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7/21/2019 Optimizacin222
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7/21/2019 Optimizacin222
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Co+ #& 3,o.o .# G-* ,"-+@o"-o#& ,#- #+ *+- -,"5
R T W X Y Z
F1 1 0 3 0 4 00F2201200=0F3101002=0F42
Ponemos como primera ecuacin aquella con coeficiente ! en este caso es " # $espu%s&acemos ' los $em(s elementos $e la primera columna! # continuamos con las si)uientescolumnas*
R
T W X Y Z F1 1 0 3 0 4 0 0 F22 0 1 2 0 0 0 F3 1 0 1 0 0
0F42210000F50
-F1 -F4+F2
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 2 0 2 0 0 0 F3 1 0 1 0 0
0
F4
2
2
1
0
0
0
0
F5
0
-F1+F3 -2F1+F4
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 2 0 2 0 0 0 F3 0 0 2 0 4
0F40270800F50
F5F2
Intercambiamos las filas 5 y 2 para mantener el orden ecalonado
RT W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 2 0 4
0F4 0270800F5
-1/2F3 -2F2+F4 2F2+F5 -2F2+F6
Se!imos "ol"iendo # las si!ientes col!mnas
ELIMINACIN GAUSSIANA(?0 P*+,o)
5
6Y 7 89 7 R 0
2; 7 9 < 2 R 0
2Z < 9 < R 0
2T 7 9 < 2R 0
=; 7 8Y < Z < T 2
2; 7Y < Z < 2T 6
7/21/2019 Optimizacin222
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R
T W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F400721424
$F3+F4
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F40002094F
1/2F4
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F4000109/22
-4F4+F6
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F4000109/22
F6+F5
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F4000109/22
5F5+F6
R
T W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F4000109/22
-F5 1/112F6
RT W X Y Z F1 1 0 3 0 4 0 0 F20 1 0 1 3 1 2 F3 0 0 1 0 2
10F4000109/22
-3F3+F1 F4+F2
7/21/2019 Optimizacin222
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R
T W X Y Z F1 1 0 0 0 2 3 0 F20 1 0 0 3 7/2 0 F3 0 0 1 0 2
10F4000109/22F
-2F5+F1 3F5+F2 2F5+F3
R
T W X Y Z F1 1 0 0 0 0 41 20 F20 1 0 0 0 121/2 30 F3 0 0 1 0 0
3920F4 000109
-41F6+F1 121/2F6+F2 3%F6+F3 %/2F6+F4 1%F6+F5
R
T W X Y Z F1 1 0 0 0 0 0 69/56 F20 1 0 0 0 0 149/112 F3 0 0 1 0 0
11/56F400010
&l res!ltado indica '!e(
)* -6%/56 *14%/112 ,*11/56 * 3$/112 .*-%/56 * 2%/56
0omprobando
4.-3,-)*#
4-%/56-311/56--6%/56 10)789
-36/56-33/56+6%/56
-36-33+6%/56
#/56*#
2-,+2)*#
23$/112-11/56+2-6%/56 2 :
3$/56-11/56+-13;/56
3$-11-13;/56
3$-14%/56
-112/56
-14/$* -2
2+,+)*#
22%/56+11/56+-6%/56 3 0)789
5;+11-6%/56
#/56*#
2-,+2)*#
214%/112-11/56+2-6%/56 4 0)789
7/21/2019 Optimizacin222
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14%-11-13;/56
#/56*#
--3.++*2
-3$/112-3-%/56+2%/56+14%/112 5 0)789
-3$+14%/112+2$/56+2%/56
112/112+2$+2%/561+1*2
2-.++2*4
23$/112--%/56+2%/56+214%/112 6 0)789
$4/112+%+2%/56+2%;/112
$4+2%;/112+3;/56
1;6/56+1%/2;
%3/2;+1%/2;
%3+1%/2;
112/2;*4
;= 6Y = Z= T 0 2 2
Y= 6; < Z7 YT 0 2
=; 7 Y 0
=; 7 Y2< 2 0
9espeamos las mismas inc=nitas en 2 ec!aciones
-*.
*-.
-* Y2
-2
* - Y2
+2
S!stit!imos !na en el otro
-.* - Y2
+2
IGUALACIN(> P*+,o)5
7/21/2019 Optimizacin222
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Y2
-.-2*#
tili?amos la f=rm!la de se!ndo rado
8*1 7* -1 0* -2
y=b b
24 ac2a
y=(1)(1)
24 ( 1 )(2)2(1)
y=19
2(1)
y=
1+32(1) * 2
y=132(1) * -1
S!stit!imos
Y2
-.-2*.-2@.+1
8; < >Y < 8Z >
6; < 6Y < 2Z ?
2; < 8Y < Z 8
-"- #+,#+.#" #Bo" #+*#"-"# &- #/*-/o+#
#/*-/+ ?
8; < >Y < 8Z >
REDUCCIN(6 P*+,o)5
7/21/2019 Optimizacin222
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#/*-/+ 2
6; < 6Y < 2Z ?
#/*-/+ 8
2; < 8Y < Z 8
-&/-o #& 3,o.o .# "#.*//+ #+,"# &- #/*-/+ ? 8
(=2) (8; < >Y < 8Z >)(8) (2; < 8Y < Z 8)
$$$$$$$$$$$$$$$$$$$$$$=>; = ?2Y < >Z =?2>; < Y < 8Z
$$$$$$$$$$$$$$$$$$$$= 8Y = 8Z =8 L- &&--"#o #/*-/+ +*#"o 6
-&/-o #& 3,o.o .# "#.*//+ #+,"# &- #/*-/+ 2 8
6; < 6Y < 2Z ?(=2) (2; < 8Y < Z 8)
$$$$$$$$$$$$$$$$$$$$$$$$$$6; < 6Y < 2Z ?=6; = >Y = 2Z =>
$$$$$$$$$$$$$$$$$$$$$$$$$$=2Y =
Ao"- 4*# ,"-!-B-o o!"# *+- o&- '-"-! o.#o .##B-" Y * @H/+,#5
=2Y =Y ==2
Y 21
Ao"- *,,*o #& '-&o" .# Y #+ &- #/*-/+ +#"o 6 -"- o!,#+#" #& '-&o" .# Z
= 8Y = 8Z =8=8(21) = 8Z =8
=K1 = 8Z = 8=8Z =8 < K1
=8Z 61Z 61=8
Z =?1
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Ao"- *,,*o #& '-&o" .# Y .# Z #+ /*-&4*#"- .# &- #/*-/o+# ?1 2 o 85 E+ #,#/-o *-"# &- #/*-/+ +*#"o 85
2; < 8Y < Z 82; < 8(21) < ?(=?1) 8
2; < K1 = ?1 82; 8 = >2; =8; =82
; =?1
R#*&,-.o:
; =?5 Y 25 Z =?5
Po.#o /o"o!-"&o *,,*#+.o &o "#*&,-.o .# ;1Y1Z #+ /*-&4*#"- .# &-
#/*-/o+# ?1218 o &o /o"o!-"# /o+ &- #/*-/+ +*#"o 8
2; < 8Y < Z 82(=?1) < 8(25) ?(=?1) 8
=8 < K1 = ?1 8
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