " %n)#/n
7
4.- DIAGRAMA D !"#
%$5-. C%DI!ICACI%&: f=inline(get(handles.edit1,'string')); a=str2num(get(handles.edit2, 'string')); b=str2num(get(handles.edit3, 'string')); h=(b-a)/2; c=0; for i=12 if i==1 !(i)=1/s"rt(3); #(i)=1; else !(i)=-1/s"rt(3); #(i)=1; end c=c$#(i)%f((b-a)/2%!(i)$(b$a)/2); if i==1 d1=c;&unto ara graficar else d2=c;&unto ara graficar end end r=h%c; set(handles.te!t, 'string',r);
'
4-. APLICACIÓN MANUAL: (e)*+o: I 7 f 8 =
1.5
∫ 1
2
e − x dx
- G%$$ "n(: )&#* (#" 1+-1H 1.5
∫ 1
e
− x
2
dx =
1
1
e 4 ∫
− 7 t + 5 8 2 16
dx
−1
- G%$$ "n(: nJ2 1 I f 8 = e 4
− N.5773+ 5 8 2 16
− −N.5773+ 5 8 2
+e
16
= N.1NT4NN3
- G%$$ "n(: nJ3
I f 8 ≈
1
− N.7745T6+ 5 8 2
N.5555556 e 4
16
+
− N+ 582
+ N.RRRRRT e
16
− −N.7745T6+ 5 8 2
+ N.555556 e
16
= N.1NT3642
4.1.- APLICACIÓN COMPUTACIONAL
9
5-. CONCLUSIONES: C,a+,ie inte/a+o aitaio de ,na ,ncin *,ede tas+adase a +a c,adat,a de Ga,ss. "a c,adat,a de Ga,ss no se *,eden ,ti+ia *aa intea ,na ,ncin dada en o)a de ta+a a ,e +os *,ntos de "eende no estn se*aados de esa o)a.
6-. RECOMENDACIONES 8e eco)ienda ,e +a ,ncin a intea no este dada en o)a de ta+a.
7-. BIBLIOGRAFIA: &ie/es A. and Do)n,e !. 2002;. Metodos numricos aplicados a la ingeniera. M<=ico: Continenta+. C>a*a 8. and Cana+e R. 2006;. Numerical methods for engineers . ?oston: McGa@-i++ i>e d,cation.Bad,cido a+ es*ao+; &aa),a 8. and Ea+)as Fe+asco %. 1992;. Mtodos numricos aplicados con software. M<=ico: Eentice a++ is*anoa)eicana.
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