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