Tarea 2
20
description
Análisis Numérico II
Transcript of Tarea 2
1
mxmA A
k 1 k m kxk 1: ,1:k kA
1: ,1:det( )k kA
1: ,1:det(U ) 0k k iiu
1: ,1: 1: ,1:det(A ) det(U )k k k k
1: ,1:det(A ) 0k k
1: ,1:A k k
1: ,1:A k k 1: ,1:k kU
0kku
1k
k m
1: 1,1: 1 1: 1,1: 1 1: 1,1: 1m m m m m mA L U
1:m,1:1: ,1:
1: 1,1: 1
m 1, 1
0
01
mm m
m m
m
U cLA
ub
2
1
1,1 1,2 1,m 1:m,1:........m m m mb a a a U
1
1:m,1: 1, 1 2, 1 ,m 1........T
m m m mc L a a a
m 1, 1mu bc
1: 1,1: 1 1:m,1: m 1, 1det( ) det( ) 0m m m mA U u
1 1 2 2A LU L U
1 1
2 1 2 1L L U U
1 1
2 1 2 1L L U U I
1 2L L
1 2U U
3
11 12
21 22
A AA
A A
11A 22A
11 12 11 12
1 1
21 11 21 22 22 21 11 12
0
0
I A A A AA
A A I A A A A A A
21A 1
22 21 11 12A A A A
11A A
21A
( ) ( )m n x m n 1
22 21 11 12A A A A
11 12
1
21 11 21 22
0I A AA
A A I A A
11 12
1 1
21 21 11 11 22 21 11 12
AA A
A A A A A A A A
11 12
1
22 21 11 12
A0
A A
A A A A
1111 12 11 12
21 ( )x(m n)21 22 22
0
0m n
LA A U U
L IA A A
4
11 11 11A L U
12 11 12A L U
21 21 11A L U
22 21 12 22A L U A
1 1 1
22 22 21 12 22 21 11 11 12 22 21 11 12A A L U A A U L A A A A A
5
2 1 1 0
4 3 3 1
8 7 9 5
6 7 9 8
A
2 1 1 0 1 0 0 0 2 1 1 0
4 3 3 1 2 1 0 0 0 1 1 1
8 7 9 5 4 3 1 0 0 0 2 2
6 7 9 8 3 4 1 1 0 0 0 2
A
1 0 0 0 8 7 9 5
3 7 9 170 0 1 0 2 1 1 0 1 0 0 04 4 4 40 0 0 1 4 3 3 11 2 6 2
0 1 0 0 8 7 9 5 1 0 0 02 7 7 7
1 0 0 0 6 7 9 81 3 1 2
1 0 0 04 7 3 3
6
det( ) det( ) 2 1 2 2 8iiA U u
7 6 2det( ) det( ) 8 8
4 7 3iiPA U u
det( ) 1P
det( ) det( ) det( ) 8PA P A
det( ) 8A
det( ) ( 1) fP
det( ) det( ) det( ) det( ) iiPA P A U u
7
1det( )
( 1)iif
A u
8
1Q 1Q
1 1L AQ
1 1 1 1... ...m mL L AQ Q U
1 1
1 1... mL L L
0 0
1 1A
0 0 1 0
1 1 1 0
b c b cA
a d ab ac d
0 0
1 1
b c
ab ac d
9
10
, 0jj jjr u
( ) ( )A A QR QR
( )A A R Q Q R
A A R R
A A U U
R R U U
R R
R U
11
mxmA
A I
1 1A
v
Av v
v
Av v v v
( ) ( )A I v v
A I
12
2
0 1 2( ) ... m
mP A c c c c
0 1 2, , ,..., mc c c c
2
0 1 2( ) ... m
mP A c c c c
( ) 0P A
( ) 0P A
v
Av v
1v A v
1 1A v v
13
1 1A
0 0
1 0A
0A
A Q Q
1A X X
I
1A XX I
14
2
tAe
15
1, 0n nh
nK n nAK K
1, ,..., k
nK b Ab A b 1 2 ...n n nK K K
Hn
Ax b
nK
1, 0n nh
1n n n n nAQ Q H Q H
0
0 ...
nHH
1
1 2, ,..., , ,...,n
n nK b Ab A b q q q
n n nAQ Q H
16
1 2 1 2... ...n n nAq Aq Aq q q q H
iAq 1,2,...,i n
1 2 ... nq q q
1 2, ,...,n nx AK Aq Aq Aq
1 2 ... nq q q
1 2, ,...,n nx K q q q
n nAK K
1
1 2, ,..., , ,...,n
n nK b Ab A b q q q
1
1 , ,..., ,n n
nK b Ab A b A b
1
1 1 2 2 ...n
n nA b c q c q c q
1
1 1 2 2( ) ( ) ( ) ... ( )n n
n nA b A A b c Aq c Aq c Aq
17
AQ Qn n nH
i nAq K 1,2,...,i n
n
nA b K
1n nK K
1n nK K 1n nK K
1 1 2, ,..., , ,...,n
n nK b Ab A b q q q
1
2 , ,..., ,n n
nK b Ab A b A b
1
1 1 2 2( ) ( ... )n n
n nA b A A b A d q d q d q
1
1 1 2 2( ) ( ) ... ( )n
n nA b d Aq d Aq d Aq
1
1
n
n nA b K K
18
1 2n nK K
1 2 ...n n nK K K
n n nAQ Q H
n n nA Q H Q
Ax x
n n nQ H Q x x
nQ
n n n n nQ Q H Q x Q x
( ) ( )n n nH Q x Q x
nH
nH
1 2 1 2... ...n n nAq Aq Aq q q q H
19
1
1 2 1 2... ...n n nAq Aq Aq H q q q
kq
1 2, ,..., nAq Aq Aq
n nK AK
1 2, ,...,n nK q q q
1 2,A ,...,An nAK Aq q q
n nAK K
n nAK K
1 n nb b q K AK
1 1 2 2 ... n nb c Aq c Aq c Aq
1 1 2 2( ... )n nb A c q c q c q
1 1 2 2 ... n nx c q c q c q
nx K
