lista1p1-solução

Prévia do material em texto

(1.2)(1.2)
 
 
 
 
(1.6)(1.6)
 
 
(1.9)(1.9)
(1.8)(1.8)
 
(1.1)(1.1)
 
(1.5)(1.5)
(1.4)(1.4)
 
 
 
(1.3)(1.3)
(1.7)(1.7)
LLLLIIIISSSSTTTTAAAA 1111
Exercício 1
with(linalg):
eqns := {2*x+3*y+5*z=5,3*x-y+2*z=2};
eqns := 2 x 3 y 5 z = 5, 3 x y 2 z = 2
A := genmatrix(eqns, [x,y,z],flag);
A :=
2 3 5 5
3 1 2 2
backsub(gausselim(A));
1 _t1 1 _t1 _t1
eqns := {4*x+3*y=7,3*x-2*y=1,2*x-7*y=-5};
eqns := 4 x 3 y = 7, 3 x 2 y = 1, 2 x 7 y = 5
A := genmatrix(eqns, [x,y],flag);
A :=
4 3 7
3 2 1
2 7 5
backsub(gausselim(A));
1 1
eqns := {4*x+3*y=7,3*x-2*y=1,2*x-7*y=-1};
eqns := 4 x 3 y = 7, 3 x 2 y = 1, 2 x 7 y = 1
A := genmatrix(eqns, [x,y],flag);
A :=
4 3 7
3 2 1
2 7 1
backsub(gausselim(A));
Error, (in linalg:-backsub) inconsistent system
Não tem solução, veja a última linha da matriz aumentada após escalonamento.
rref(A);
(1.9)(1.9)
 
 
(2.2)(2.2)
 
 
(2.3)(2.3)
(2.1)(2.1)
(2.4)(2.4)
(2.5)(2.5)
 
1 0 0
0 1 0
0 0 1
Exercício 2
a:=matrix([[1,2,3],[2,3,5],[3,5,7]]);
a :=
1 2 3
2 3 5
3 5 7
u:=LUdecomp(a, L='l', P='p'); evalm(l);
u :=
1 2 3
0 1 1
0 0 1
1 0 0
2 1 0
3 1 1
evalm(a-l&*u);
0 0 0
0 0 0
0 0 0
a:=matrix([[1,2,3],[2,4,5],[3,5,7]]);
a :=
1 2 3
2 4 5
3 5 7
u:=LUdecomp(a, L='l',P='p'); evalm(l);
u :=
1 2 3
0 1 2
0 0 1
(2.6)(2.6)
(2.9)(2.9)
 
 
 
 
(2.8)(2.8)
(2.7)(2.7)
1 0 0
3 1 0
2 0 1
evalm(a-l&*u);
0 0 0
1 1 2
1 1 2
O que deu errado? Precisamos da permutação!
evalm(p);evalm(p&*a-l&*u);
1 0 0
0 0 1
0 1 0
0 0 0
0 0 0
0 0 0
a:=matrix([[1,2,0,0,0],[1,1,0,0,0],[1,1,1,0,0],[1,1,1,1,0],
[1,1,1,1,1]]);
a :=
1 2 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1
u:=LUdecomp(a, L='l',P='p'); evalm(l);
u :=
1 2 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
(3.1)(3.1)
 
 
(3.3)(3.3)
(2.10)(2.10)
(3.2)(3.2)
 
 
 
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1
evalm(a-l&*u);
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Exercício 3
a:=matrix([[3,2,2],[0,2,3],[2,1,1]]);
a :=
3 2 2
0 2 3
2 1 1
e1:=vector([1,0,0]):e2:=vector([0,1,0]):e3:=vector([0,0,1]):
u:=LUdecomp(a, L='l', P='p'); evalm(l);
u :=
3 2 2
0 2 3
0 0 16
1 0 0
0 1 0
2
3
1
6 1
y1:=forwardsub(augment(l,e1));x1:=backsub(augment(u,y1));
#conferindo 
evalm(a&*x-e1);
 
 
(3.6)(3.6)
(3.4)(3.4)
(3.3)(3.3)
(3.5)(3.5)
 
y1 := 1 0 23
x1 := 1 6 4
0 0 0
y2:=forwardsub(augment(l,e2));x2:=backsub(augment(u,y2));
#conferindo 
evalm(a&*x2-e2);
y2 := 0 1 16
x2 := 0 1 1
0 0 0
y3:=forwardsub(augment(l,e3));x3:=backsub(augment(u,y3));
#conferindo 
evalm(a&*x3-e3);
y3 := 0 0 1
x3 := 2 9 6
0 0 0
inversa:=augment(x1,x2,x3); #conferindo
evalm(inverse(a)-inversa);
inversa :=
1 0 2
6 1 9
4 1 6
0 0 0
0 0 0
0 0 0