OBSERVAÇÕES:
k= número de interações que demora para convergir à solução
do método
norma_r= resíduo
vetor zero no Scilab: zeros(n,1)
matriz identidade (In): eye(n,n)
Rodar os seguintes scripts no SCILAB: sor, jacobi e lu.sem
Exercício 1
CRIAR MATRIZ NO SCILAB
-->A=[-1 0 1;1 0 4; 4 2 1]
A =
- 1. 0.
1.
1. 0.
4.
4. 2.
1.
-->b=[0;0;1;1]
b =
0.
0.
1.
1.
JACOBI
-->[x,norma_r,k]=jacobi(A,b,[0;0;0;0],1e-10,100)
k=1 norma_r=1.41421
k=2 norma_r=0.5
k=3 norma_r=0.25
k=4 norma_r=0.125
k=5 norma_r=0.0625
k=6 norma_r=0.03125
k=7 norma_r=0.015625
k=8 norma_r=0.0078125
k=9 norma_r=0.00390625
k=10 norma_r=0.00195313
k=11 norma_r=0.000976563
k=12 norma_r=0.000488281
k=13 norma_r=0.000244141
k=14 norma_r=0.00012207
k=15 norma_r=6.10352e-05
k=16 norma_r=3.05176e-05
k=17 norma_r=1.52588e-05
k=18 norma_r=7.62939e-06
k=19 norma_r=3.8147e-06
k=20 norma_r=1.90735e-06
k=21 norma_r=9.53674e-07
k=22 norma_r=4.76837e-07
k=23 norma_r=2.38419e-07
k=24 norma_r=1.19209e-07
k=25 norma_r=5.96046e-08
k=26 norma_r=2.98023e-08
k=27 norma_r=1.49012e-08
k=28 norma_r=7.45058e-09
k=29 norma_r=3.72529e-09
k=30 norma_r=1.86265e-09
k=31 norma_r=9.31323e-10
k=32 norma_r=4.65661e-10
k=33 norma_r=2.32831e-10
k=34 norma_r=1.16415e-10
k=35 norma_r=5.82077e-11
k =
35.
norma_r =
5.821D-11
x =
0.125
0.125
0.375
0.375
EXERCÍCIO 2
-->A=[1 0 0;0 1 0;1 0 1]
A =
1. 0.
0.
0. 1.
0.
1. 0.
1.
-->b=[1;0;1]
b =
1.
0.
1.
JACOBI
-->[x,norma_r,k]=jacobi(A,b,[0;0;0],1e-10,100)
k=1 norma_r=1.41421
k=2 norma_r=1
k=3 norma_r=0
k =
3.
norma_r =
0.
x =
1.
0.
0.
-->[x,norma_r,k]=jacobi(A,b,[0;0;0],0.1,100)
k=1 norma_r=1.41421
k=2 norma_r=1
k=3 norma_r=0
k =
3.
norma_r =
0.
x =
1.
0.
0.
GAUSS-SEIDEL
-->[x,norma_r,k]=sor(A,b,[0;0;0],1,1e-10,100)
k=1 norma_r=1.41421
k=2 norma_r=0
k =
2.
norma_r =
0.
x =
1.
0.
0.
EXERCÍCIO 3
-->A=[2 1 3;4 2 4;2 0.99 6]
A =
2. 1. 3.
4. 2. 4.
2. 0.99
6.
-->b=[0;2;-3]
b =
0.
2.
- 3.
JACOBI
-->[x,norma_r,k]=jacobi(A,b,[0;0;0],1e-10,100)
k=1 norma_r=3.60555
k=2 norma_r=2.28694
k=3 norma_r=1.60957
k=4 norma_r=2.30302
k=5 norma_r=3.25878
k=6 norma_r=5.10364
k=7 norma_r=7.78771
k=8 norma_r=12.0224
k=9 norma_r=18.4758
k=10 norma_r=28.4511
k=11 norma_r=43.7692
k=12 norma_r=67.3697
k=13 norma_r=103.665
k=14 norma_r=159.542
k=15 norma_r=245.511
k=16 norma_r=377.831
k=17 norma_r=581.439
k=18 norma_r=894.796
k=19 norma_r=1377
k=20 norma_r=2119.1
k=21 norma_r=3261.11
k=22 norma_r=5018.57
k=23 norma_r=7723.15
k=24 norma_r=11885.3
k=25 norma_r=18290.4
k=26 norma_r=28147.4
k=27 norma_r=43316.4
k=28 norma_r=66660.3
k=29 norma_r=102584
k=30 norma_r=157869
k=31 norma_r=242946
k=32 norma_r=373874
k=33 norma_r=575360
k=34 norma_r=885430
k=35 norma_r=1.3626e+06
k=36 norma_r=2.09693e+06
k=37 norma_r=3.22699e+06
k=38 norma_r=4.96606e+06
k=39 norma_r=7.64235e+06
k=40 norma_r=1.17609e+07
k=41 norma_r=1.80991e+07
k=42 norma_r=2.78529e+07
k=43 norma_r=4.28632e+07
k=44 norma_r=6.59628e+07
k=45 norma_r=1.01511e+08
k=46 norma_r=1.56217e+08
k=47 norma_r=2.40405e+08
k=48 norma_r=3.69962e+08
k=49 norma_r=5.6934e+08
k=50 norma_r=8.76166e+08
k=51 norma_r=1.34834e+09
k=52 norma_r=2.07499e+09
k=53 norma_r=3.19323e+09
k=54 norma_r=4.91411e+09
k=55 norma_r=7.56239e+09
k=56 norma_r=1.16379e+10
k=57 norma_r=1.79097e+10
k=58 norma_r=2.75615e+10
k=59 norma_r=4.24148e+10
k=60 norma_r=6.52727e+10
k=61 norma_r=1.00449e+11
k=62 norma_r=1.54583e+11
k=63 norma_r=2.37889e+11
k=64 norma_r=3.66092e+11
k=65 norma_r=5.63384e+11
k=66 norma_r=8.66999e+11
k=67 norma_r=1.33424e+12
k=68 norma_r=2.05328e+12
k=69 norma_r=3.15982e+12
k=70 norma_r=4.86269e+12
k=71 norma_r=7.48327e+12
k=72 norma_r=1.15161e+13
k=73 norma_r=1.77223e+13
k=74 norma_r=2.72731e+13
k=75 norma_r=4.1971e+13
k=76 norma_r=6.45898e+13
k=77 norma_r=9.93982e+13
k=78 norma_r=1.52965e+14
k=79 norma_r=2.35401e+14
k=80 norma_r=3.62261e+14
k=81 norma_r=5.57489e+14
k=82 norma_r=8.57929e+14
k=83 norma_r=1.32028e+15
k=84 norma_r=2.0318e+15
k=85 norma_r=3.12676e+15
k=86 norma_r=4.81182e+15
k=87 norma_r=7.40498e+15
k=88 norma_r=1.13956e+16
k=89 norma_r=1.75369e+16
k=90 norma_r=2.69878e+16
k=91 norma_r=4.15319e+16
k=92 norma_r=6.39141e+16
k=93 norma_r=9.83583e+16
k=94 norma_r=1.51365e+17
k=95 norma_r=2.32938e+17
k=96 norma_r=3.58471e+17
k=97 norma_r=5.51657e+17
k=98 norma_r=8.48953e+17
k=99 norma_r=1.30647e+18
k=100 norma_r=2.01054e+18
k =
100.
norma_r
=
2.011D+18
x =
10^17 *
2.5055056
4.6028788
1.0362131
eliminação gaussiana com pivotamento
-->[L,U,P]=lu(A)
P =
0. 1. 0.
0. 0. 1.
1. 0. 0.
U =
4. 2. 4.
0. - 0.01 4.
0. 0. 1.
L =
1. 0. 0.
0.5 1. 0.
0.5 0. 1.
-->A
A =
2. 1. 3.
4. 2. 4.
2. 0.99 6.
-->P*A
ans =
4.
2. 4.
2.
0.99 6.
2.
1. 3.
-->L*U
ans =
4.
2. 4.
2.
0.99 6.
2.
1. 3.
-->L*U-P*A
ans =
0.
0. 0.
0.
0. 0.
0.
0. 0.
-->y=L\(P*b)
y =
2.
- 4.
- 1.
-->x=U\y
x =
1.5
0.
- 1.
-->A*x
ans =
0.
2.
- 3.
-->b
b =
0.
2.
- 3.
EXERCÍCIO 4
-->A=[1 -1 0;1 1 -1;0 1 1]
A =
1. - 1. 0.
1. 1. - 1.
0. 1. 1.
-->b=[1;0;1]
b =
1.
0.
1.
-->[x,norma_r,k]=sor(A,b,[0;0;0],0.5,1e-10,100)
k=1 norma_r=1.41421
k=2 norma_r=1.11803
k=3 norma_r=0.612372
k=4 norma_r=0.414578
k=5 norma_r=0.233854
k=6 norma_r=0.143205
k=7 norma_r=0.0796722
k=8 norma_r=0.0462194
k=9 norma_r=0.0253154
k=10 norma_r=0.014219
k=11 norma_r=0.00768946
k=12 norma_r=0.00422864
k=13 norma_r=0.00226407
k=14 norma_r=0.00122679
k=15 norma_r=0.000651677
k=16 norma_r=0.00034929
k=17 norma_r=0.000184373
k=18 norma_r=9.80013e-05
k=19 norma_r=5.14631e-05
k=20 norma_r=2.71755e-05
k=21 norma_r=1.42094e-05
k=22 norma_r=7.46368e-06
k=23 norma_r=3.88849e-06
k=24 norma_r=2.03356e-06
k=25 norma_r=1.0562e-06
k=26 norma_r=5.50335e-07
k=27 norma_r=2.85076e-07
k=28 norma_r=1.48077e-07
k=29 norma_r=7.65273e-08
k=30 norma_r=3.96442e-08
k=31 norma_r=2.04467e-08
k=32 norma_r=1.05675e-08
k=33 norma_r=5.44047e-09
k=34 norma_r=2.80607e-09
k=35 norma_r=1.44233e-09
k=36 norma_r=7.42576e-10
k=37 norma_r=3.81138e-10
k=38 norma_r=1.95911e-10
k=39 norma_r=1.00424e-10
k=40 norma_r=5.15452e-11
k =
40.
norma_r
=
5.155D-11
x =
1.
- 1.819D-12
1.
OU
-->[x,norma_r,k]=sor(A,b,[0;0;0],0.5,1e-1,100)
(VAI DAR MENOS INTERAÇÕES)
EXERCÍCIO 5
-->A=[7.0857 0 0 0;-0.2306
6.9014 0.1961 -3.8461e-4;-1.1529 -1.4669 0.9806 -1.9230e-3;1175.93 776.765
-490.2900 1.96154]
A =
7.0857 0.
0. 0.
- 0.2306
6.9014 0.1961 - 0.0003846
- 1.1529
- 1.4669 0.9806 - 0.001923
1175.93
776.765 - 490.29 1.96154
-->[L,U,P]=lu(A)
P =
0. 0. 0. 1.
0. 1. 0.
0.
1. 0. 0.
0.
0. 0. 1.
0.
U =
1175.93 776.765 - 490.29 1.96154
0. 7.0537237 0.0999541
4.821D-08
0. 0. 3.0206225 - 0.0118195
0. 0. 0. 0.0019954
L =
1. 0. 0. 0.
- 0.0001961
1. 0. 0.
0.0060256
- 0.6635482 1. 0.
- 0.0009804
- 0.0999965 0.1688086 1.
-->B=[1
0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]
B =
1.
0. 0. 0.
0.
1. 0. 0.
0.
0. 1. 0.
0.
0. 0. 1.
-->y=L\(P*B)
; x=U\y
x =
0.1411293 - 2.417D-17
4.532D-17 - 3.257D-20
0.0000003 0.1389902 - 0.0277918
6.841D-09
0.0000214 0.1961088 1.9610146
0.0019609
- 84.600831 - 6.0220804 501.16414
0.9999401
CONFERÊNCIA (os valores devem
estar bem próximos de zero)
-->A*x
ans =
1.
- 1.713D-16 3.211D-16
- 2.308D-19
6.939D-18
1. 0. - 5.421D-20
0.
0. 1. 2.168D-19
0.
- 3.553D-15 0. 1.
