ÿþM a r e k S k o w r o Ds k i
U k Ba d z a m k n i t y
P r o s t y u k Ba d p o m p o w y
O b l i c z a n i e u k Ba d ó w p o m p o w y c h
G r a f o b i e k t u p r z e p By w o w e g o
C h a r a k t e r y s t y k i o b i e k t ó w p r z e p By w o w y c h
S u m a s z e r e g o w a
P r o s t y u k Ba d p o m p o w y
C h a r a k t e r y s t y k a z a s t p c z a
S u m a r ó w n o l e g Ba
P o Bc z e n i a
U k Ba d y d r z e w i a s t e d e n d r y t y
A l g o r y t m o b l i c z a n i a d e n d r y t ó w
O b l i c z e n i a s i e c i p i e r [c i e n i o w y c h
1 2
U k Ba d z a m k n i t y P r o s t y u k Ba d p o m p o w y
C i e p Bo
o d p r o w a d z a n e
z u k Ba d u
H g Áz a Áp
o
T z T p
a o
Q
P k
H
M o c c i e p l n a
k o t Ba
Q
G r a f o b i e k t u p r z e p By w o w e g o G r a f o b i e k t u p r z e p By w o w e g o
Q 1 2
Q i j
1 2
h 1 2
w z e B i w z e B j
o b i e k t
H 1
H 2
0 p o z i o m o d n i e s i e n i a
H 1 - H = h 1 2
2
h 1 2 = f 1 2 ( Q 1 2 )
C h a r a k t e r y s t y k i o b i e k t ó w p r z e p By w o w y c h
H
P o m p a
Q
( - ) H
O p e r a c j e p o d s t a w o w e
Z a w ó r
Q
( - ) H
R u r a
Q
S u m a s z e r e g o w a
( - ) H ( - ) H
R u r a 2
H
Q Q
Q 1 2 Q 2 3
R u r a 1
S u m a s z e r e g o w a r u r
Q = Q 1 2 = Q 2 3
( - ) H ( - ) H
"h = "h 1 2 + "h 2 3
R u r a 2
R u r a 1
Q
Q Q
C h a r a k t e r y s t y k a w g a Bz i
P r o s t y u k Ba d p o m p o w y
C h a r a k t e r y s t y k a z a s t p c z a
H H
( - ) H
Q
Q Q
H
Q
S u m a r ó w n o l e g Ba S u m a r ó w n o l e g Ba
Q 1 Q 1
H 3
H 2
Q Q Q
Q 2 Q 2
Q = Q 1 + Q 2 H Q = Q 1 + Q 2 H
"h = "h 1 = "h 2 "h = "h 1 = "h 2
H 3 H 2
Q
Q
H + h 1 2 = H + h 1 3
2 3
( - ) H ( - ) H
P o Bc z e n i a
R u r a 2
Q 0 1 - D 1 = Q 1 2 + Q 1 3
H 1 - H = h 1 2 Q 1 2 = Q 2 3 - D 2
2
Q Q
H - H 3 = h 2 3
2
Q 0 1 + Q 2 1 + Q 3 1 = D 1
R u r a 1
S u m a r ó w n o l e g Ba r u r
( - ) H
R u r a 2
Q
R u r a 1
h 1 2 + h 2 3 + h 3 1 = 0
A l g o r y t m o b l i c z a n i a d e n d r y t ó w
W y b i e r z p i e D d r z e w a
O b l i c z c h a r a k t e r y s t y k
k o r o n y b e z p n i a
O b l i c z c h a r a k t e r y s t y k k o r z e n i b e z
p n i a
D o c h a r a k t e r y s t y k i k o r o n y d o d a j
c h a r a k t e r y s t y k p n i a
c h a r a k t e r y s t y k a k o r o n y
p i e D w z e B c e n t r a l n y
O b l i c z p u n k t p r z e c i c i a c h a r a k t e r y s t y k
( k o r o n a + p i e D) i k o r z e n i e
c h a r a k t e r y s t y k a k o r z e n i
W y s o k o [ c i [n i e n i a w p u n k c i e c e n t r a l n y m H
N a t \e n i e p r z e p By w u w p n i u d r z e w a Q
O b l i c z n a t \e n i a p r z e p By w u w g a Bz i a c h
O b l i c z w y s o k o [c i c i [n i e n i a w z Ba c h
U k Ba d y d r z e w i a s t e d e n d r y t y
K O N I E C
C h a r a k t e r y s t y k a o b i e k t u p r z e p By w o w e g o S u m a c h a r a k t e r y s t y k
H j G j k
k
Q i j
Q j k
- h
h k
H i H j
i
Q i j
i j
Q
Q
"
n
îø
"
G ( Q ) = h j k ( Q ) + H ( Q ) H ( Q i j ) = ( H ) ùø
j k j k j k k j k j "G j k
ïø úø
ðø k = 1 ûø
h i j ( Q i j ) = H i ( Q i j ) - H ( Q i j )
j
S u m a s z e r e g o w a S u m a r ó w n o l e g Ba
( c h a r a k t e r y s t y k a g a Bz i ) ( c h a r a k t e r y s t y k a w w zl e )
S u m a s z e r e g o w a
S u m a r ó w n o l e g Ba
( c h a r a k t e r y s t y k a g a Bz i )
( c h a r a k t e r y s t y k a w w zl e )
G
H
G j 1 G j 2 G j 1 G j 2
H k H
H
H h j k Q
H j
G j k
H * j
Q j k
G * j 2
h
G * j 1
Q
Q H
Q
G ( Q ) = h j k ( Q ) + H ( Q )
j k j k j k k j k
"
n
îø
"
S u m a c h a r a k t e r y s t y k i w w zl e k o Dc o w y m H ( Q i j ) = ( H ) ùø
j "G j k
ïø úø
ðø k = 1 ûø
i
c h a r a k t e r y s t y k i o b i e k t u
E t a p y o b l i c z e D u k Ba d u
O b l i c z a n i e c h a r a k t e r y s t y k i
k o r o n y
W y b i e r z w z e B k o Dc o w y
A l g o r y t m Bc z e n i a
O b l i c z c h a r a k t e r y s t y k
W z e B s t a r t o w y
s i e c i d l a w z Ba k o Dc o w e g o
R e k u r e n c j a
O b l i c z c h a r a k t e r y s t y k ( w z e B s t a r t o w y )
W y z n a c z p u n k t p r z e c i c i a c h a r a k t e r y s t y k i s i e c i i l i n i i w y s o k o [c i
S t w ó r z o b i e k t c h a r a k t e r y s t y k i c i [n i e n i a w z Ba k o Dc o w e g o
p i e D p i e D
w z Ba s t a r t o w e g o
w z e B k o Dc o w y w z e B k o Dc o w y
O b l i c z n a t \e n i e p r z e p By w u
w p n i u d r z e w a Q
D o d a j d o c h a r a k t e r y s t y k i
N I E C z y j e s t T A K w z Ba s t a r t o w e g o
n a s t p n a g a Bz
O b l i c z n a t \e n i a p r z e p By w u w g a Bz i a c h
J e s t O b l i c z w y s o k o [c i c i [n i e n i a w z Ba c h
N I E C z y m a T A K
c h a r a k t e r y s t y k a
c h a r a k t e r y s t y k
U s t a l w z e B s t a r t o w y
K O N I E C
n a k o Dc u g a Bz i
K O N I E C
O b l i c z e n i a s i e c i p i e r [c i e n i o w y c h
D r u g i e p r a w o K i r h o f f a
( r e g u Ba p i e r [c i e n i a )
P i e r w s z e p r a w o K i r h o f f a
Q 2
( r e g u Ba w z Bó w )
Q 2
Q 1
"h 1 2 2
Q 1
"h 2 3
"Q = 0
1
Q 3
"h 3 1 Q 3
3
Q 2
Q 2
Q 1 Q 4 ""h = 0
Q 1
Q 3
Q 3
M e t o d y o b l i c z a n i a u k Ba d ó w p i e r [c i e n i o w y c h
M e t o d a H a r d y C r o s s
M e t o d a C R O S S A
M e t o d a I s a a c N e w t o n
M e t o d a
E z i o T o d i n i , S t e f a n o P i l a t i
Z a s a d a m e t o d y C R O S S A
P o j e d y n c z y p i e r [c i e D s i e c i
q 2
= 0
"Q i
Q 1 2 2
q 1
Q 2 3
1
q 3
Q 3 1
3
"h 1 2 + "h 2 3 + "h 3 1 = 0
"h = a Å" s g n ( Q ) Å" Q 2
`" 0
""h i j
M e t o d a H . C r o s s a
2
"h = a Å" s g n ( Q + "Q ) Å"( Q + "Q )
q 2
"h = a Å"s g n ( Q + "Q ) Å"( Q 2 + 2 Q Å" "Q + "Q 2 )
Q 1 2 2
q 1 "Q 2 H" 0
Q 2 3
1
M e t o d a N E W T O N A
s g n ( Q + "Q ) ’! s g n ( Q )
Q 3 1 q 3
3
a Å"s g n ( Q ) Å"Q 2 Å" 2 "Q
"h = a Å"s g n ( Q ) Å"Q 2 +
Q
-
""h i
"Q =
"h i
2 "
"
Q 1
M e t o d a N E W T O N A
M e t o d a N E W T O N A
" Q j = 0
Q 2
= 0
"Q i
"h i = f i ( Q i )
"h 1 2 2
Q 1
"h 2 3
1
" "h k = 0
"h 3 1 Q 3
3
" f i ( Q i + "Q k ) = 0
k - p i e r [c i e n i
" f i ( Q i + "Q 1 ) = 0
k - r ó w n a D
" f i ( Q i + "Q 2 ) = 0
. . .
k - p o p r a w e k
" f i ( Q i + "Q k ) = 0
`" 0
""h i j
M e t o d a N E W T O N A
f ( Q n ) = t g ( ±) = f ( Q n ) / ( Q n Q n + 1 )
M e t o d a G r a d i e n t o w a
T o d i n i , P i l a t t i
"Q n = Q n + 1 - Q n
'
f ( Q n ) Å" "Q n = - f ( Q n )
M e t o d a g r a d i e n t o w a
M e t o d a G R A D I E N T O W A
( T o d i n i , P i l a t t i )
H 3
3
Q 1 3
`" 0
`" 0 ""h i j
"Q i
2 2 H 1
H i - H = h i j = r "Q i j + m "Q i j H 2 Q 1 2
j
D 1
2 1
Q 1 4
4
- D i = 0 d l a i = 1 , . . . N .
"Q i j H 4
j
D 1 = ( Q 1 2 - "Q 1 2 ) + ( Q 1 3 - "Q 1 3 ) + ( Q 1 4 - "Q 1 4 )
M e t o d a g r a d i e n t o w a ( T o d i n i , P i l a t t i ) M e t o d a g r a d i e n t o w a ( T o d i n i , P i l a t t i )
- Q i j - D i = 0 d l a j = 1 , . . . N
"
d h i j
j
= 2 r Å" Q i j + 2 m Å" Q i j
d Q i j
Q n + 1 = Q n + "Q n
1 1
p i j = =
d h i j 2 r Å" Q i j + 2 m Å" Q i j
1
d Q i j
"Q n = Å"( "H n - h n )
d h
d Q n
M e t o d a g r a d i e n t o w a ( T o d i n i , P i l a t t i )
M e t o d a g r a d i e n t o w a ( T o d i n i , P i l a t t i )
( p 1 2 + p 1 3 ) H 1 - p 1 2 Å" H 2 = - Q 1 2 - Q 1 3 - D 1 + p 1 2 Å" h 1 2 + p 1 3 Å" h 1 3 + p 1 3 Å" H 3
- p 2 1 Å" H 1 + ( p 2 1 + p 2 3 ) H 2 = - Q 2 1 - Q 2 3 - D 2 + p 2 1 Å" h 2 1 + p 2 3 Å" h 2 3 + p 2 3 Å" H 3
( Q 1 2 + "Q 1 2 ) + ( Q 1 3 + "Q 1 3 ) + D 1 = 0
( p 1 2 + p 1 3 ) - p 1 2 H 1 - Q 1 2 - Q 1 3 - D 1 + p 1 2 Å" h 1 2 + p 1 3 Å" h 1 3 + p 1 3 Å" H
îø ùø îø ùø îø ùø
3
( Q 2 1 + "Q 2 1 ) + ( Q 2 3 + "Q 2 3 ) + D 2 = 0
ïø úø
- p 2 1 ( p 2 1 + p 2 3 ) úø × ïøH úø = ïø- Q 2 1 - Q 2 3 - D 2 + p 2 1 Å" h 2 1 + p 2 3 Å" h 2 3 + p 2 3 Å" H
ðø ûø ðø 2 ûø ðø 3 ûø
A H = F
1
"Q i j = ( "H i j - h i j ) = p i j ( "H i j - h i j ) = p i j ( H i - H ) - p i j Å" h i j
A s y m e t r y c z n a m a c i e r z J a k o b i e g o ( N x N ) ,
j
d h i j
H - w e k t o r n i e z n a n y c h w y s o k o [c i w w z Ba c h ( N x 1 ) ,
d Q i j F - w e k t o r w a r u n k u p r a w o s t r o n n e g o ( N x 1 ) .
M e t o d a g r a d i e n t o w a ( T o d i n i , P i l a t t i )
A H = F
D i a g o n a l n e e l e m e n t y m a c i e r z y
A i i = p i j
"
j
P o z o s t a Be n i e z e r o w e e l e m e n t y
A i j = - p i j
F i = Q i j - D i + p i j Å" h i j + p i f Å" H
"- " " f W y r a \e n i e p r a w o s t r o n n e
j j f