Tema de casa Metode numerice Universitatea: Politehnica Bucuresti Facultatea: I.S.B. Student: Krancevik Valentin Grupa: 714 Numar condica: 16
Tema de casa
Metode numerice
Universitatea: Politehnica Bucuresti
Facultatea: I.S.B.
Student: Krancevik Valentin
Grupa: 714
Numar condica: 16
Aplicatia 1
Sa se calculeze erorile absolute si relative ale marimii x , daca ncat este pozitia in catalog a
studentului iar:
Valoarea exacta
valoarea aproximativa
Aplicatia 2
Sa se calculeze erorile absolute si relative ale marimii x , daca ncat este pozitia in catalog
a studentului iar:
Valoarea exacta
valoarea aproximativa
xapr 0.9 ncat
Eabs xex xapr
Eabs 0.1
Erel
Eabs
xex
Erel 5.882 103
xex 0.0004 ncat
xapr 0.0003 ncat
Eabs xex xapr
Eabs 10 105
ncat 16
Aplicatia 3
Sa se calculeze valoarea expresiei f = x* y* / z* , pentru :
f = x* y* / z*
x* = 5+/- 0.100 , y* = 6+/- 0.150 , z* = 10+/- 0.200 (valorile aproximative ale marimilor)
(erorile absolute limita)
(valorile exacte ale parametrilor)
ffin = fex +/- Δf
Erel
Eabs
xex
Erel 6.25 106
x 0.100 y 0.150 z 0.200
x 5 ncat y 6 ncat z 10 ncat
x 21 y 22 z 26
fex xy
z
fex 17.769
xy y x x y
xy 5.35
f z xy x y z
f 231.5
f
x
x
y
y
z
z
f 0.019
ffin fex f 0if
fex f fex f
Aplicatia 4:
x* = 50+ncat [Hz]
Aplicatia 5:
ffin 213.731 249.269( )
0.05
xaprox 50 ncat
x xaprox
x 66.05
xexact xaprox 0if
xaprox x xaprox x
xexact 0.05 132.05( )
M ncat
8
7
3
2
7
1
0
4
3
0
5
9
2
4
9
2
M1
1.282 104
8.219 103
3.218 103
1.83 103
8.219 103
9.723 103
1.679 103
3.672 103
3.218 103
1.679 103
1.189 103
5.211 103
1.83 103
3.672 103
5.211 103
2.285 103
MT
128
112
48
32
112
16
0
64
48
0
80
144
32
64
144
32
Aplicatia 6:
M 3.513 108
max M( ) 144
min M( ) 80
eigenvecsM( )
0.144
0.098
0.804
0.569
0.495
0.838
0.142
0.181
0.766
0.536
0.047
0.352
0.385
0.026
0.576
0.721
eigenvalsM( )
190.535
63.897
218.138
132.294
A ncat
1
5
7
2
5
8
3
6
9
B ncat
1
7
13
7
9
15
5
11
17
S A B D A B
D
0
32
320
80
224
112
32
272
128
S
32
192
96
144
64
368
128
80
416
5 E A
E
80
400
560
160
400
640
240
480
720
P A BP
6.144 103
2.765 104
1.382 104
1.792 104
2.048 104
6.554 104
8.704 103
4.659 104
2.56 104
Aplicatia 7:
Aplicatia 8:
B4
1.929 109
1.424 109
3.802 109
2.204 108
2.814 109
7.556 108
2.348 109
2.133 108
4.8 109
M ncat
8
7
3
2
7
1
0
4
3
0
5
9
2
4
9
2
r 0 rows M( ) c 0 cols M( ) 5
Matricea M r c( ) B MT
Br
MT
r
MT
c
BT
Matricea M 0 2( )
112
112
48
32
112
16
0
64
448
0
80
144
752
64
144
32
f x( ) x5
20x3
15ncat x 1.5 5
x5
20x3
15ncat 0
x 3
Given
Aplicatia 9:
Aplicatia 10:
S Find x( )
S 0
g x( ) 0.01ncat 11.09 24.13 e0.1 x
e0.5 x
x 5 5
g' x( )x
g x( )d
d
N 10
xo 4
NR g x N( )
x xg x( )
xg x( )
d
d
i 0 Nfor
x
NR g 5 5( ) 0.195
h x( ) x4
4x3
4x2
0.5 ncat x 5 5
N 10
x0 4
Secanta h x n( )
x1 x
x2 x 106
x x1 h x1 x2 x1
h x2 h x1
i 0 Nfor
x
Aplicatia 11:
Aplicatia 12:
Secanta h 5 5( ) 0.957
q 0.4 Mt 9.188 105
adm 86.16 x 15 55
p x( )ncat
2adm
5.1 Mt
x3
1 q4
N 10
Bisectiep a b N( ) x1 a
xu b
xm
x1 xu
2
x1 xm p x1 p xm 0if
xu xm p x1 p xm 0if
i 0 Nfor
x1 xu
2
Bisectiep 15 55 N( ) 19.111
5x1 3x2 2x3 6ncat
4 x1 7x2 x3 4ncat
3 x1 2x1 6x 5ncat
M
5
4
3
3
7
2
2
1
6
v
6ncat
4ncat
5ncat
v
96
64
80
Aplicatia 13:
Solutia Matricii Inverse:
x1
x2
x3
lsolveM v( )
x1
x2
x3
16
16
16
0.3w 0.2x 6.6y 1.1z 1ncat
4.5w 1.8x 0.3y 6.5z 0.1ncat
7.3 w 9.7x 10.9y 4.1z 0.01ncat
8.1w 2.7x 8.7y 8.9z 0.001ncat
M
0.3
4.5
7.3
8.1
0.2
1.8
9.7
2.7
6.6
0.3
10.9
8.7
1.1
6.5
4.1
8.9
v
1
0.1
0.01
0.001
x M1
x
3.538
2.697
0.683
1.734
4.023
2.813
0.633
2.189
0.026
0.133
2.068 103
0.019
2.513
1.783
0.379
1.263
w
x
y
z
lsolveM v( )
w
x
y
z
3.937
2.975
0.746
1.952
Solutia Gauss-Siedel:
ORIGIN 1
n rows M( )
C augmentM v( )
C
0.3
4.5
7.3
8.1
0.2
1.8
9.7
2.7
6.6
0.3
10.9
8.7
1.1
6.5
4.1
8.9
1
0.1
0.01
1 103
EliminareNecNpasi( ) C C
miuCi k
Ck k
Ci j Ci j miuCk j
j k n 1for
Uk C
i k 1( ) nfor
k 1 n 1for
UNpasi
Npasi 1 n 1
SupTriunghi submatrixEliminareNecn 1( ) 1 n 1 n( )
vnou submatrixEliminareNecn 1( ) 1 n n 1 n 1( )
SubstitutiaInapoi m SupTriunghi
v vnou
xnvn
mn n
suma 0
suma suma mi j xj
j i 1( ) nfor
xi
vi suma
mi i
i n 1 n 2 1for
x
Aplicatia 14:
Aplicatia 15:
Solutiasistem SubstitutiaInapoi
Solutiasistem
0
0
0
1.952
L 1m
F1 ncat 200 N F2 ncat 150( ) N
x1 0.152 m x2 0.61 m
R1 0 N R2 0 N
Given
R1 R2 F1 F2 0
R2 L F1 x1 F2 x2 0
R1
R2
Find R1 R2
R1 1.778 104
R2 9.776 103
l 0.4 1.1 P ncat
Given
Aplicatia 16:
Aplicatia 17:
P 4l( ) sin( ) l sin 2( ) 0
P 2l( ) l cos ( ) l2
cos 2( ) 0
l
Find l ( )
l
0
0
x 1 y 1 z 1
Given
sin x( ) y2
ln z( ) 7 0.05ncat 0
3x 2y
z3
1 0.05 ncat 0
x y z 5 0.05ncat 0
SolutiaSistem Find x y z( )
SolutiaSistem
0.012
2.246
1.766
X
1.1
1.2
1.3
1.4
1.5
ncat Y
1.102
1.332
1.445
1.697
1.923
ncat
N rows X( )
Aplicatia 18:
Origin 1
Dif_finiteN X Y( )
Si 1 Xi
Si 2 Yi
i 1 Nfor
Si j Si j 1 Si 1 j 1
S i 1 ( ) Si j 2 1 0
i j 1 Nfor
j 3 N 1for
S
DiffFin Dif_finiteN X Y( )
Dif_finiteN X Y( )
17.6
19.2
20.8
22.4
24
17.632
21.312
23.12
27.152
30.768
0
3.68
1.808
4.032
3.616
0
0
1.872
2.224
0.416
0
0
0
4.096
2.64
0
0
0
0
6.736
x
2.441 103
3.346 103
4.291 103
5.079 103
5.709 103
5.984 103
6.22 103
6.457 103
y
206.66 ncat
184.44 ncat
140 ncat
106.66 ncat
62.22 ncat
17.77 ncat
4.44 ncat
26.66 ncat
Aplicatie 19:
F x ( ) x 1
e x
n last y( ) i 0 n
sse ( )
i
yi F xi 2
0.8 1
Given
sse ( ) 0
Minerr ( )
0.8
1
f x( ) ncat e2x 1
i 1 9 h 0.01
x0 1.5 5 h xi x0 i h
yi f xi
i 5 xi 1.5
der1syi yi 1
h der1d
yi 1 yi
h der1c
yi 1 yi 1
2 h
der1s 234.101 der1d 238.83 der1c 236.466