7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

1/64

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P W D

M D

D. A MS M F

F E C U

FALL 2014

D. A M N A F 2014

N

.

A

.

T

.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

2

D. A M N A F 2014

1. A ( ):

E .

A , ,

.

L

R , ,

.

D. A M N A F 2014

2. G :

C

U

N .

E . L

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

3

D. A M N A F 2014

3. C:

I

A .

B .

C :

D. A M N A F 2014

W N :

A .

C ,

, .

D ,

.

M

. Y

.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

4

D. A M N A F 2014

R

S

O

C

I

O

P

D. A M N A F 2014

R

T

.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

5

D. A M N A F 2014

S

T .

I ,

.

D. A M N A F 2014

O

D

.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

6

D. A M N A F 2014

C

F .

T : R I.

R

.

E., E .

D. A M N A F 2014

C

I

.

E., .

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

7

D. A M N A F 2014

I

A

.

D. A M N A F 2014

O

M

.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

8

D. A M N A F 2014

P

A

.

D. A M N A F 2014

S

R

C

O

I

O

P

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

9

D. A M N A F 2014

B}{[A]{X}=

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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10

D. A M N A F 2014

C 2

R 1

R

C

D. A M N A F 2014

R , = 1

C , = 1

S , =

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

11/64

24 9 2014

1

D. A M N A F 2014

S

D

I

11000

0100

0080

0002

1000

01000010

0001

D. A M N A F 2014

U

B

L

E., B = 3( )

11000

9100

4080

6341

11672

0127

0085

0001

11600

6120

0385

0041

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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12

D. A M N A F 2014

=

438

2127

541

438

2127

541

10382127

541

4382127

541

D. A M N A F 2014

=

+

152

123

90

82

62

41

=

3241

9082

6241

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

13

D. A M N A F 2014

=

+

152123

9082

6241

=

+

152

123

62

41

90

82

D. A M N A F 2014

=

8616

42414

1082

438

2127

541

2x

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

14

D. A M N A F 2014

D. A M N A F 2014

[ ][ ] [ ]CBA =

[ ]?27

95

40

68

13

=

3 2 2 2

3 2

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

15

D. A M N A F 2014

+

=

??

??

?7x13x5

27

95

40

68

13

+

=

??

??

1x23x922

27

95

40

68

13

+=

??

?6x78x5

2922

27

95

40

68

13

=

828

8482

2922

27

95

40

68

13

D. A M N A F 2014

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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16

D. A M N A F 2014

A1 A

I A :

=

27

95AEx.,

=

57-

9-2

7925

1A

1

xx

D. A M N A F 2014

AT A

I A :

=

1098

765

432

AEx.,

=

1074

963

852

AT

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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17

D. A M N A F 2014

A A

=

1098

765

432

AEx., 181062tra[A] =++=

D. A M N A F 2014

F

D:

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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18

D. A M N A F 2014

F 2 2

E.,

D. A M N A F 2014

F 3 3

E.,

D = 0.3 0.52 + 1

D = 0.3 (10.5 0.3 1.9) 0.52 0.06 + 1 0.05 = 0.0022

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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19

D. A M N A F 2014

A

D. A M N A F 2014

[A]

B}{{X} =

N M

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

20

D. A M N A F 2014

D. A M N A F 2014

S

T G M

C R

T E U

G E

GJ

LU D

G S

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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2

D. A M N A F 2014

S ( 3)

D. A M N A F 2014

E., S

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22

D. A M N A F 2014

E., S

D. A M N A F 2014

E., S

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23

D. A M N A F 2014

I ?

P L : N (S )

D. A M N A F 2014

I ?

C L : I (S )

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24

D. A M N A F 2014

I ?

I

D. A M N A F 2014

T

. F , 1

:

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25

D. A M N A F 2014

E.,

D. A M N A F 2014

E.,

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26

D. A M N A F 2014

E.,

D. A M N A F 2014

E.,

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27

D. A M N A F 2014

T

.

T

.

T

.

D. A M N A F 2014

21

11

C

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28

D. A M N A F 2014

E.,

D. A M N A F 2014

T :

1. T

.T

.

2. C,

.

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29

D. A M N A F 2014

D. A M N A F 2014

E.,

4.71102.03.0

3.193.071.0

85.72.01.03

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30

D. A M N A F 2014

4.71102.03.0

3.193.071.0

85.72.01.03 0.1/3

4.71102.03.0

5617.1929333.000333.70

85.72.01.03 0.3/3

615.7002.1019.00

5617.1929333.000333.70

85.72.01.03

P

P 0.19/7.003333

D. A M N A F 2014

0843.700120.1000

5617.1929333.000333.70

85.72.01.03

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

31/64

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3

D. A M N A F 2014

0843.700120.1000

5617.1929333.000333.7085.72.01.03

D. A M N A F 2014

0843.700120.1000

5617.1929333.000333.70

85.72.01.03

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

32

D. A M N A F 2014

1. D ( )

S: P

P=0

D. A M N A F 2014

2. R

I 100

Y

.

S: &

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33

D. A M N A F 2014

E.,

211

1000001000002

49998499990

1000001000002

199998.049999-

49998x 2 =

=

000,1001000,1002x1 =+ x 0x1=

99998.0x,00002.1x 21 ==

C

D. A M N A F 2014

U

211

1000001000002

49998499990

1000001000002

99998.049999-

49998x 2 =

=

000,10099998.0000,1002x1 =+ x 1x1=

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

34

D. A M N A F 2014

U 1

211

1100002.0

1100002.0

211P

99996.099998.00

21199998.0

0.99998

99969.0x 2 ==

00002.1x1 =

D. A M N A F 2014

3. S (D = 0)

4. I

I

.

B

,

.

D 0 ( )

S:

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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35

D. A M N A F 2014

E.,

4.1021.1

1021

6.02.00

1021

30.2

0.6x 2 =

=

1032x1 =+ x 4x1=

D. A M N A F 2014

R 1.1 1.05

4.10205.1

1021

1.01.00

1021

10.1

0.1x 2 =

=

1012x1 =+ x 8x1 =

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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36

D. A M N A F 2014

R 1.1 1.05

1x,8x 21 ==3x,4x 21 ==

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10

1+22=101.11+22=10.4

I

D. A M N A F 2014

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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37

D. A M N A F 2014

T GJ G .

T

GJ ,

.

I ,

. T,

.

H G E.

D. A M N A F 2014

E.,

/3

0.1 0.3

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38

D. A M N A F 2014

/7.00333

0.0333333 0.19

/10.012

D. A M N A F 2014

0.0680629 0.0418848

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

39

D. A M N A F 2014

T

, .

I

.

O

. T

.

I .

D. A M N A F 2014

U

U

A L

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

40

D. A M N A F 2014

D. A M N A F 2014

A :

1. LU . A

L U .

2. S . L U

X B. T

:

G D .

T, X.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

41/64

24 9 2014

4

D. A M N A F 2014

A =

(21/11) (31/11)

'

33

'

32

'

23

'

22

131211

0

0

aa

aa

aaa

(32/22)

D. A M N A F 2014

C

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42

D. A M N A F 2014

F

B

D. A M N A F 2014

A

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43

D. A M N A F 2014

D. A M N A F 2014

L = B =

D

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44

D. A M N A F 2014

C # 59

D. A M N A F 2014

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45

D. A M N A F 2014

D

X

F

A1

D. A M N A F 2014

T , :

1

0

0

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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46

D. A M N A F 2014

C AA1 = I

D. A M N A F 2014

F :

1. S A

1. I

A1 ,

.

H M

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47

D. A M N A F 2014

F :

2. M

. I ,

.

3. I

. I ,

.

D. A M N A F 2014

=

?00

??0

???

1??

01?

001

=

100

?10

??1

???

0??

00?

=

c

b

a

c

b

a

00

?0

??

??

0?

00

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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48

D. A M N A F 2014

D

=

33

22

11

33

22

11

00

?0

??

??

0?

00

l

l

l

l

l

l

D. A M N A F 2014

= 1, =

= 2, = 1

= 3, = 1

= 2, =

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

49

D. A M N A F 2014

= 3, = 2

= 3, =

D. A M N A F 2014

T

.

H,

.

B , , , C

.

P : > 0, X

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

50

D. A M N A F 2014

[ ] [ ] 010

0122>+=

=

ba

b

aba

b

aba

I I

=

210

121

012

M

[ ] 0)()(210

121

012

2222 >+++=

ccbbaa

c

b

a

cba

M

D. A M N A F 2014

=

12

21N

[ ] )0(412

2122

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

51/64

24 9 2014

5

D. A M N A F 2014

BW:

HBW:

BW = 2HBW + 1

G

, .

N

. T,

.

A : T A

D. A M N A F 2014

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52

D. A M N A F 2014

U = L =

D

O

LU

D. A M N A F 2014

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53

D. A M N A F 2014

R =

F

R =

D. A M N A F 2014

T =

B

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54

D. A M N A F 2014

A I

.

T

.

T

.

D. A M N A F 2014

1. A .

2. S 1.

3. S 1

3 2.

4. R 3.

5. R 1 4 :

(1)

(2)

(3)

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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55

D. A M N A F 2014

T

D. A M N A F 2014

1 0 0 0

2 2.616667 2.794524 7.005610

E 100% 100% 100%

3 2.990557 2.499625 7.000291

12.5% 11.8% 0.076%

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7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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57

D. A M N A F 2014

1. S, .

=

911

1311][A

(N)

11 < 13

9 < 11

1 0 0

2 26 20.77778

3 1.444442 9.23457

4 36.91358 34.1166

5 14.3196 28.5018

6 59.68395 61.94704

7 47.2101 68.7013

8 107.1924 120.013

D. A M N A F 2014

1. S, .

=

1311

911][A

C

11 > 9

13 > 11

1 0 0

2 9 14.38462

3 2.76923 24.3432

4 10.9172 31.2376

5 16.558 36.01065

6 20.4633 39.31507

7 23.1669 41.60274

8 25.0386 43.18651

B

!

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58

D. A M N A F 2014

2. W , .

GS

.

A ,

:

I= 0, .

I= 0 1,

I= 1 2, , (SOR)

D. A M N A F 2014

A GS ,

. T, ,

.

A ,

. R

,

.

T, ,

.

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

59/64

24 9 2014

59

D. A M N A F 2014

/

G S J I

D. A M N A F 2014

E F

S R

(R I F)

S

(G, ,..)

B}{[A]{X}=

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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24 9 2014

60

D. A M N A F 2014

F .

F1, F2, F3

.

H2, V2, V3

().

1000 I (F1,)

D. A M N A F 2014

T

.

F

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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6

D. A M N A F 2014

A (1):

A (2):

A (3):

D. A M N A F 2014

U

(LU , ?)(P )

v

h

v

h

v

h

F

F

F

F

F

F

,3

,3

,2

,2

,1

,1

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

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62

D. A M N A F 2014

0

0

0

1000

0

1000

B =

0

1000

0

0

0

1000

B =

v

h

v

h

v

h

F

F

F

F

F

F

,3

,3

,2

,2

,1

,1

B =

D. A M N A F 2014

v

h

v

h

v

h

F

F

F

FF

F

,3

,3

,2

,2

,1

,1

7/25/2019 Part I_Linear Algebraic Equations_Computers and Numerical Analysis

63/64

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63

D. A M N A F 2014

T

.

E

. F

, 132 (3)

0.866

(1). T,

1,3 0.866.

D. A M N A F 2014

T 0

. F 113 = 0

1 2,.

T

, , ,

.I ,

.

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D. A M N A F 2014

M

M

E