ÿþA d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
W y k Ba d 1 0 : M e t o d y n u m e r y c z n e : M e t o d a e l i m i n a c j i G a u s s a
1 . T e r m i n y e g z a m i n u :
1 . K o l o k w i u m : ?
2 . E g z a m i n :
1 . ?
2 . ?
3 . ( p o p r . ) ?
2 . O b l i c z a n i e w a r t o [c i w y z n a c z n i k a
( p r o s z e o b e j r z e i z a s t a n o w i s i e n a d t r e [c i a m i f u n k c j i , s k r y p t ó w , . . . n a d e s Ba n y c h p r z e z s t u -
d e n t ó w )
" K a t a r z y n a Z i m a :
% p r o g r a m n a l i c z e n i e w y z n a c z n i k a p r z e z d o p r o w a d z e n i e m a c i e r z y d o m a c i e r z y t r ó j k at n e j
f u n c t i o n r = w y z n ( A )
[ n , m ] = s i z e ( A )
k = 2 ;
i f n ~ = m
r = n ~ = m ; b r e a k
e n d
i f n = = 1
r = A ( 1 , 1 ) ;
e l s e
f o r j = 1 : ( m - 1 )
f o r i = k : n
A ( i , : ) = A ( i , : ) - A ( j , : ) * A ( i , j ) / A ( j , j ) ;
e n d
k = k + 1 ;
e n d
s = 1 ;
f o r i = 1 : n
s = s * A ( i , i ) ;
e n d
r = s
e n d
" M a r t a S t ó j :
c l c , c l e a r , c l o s e a l l
% O b l i c z a n i e w y z n a c z n i k a z a p o m o c a m e t o d y e l i m i n a c j i
% " A " j e s t t o m a c i e r z d i a g o n a l n a
% " w " j e s t w a r t o s c i a w y z n a c z n i k a
f p r i n t f ( P o d a j m a c i e r z A w p o d a n e j p o n i z e j p o s t a c i \ n ) ;
f p r i n t f ( [ a 1 1 , . . . , a 1 n ; . . . ; a n 1 , . . . , a n n ] \ n ) ;
A = i n p u t ( A = ) ;
n = l e n g t h ( A ) ;
f o r i = 2 : n
j = 1 ;
w h i l e j < i
w h i l e A ( i , j ) ~ = 0
A ( i , j : n ) = A ( 1 , j : n ) - A ( 1 , j ) . / A ( i , j ) . * A ( i , j : n ) ;
e n d
j = j + 1 ;
1
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
e n d
e n d
A
w = A ( 1 , 1 ) ;
f o r l = 2 : n
w = w * A ( l , l ) ;
e n d
w
" S e b a s t i a n S z c z e p k o w i c z :
f u n c t i o n z = w y z n ( A )
% J e s t t o p r o g r a m d o o b l i c z a n i a w y z n a c z n i k a
% m a c i e r z y m e t o d a p o d a n a n a w y k l a d z i e
[ n m ] = s i z e ( A ) ;
i f ( n ~ = m )
d i s p ( P o d a n a m a c i e r z n i e j e s t k w a d r a t o w a ! )
b r e a k
e n d
f o r ( k = 1 : ( n - 1 ) )
f o r ( i = ( k + 1 ) : n )
A ( i , : ) = A ( i , : ) - A ( k , : ) . * ( A ( i , k ) / A ( k , k ) ) ;
e n d
e n d
A
z = 1 ;
d i s p ( W a r t o s c w y z n a c z n i k a m a c i e r z y A w y n o s i : )
f o r ( i = 1 : n )
z = z * A ( i , i ) ;
e n d
" D a w i d B o g d a Ds k i :
% D a w i d B o g d a Ds k i
% s k r y p t t e n s l u z y d o w y l i c z e n i a w y z n a c z n i k a m a c i e r z y [ n * n ]
% n a l e z y p o d a c m a c i e r z n p [ 1 2 3 ; 4 5 6 ; 7 8 9 ]
% Z M I E N N E :
% A - m a c i e r z
% i - n u m e r w i e r s z a
% j - n u m e r k o l u m n y
% k - n u m e r k o l u m n y w p e t l i
% w s p - w s p o l c z y n n i k w o d e j m o w a n i u w i e r s z y
% n , m - w y m i a r m a c i e r z y
% w y z n a c z n i k - b e z k o m e n t a r z a
c l c ; c l e a r ; c l o s e a l l ;
A = i n p u t ( P o d a j m a c i e r z )
[ n m ] = s i z e ( A ) ;
i f n ~ = m , % a b y p o l i c z y c w y z n a c z n i k m a c i e r z m u s i b y c n * n
d i s p ( T o n i e j e s t m a c i e r z k w a d r a t o w a )
b r e a k
e n d
f p r i n t f ( W y z n a c z n i k : % d \ n \ n , d e t ( A ) )
f o r j = 1 : n - 1 % b i o r e k o l u m n y o d 1 - w s z e j d o n - 1
f o r i = j + 1 : n
w s p = A ( i , j ) / A ( j , j ) ;
f o r k = j : n % k - s l u z y d o o k r e s l a n i a k o l u m n
A ( i , k ) = A ( i , k ) - ( A ( j , k ) * w s p ) ;
e n d
2
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
e n d
e n d
f p r i n t f ( T a k w y g l a d a m a c i e r z p o p r z e k s z t a l c e n i u : \ n \ n )
d i s p ( A )
w y z n a c z n i k = A ( 1 , 1 ) ; % l i c z e w y z n a c z n i k
f o r i = 2 : n
w y z n a c z n i k = w y z n a c z n i k * A ( i , i ) ;
e n d
f p r i n t f ( a j e j w y z n a c z n i k w y n o s i : % d \ n , w y z n a c z n i k )
2 a . O b l i c z a n i e w y z n a c z n i k a m a c i e r z y t r ó j k a t n e j
( p o m y s B p . B o g d a Ds k i e g o )
" s k r y p t
% n i e w i e m k t o r a m e t o d a l i c z e n i a w y z n a c z n i k a j e s t l e p s z a b o n a w e t
% z m a c i e r z a 1 0 0 * 1 0 0 c z a s l i c z e n i a w y z n a c z n i k a u m n i e n a k o m p u t e r z e
% b y l r o w n y 0 . N a p r a c o w n i w i n s t y t u c i e w y s z l o m i z e c h y b a t e n d r u g i
% s p o s o b j e s t s z y b s z y . % m o z n a j e s z c z e z m i e r z y c c z a s d z i a l a n i a c a l e g o
% p r o g r a m u i p o r o w n a c g o z c z a s e m w y k o n y w a n i a f u n k c j i d e t ( A ) - n a
% p r a c o w n i w y s z l o m i z e m o j a m e t o d a j e s t o k 1 0 r a z y w o l n i e j s z a o d d e t ( A )
c l c ; c l e a r ; c l o s e a l l ;
A = i n p u t ( P o d a j m a c i e r z )
[ n m ] = s i z e ( A ) ;
i f n ~ = m ,
d i s p ( T o n i e j e s t m a c i e r z k w a d r a t o w a )
b r e a k
e n d
t i c ;
B = m a c ( A , n ) ;
f p r i n t f ( T a k w y g l a d a m a c i e r z p o p r z e k s z t a l c e n i u : \ n \ n )
d i s p ( B )
k t o r a = i n p u t ( K t ó r a m e t o d a p o l i c z y ( 1 c z y 2 ) ? - ) ;
i f k t o r a = = 1
t i c ;
w = w y z n 1 ( B , n ) ;
t 1 = t o c ;
e l s e i f k t o r a = = 2
t i c ;
w = w y z n 2 ( B , n ) ;
t 1 = t o c ;
e l s e b r e a k
e n d
d i s p ( [ C z a s l i c z e n i a w y z n a c z n i k a : , n u m 2 s t r ( t 1 ) ] )
f p r i n t f ( W y z n a c z n i k w y n o s i : % d \ n , w )
" f u n k c j e :
f u n c t i o n w = w y z n a c z n i k 1 ( A , n )
w y z n a c z n i k 1 = A ( 1 , 1 ) ;
f o r i = 2 : n
w y z n a c z n i k 1 = w y z n a c z n i k 1 * A ( i , i ) ;
e n d
w = w y z n a c z n i k 1 ; % = = = = = = = = = = = = = = = = %
f u n c t i o n w = w y z n a c z n i k ( A , n )
w = p r o d ( d i a g ( A ) ) ; % = = = = = = = = = = = = = = = = %
f u n c t i o n B = m a c i e r z ( A , n )
3
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
f o r j = 1 : n - 1
f o r i = j + 1 : n
w s p = A ( i , j ) / A ( j , j ) ;
f o r k = j : n
A ( i , k ) = A ( i , k ) - ( A ( j , k ) * w s p ) ;
e n d
e n d
e n d
B = A ; % = = = = = = = = = = = = = = = = %
2 b . O b l i c z a n i e w y z n a c z n i k a m a c i e r z y p r z e k a t n i o w e j
( t e s t c z a s u o p e r a c j i m a c i e r z o w y c h )
" t e s t c z a s u o b l i c z e n
% t e s t c z a s u
% a s z 1 3 . 1 2 . 0 2
% o t r z y m a n e w y n i k i : 1 . 5 4 i 1 1 . 7 , 1 . 6 5 i 1 1 . 4 8 . . . .
%
c l c ; c l e a r ; c l o s e a l l ;
i l e = 1 0 2 4 ^ 2 ;
A = h i l b ( 1 0 ) ;
[ m n ] = s i z e ( A ) ;
d i s p ( )
% m e t o d a 1 :
t i c
f o r i i = 1 : i l e
w y z n a c z n i k = A ( 1 , 1 ) ;
f o r i = 2 : n
w y z n a c z n i k = w y z n a c z n i k * A ( i , i ) ;
e n d
w = w y z n a c z n i k ;
e n d
t 1 = t o c ;
d i s p ( [ C z a s l i c z e n i a w y z n a c z n i k a m e t . 1 : , n u m 2 s t r ( t 1 ) ] )
d i s p ( )
% m e t o d a 2 :
t i c
f o r i i = 1 : i l e
w = p r o d ( d i a g ( A ) ) ;
e n d
t 1 = t o c ;
d i s p ( [ C z a s l i c z e n i a w y z n a c z n i k a m e t . 2 : , n u m 2 s t r ( t 1 ) ] )
d i s p ( )
3 . R o z w i a z y w a n i e u k Ba d u r ó w n a D l i n i o w y c h
1 . p r o g r a m
f u n c t i o n [ x , e r r ] = r o z w u k l r l i n ( a , b , w y b o r )
% D a n e :
% r o z w i a z u j e m y u k l . r . l i n . a x = b z m a c i e r z a k w a d r a t o w a a .
% w y b o r = 1 - z w y k l a m e t o d a e l i m i n a c j i G a u s s a ,
% w y b o r = 2 - z c z e s c i o w y m w y b o r e m e l e m e n t o w g l o w n y c h ,
% w y b o r = 3 - z p e l n y m w y b o r e m e l e m e n t o w g l o w n y c h .
% W y n i k i :
% e r r = 0 - r o z w i a z a n i e p o p r a w n e ,
4
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
% e r r = 1 - b r a k r o z w i a z a n i a ;
% m a c i e r z u k l a d u j e s t o s o b l i w a l u b z l e s a r o z m i a r y d a n y c h .
% x - r o z w i a z a n i e o t r z y m a n e g d y e r r = 0 .
%
e r r = 1 ;
n = s i z e ( a , 1 ) ;
i f n ~ = s i z e ( a , 2 ) | n ~ = s i z e ( b , 1 ) , x = 1 ; r e t u r n ; e n d
a = [ a b ] ; n 1 = n + 1 ;
n u m e r = 1 : n ;
e r r = 0 ;
% e l i m i n a c j a :
f o r i = 1 : n - 1
i f w y b o r = = 2 % c z e s c i o w y
[ r , w ] = m a x ( a b s ( a ( i : n , i ) ) ) ;
w = w + i - 1 ;
i f i ~ = w
r = a ( i , : ) ; a ( i , : ) = a ( w , : ) ; a ( w , : ) = r ;
e n d
e l s e i f w y b o r = = 3 % p e l n y
[ r , w ] = m a x ( a b s ( a ( i : n , i : n ) ) ) ;
[ r , k ] = m a x ( r ) ;
w = w ( k ) + i - 1 ;
k = k + i - 1 ;
n u m e r ( i ) = k ;
i f i ~ = w
r = a ( i , : ) ; a ( i , : ) = a ( w , : ) ; a ( w , : ) = r ;
e n d
i f i ~ = k
r = a ( : , i ) ; a ( : , i ) = a ( : , k ) ; a ( : , k ) = r ;
e n d
e n d
i f a b s ( a ( i , i ) ) < 1 e - 3 0 , e r r = 1 ; b r e a k ; e n d
i 1 = i + 1 ;
f o r w = i 1 : n
a ( w , i 1 : n 1 ) = a ( w , i 1 : n 1 ) - a ( w , i ) / a ( i , i ) * a ( i , i 1 : n 1 ) ;
e n d
e n d % i
i f e r r , x = 1 ; r e t u r n ; e n d
% p o s t e p o w a n i e o d w r o t n e :
x = z e r o s ( n , 1 ) ;
f o r w = n : - 1 : 1
r = a ( w , n 1 ) ;
f o r j = n : - 1 : w + 1
r = r - a ( w , j ) * x ( j ) ;
e n d
i f a b s ( a ( w , w ) ) < 1 e - 3 0 , e r r = 1 ; x = 1 ; r e t u r n ; e n d
x ( w ) = r / a ( w , w ) ;
e n d
f o r w = n : - 1 : 1
i f w ~ = n u m e r ( w )
i = n u m e r ( w ) ;
r = x ( i ) ; x ( i ) = x ( w ) ; x ( w ) = r ;
e n d
e n d
% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = %
c l e a r ; c l c
f o r n = 1 0 : 1 3
5
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
f p r i n t f ( n = % i , n ) ;
x 0 = o n e s ( n , 1 ) ;
% x 0 = [ n : - 1 : 1 ] ;
A = h i l b ( n ) ; A = r o t 9 0 ( r o t 9 0 ( A ) ) ;
b = A * x 0 ;
f o r m a t l o n g e
f p r i n t f ( d e t ( A ) = % 1 0 . 5 e \ n , d e t ( A ) ) ;
x = A \ b ;
f p r i n t f ( d z i e l e n i e l e w e : x = A \ \ b \ n ) ;
[ a b s e r r , i ] = m a x ( a b s ( x - x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , a b s e r r , i ) ;
[ r e l e r r , i ] = m a x ( a b s ( ( x - x 0 ) . / x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , r e l e r r , i ) ;
f p r i n t f ( r o z w : n o r m ( x ( i ) - x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( x - x 0 ) ) ;
f p r i n t f ( r o z w : n o r m ( ( x ( i ) - x 0 ( i ) ) / x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( ( x - x 0 ) . / x 0 ) ) ;
r e s = A * x - b ;
[ a b s r e s , i ] = m a x ( a b s ( r e s ) ) ;
f p r i n t f ( r e s i d u a : m a x | r e s ( i ) | = % 1 0 . 5 e d l a i = % i \ n \ n , a b s r e s , i ) ;
x = i n v ( A ) * b ;
f p r i n t f ( m a c i e r z o d w r o t n a : x = i n v ( A ) * b \ n ) ;
[ a b s e r r , i ] = m a x ( a b s ( x - x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , a b s e r r , i ) ;
[ r e l e r r , i ] = m a x ( a b s ( ( x - x 0 ) . / x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , r e l e r r , i ) ;
f p r i n t f ( r o z w : n o r m ( x ( i ) - x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( x - x 0 ) ) ;
f p r i n t f ( r o z w : n o r m ( ( x ( i ) - x 0 ( i ) ) / x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( ( x - x 0 ) . / x 0 ) ) ;
r e s = A * x - b ;
[ a b s r e s , i ] = m a x ( a b s ( r e s ) ) ;
f p r i n t f ( r e s i d u a : m a x | r e s ( i ) | = % 1 0 . 5 e d l a i = % i \ n \ n , a b s r e s , i ) ;
x = i n v h i l b ( n ) * b ;
f p r i n t f ( m a c i e r z o d w r o t n a d o m . H i l b e r t a : x = i n v h i l b ( n ) * b \ n ) ;
[ a b s e r r , i ] = m a x ( a b s ( x - x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , a b s e r r , i ) ;
[ r e l e r r , i ] = m a x ( a b s ( ( x - x 0 ) . / x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , r e l e r r , i ) ;
f p r i n t f ( r o z w : n o r m ( x ( i ) - x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( x - x 0 ) ) ;
f p r i n t f ( r o z w : n o r m ( ( x ( i ) - x 0 ( i ) ) / x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( ( x - x 0 ) . / x 0 ) ) ;
r e s = A * x - b ;
[ a b s r e s , i ] = m a x ( a b s ( r e s ) ) ;
f p r i n t f ( r e s i d u a : m a x | r e s ( i ) | = % 1 0 . 5 e d l a i = % i \ n \ n , a b s r e s , i ) ;
f o r w = 1 : 3
f p r i n t f ( w a r i a n t = % i \ n , w ) ;
[ x , e r r ] = r o z w u k l r l i n ( A , b , w ) ;
i f e r r , d i s p ( b l a d ) ; r e t u r n ; e n d
[ a b s e r r , i ] = m a x ( a b s ( x - x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , a b s e r r , i ) ;
[ r e l e r r , i ] = m a x ( a b s ( ( x - x 0 ) . / x 0 ) ) ;
f p r i n t f ( r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = % 1 0 . 5 e d l a i = % i \ n , r e l e r r , i ) ;
f p r i n t f ( r o z w : n o r m ( x ( i ) - x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( x - x 0 ) ) ;
f p r i n t f ( r o z w : n o r m ( ( x ( i ) - x 0 ( i ) ) / x 0 ( i ) ) = % 1 0 . 5 e \ n , n o r m ( ( x - x 0 ) . / x 0 ) ) ;
r e s = A * x - b ;
[ a b s r e s , i ] = m a x ( a b s ( r e s ) ) ;
f p r i n t f ( r e s i d u a : m a x | r e s ( i ) | = % 1 0 . 5 e d l a i = % i \ n \ n , a b s r e s , i ) ;
e n d % w
e n d % n
6
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
2 . w y n i k i
n = 1 0 d e t ( A ) = 2 . 1 6 4 4 3 e - 0 5 3
d z i e l e n i e l e w e : x = A \ b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 6 . 5 9 4 4 0 e - 0 0 4 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 6 . 5 9 4 4 0 e - 0 0 4 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 1 . 0 6 9 4 4 e - 0 0 3
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 1 . 0 6 9 4 4 e - 0 0 3
r e s i d u a : m a x | r e s ( i ) | = 2 . 2 2 0 4 5 e - 0 1 6 d l a i = 6
m a c i e r z o d w r o t n a : x = i n v ( A ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 1 . 2 6 9 1 2 e - 0 0 3 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 1 . 2 6 9 1 2 e - 0 0 3 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 1 . 8 6 8 2 3 e - 0 0 3
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 1 . 8 6 8 2 3 e - 0 0 3
r e s i d u a : m a x | r e s ( i ) | = 3 . 8 3 1 5 0 e - 0 0 4 d l a i = 1 0
m a c i e r z o d w r o t n a d o m . H i l b e r t a : x = i n v h i l b ( n ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 3 . 2 6 3 7 8 e - 0 0 4 d l a i = 2
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 3 . 2 6 3 7 8 e - 0 0 4 d l a i = 2
r o z w : n o r m ( x - x 0 ) = 4 . 2 1 1 3 7 e - 0 0 4
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 4 . 2 1 1 3 7 e - 0 0 4
r e s i d u a : m a x | r e s ( i ) | = 2 . 8 5 1 0 8 e - 0 0 5 d l a i = 1 0
w a r i a n t = 1
r o z w : m a x | x ( i ) - x 0 ( i ) | = 4 . 5 1 4 0 6 e - 0 0 4 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 4 . 5 1 4 0 6 e - 0 0 4 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 7 . 3 0 8 8 6 e - 0 0 4
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 7 . 3 0 8 8 6 e - 0 0 4
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 9
w a r i a n t = 2
r o z w : m a x | x ( i ) - x 0 ( i ) | = 4 . 2 0 1 5 4 e - 0 0 4 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 4 . 2 0 1 5 4 e - 0 0 4 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 6 . 8 0 4 5 2 e - 0 0 4
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 6 . 8 0 4 5 2 e - 0 0 4
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 5
w a r i a n t = 3
r o z w : m a x | x ( i ) - x 0 ( i ) | = 1 . 1 2 8 0 3 e - 0 0 4 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 1 . 1 2 8 0 3 e - 0 0 4 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 1 . 8 2 5 0 9 e - 0 0 4
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 1 . 8 2 5 0 9 e - 0 0 4
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 1 0
n = 1 1 d e t ( A ) = 3 . 0 2 7 6 0 e - 0 6 5
d z i e l e n i e l e w e : x = A \ b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 2 . 4 0 9 9 0 e - 0 0 2 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 2 . 4 0 9 9 0 e - 0 0 2 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 3 . 8 1 4 3 5 e - 0 0 2
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 3 . 8 1 4 3 5 e - 0 0 2
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 1 0
m a c i e r z o d w r o t n a : x = i n v ( A ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 3 . 8 3 4 2 7 e - 0 0 2 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 3 . 8 3 4 2 7 e - 0 0 2 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 5 . 7 4 0 6 9 e - 0 0 2
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 5 . 7 4 0 6 9 e - 0 0 2
7
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
r e s i d u a : m a x | r e s ( i ) | = 6 . 6 7 9 9 4 e - 0 0 3 d l a i = 1 1
m a c i e r z o d w r o t n a d o m . H i l b e r t a : x = i n v h i l b ( n ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 1 . 9 5 7 7 9 e - 0 0 2 d l a i = 3
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 1 . 9 5 7 7 9 e - 0 0 2 d l a i = 3
r o z w : n o r m ( x - x 0 ) = 3 . 4 6 8 4 7 e - 0 0 2
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 3 . 4 6 8 4 7 e - 0 0 2
r e s i d u a : m a x | r e s ( i ) | = 1 . 8 9 1 1 8 e - 0 0 5 d l a i = 1 1
w a r i a n t = 1
r o z w : m a x | x ( i ) - x 0 ( i ) | = 5 . 7 5 9 5 8 e - 0 0 3 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 5 . 7 5 9 5 8 e - 0 0 3 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 9 . 1 1 7 8 5 e - 0 0 3
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 9 . 1 1 7 8 5 e - 0 0 3
r e s i d u a : m a x | r e s ( i ) | = 2 . 2 2 0 4 5 e - 0 1 6 d l a i = 6
w a r i a n t = 2
r o z w : m a x | x ( i ) - x 0 ( i ) | = 3 . 6 2 3 8 6 e - 0 0 2 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 3 . 6 2 3 8 6 e - 0 0 2 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 5 . 7 3 5 7 1 e - 0 0 2
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 5 . 7 3 5 7 1 e - 0 0 2
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 1 0
w a r i a n t = 3
r o z w : m a x | x ( i ) - x 0 ( i ) | = 1 . 6 4 5 8 8 e - 0 0 2 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 1 . 6 4 5 8 8 e - 0 0 2 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 2 . 6 0 5 0 4 e - 0 0 2
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 2 . 6 0 5 0 4 e - 0 0 2
r e s i d u a : m a x | r e s ( i ) | = 2 . 2 2 0 4 5 e - 0 1 6 d l a i = 6
n = 1 2 d e t ( A ) = 2 . 6 0 3 1 1 e - 0 7 8
W a r n i n g : M a t r i x i s c l o s e t o s i n g u l a r o r b a d l y s c a l e d .
R e s u l t s m a y b e i n a c c u r a t e . R C O N D = 2 . 7 3 5 3 6 5 e - 0 1 7 .
( T y p e " w a r n i n g o f f M A T L A B : n e a r l y S i n g u l a r M a t r i x " t o s u p p r e s s t h i s w a r n i n g . )
> I n E : \ 0 1 0 2 \ d y d a k t y k a \ 0 2 3 - m a t - o b l i c z e n i o w a \ 2 0 0 3 \ w 9 1 0 \ t e s t r o z w u k l r l i n . m a t l i n e 1 0
d z i e l e n i e l e w e : x = A \ b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 1 . 3 4 1 5 5 e - 0 0 2 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 1 . 3 4 1 5 5 e - 0 0 2 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 2 . 1 3 3 4 5 e - 0 0 2
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 2 . 1 3 3 4 5 e - 0 0 2
r e s i d u a : m a x | r e s ( i ) | = 2 . 2 2 0 4 5 e - 0 1 6 d l a i = 6
W a r n i n g : M a t r i x i s c l o s e t o s i n g u l a r o r b a d l y s c a l e d .
R e s u l t s m a y b e i n a c c u r a t e . R C O N D = 2 . 3 9 7 4 6 9 e - 0 1 7 .
> I n E : \ 0 1 0 2 \ d y d a k t y k a \ 0 2 3 - m a t - o b l i c z e n i o w a \ 2 0 0 3 \ w 9 1 0 \ t e s t r o z w u k l r l i n . m a t l i n e 2 1
m a c i e r z o d w r o t n a : x = i n v ( A ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 8 . 9 8 9 9 1 e - 0 0 1 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 8 . 9 8 9 9 1 e - 0 0 1 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 1 . 2 5 0 5 6 e + 0 0 0
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 1 . 2 5 0 5 6 e + 0 0 0
r e s i d u a : m a x | r e s ( i ) | = 5 . 3 3 7 8 0 e - 0 0 2 d l a i = 1 2
m a c i e r z o d w r o t n a d o m . H i l b e r t a : x = i n v h i l b ( n ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 4 . 9 2 1 2 0 e - 0 0 1 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 4 . 9 2 1 2 0 e - 0 0 1 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 6 . 7 6 7 2 6 e - 0 0 1
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 6 . 7 6 7 2 6 e - 0 0 1
8
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
r e s i d u a : m a x | r e s ( i ) | = 5 . 1 0 9 0 9 e - 0 0 2 d l a i = 1 2
w a r i a n t = 1
r o z w : m a x | x ( i ) - x 0 ( i ) | = 2 . 4 9 1 7 4 e - 0 0 1 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 2 . 4 9 1 7 4 e - 0 0 1 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 4 . 0 2 6 2 2 e - 0 0 1
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 4 . 0 2 6 2 2 e - 0 0 1
r e s i d u a : m a x | r e s ( i ) | = 1 . 1 1 0 2 2 e - 0 1 6 d l a i = 5
w a r i a n t = 2
r o z w : m a x | x ( i ) - x 0 ( i ) | = 1 . 0 4 3 6 0 e + 0 0 0 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 1 . 0 4 3 6 0 e + 0 0 0 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 1 . 6 8 3 8 2 e + 0 0 0
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 1 . 6 8 3 8 2 e + 0 0 0
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 8
w a r i a n t = 3
r o z w : m a x | x ( i ) - x 0 ( i ) | = 3 . 6 9 3 4 8 e - 0 0 3 d l a i = 5
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 3 . 6 9 3 4 8 e - 0 0 3 d l a i = 5
r o z w : n o r m ( x - x 0 ) = 5 . 6 9 9 2 4 e - 0 0 3
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 5 . 6 9 9 2 4 e - 0 0 3
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 1 2
n = 1 3 d e t ( A ) = 1 . 8 9 3 5 5 e - 0 9 2
W a r n i n g : M a t r i x i s c l o s e t o s i n g u l a r o r b a d l y s c a l e d .
R e s u l t s m a y b e i n a c c u r a t e . R C O N D = 9 . 8 6 4 9 5 9 e - 0 1 9 .
( T y p e " w a r n i n g o f f M A T L A B : n e a r l y S i n g u l a r M a t r i x " t o s u p p r e s s t h i s w a r n i n g . )
> I n E : \ 0 1 0 2 \ d y d a k t y k a \ 0 2 3 - m a t - o b l i c z e n i o w a \ 2 0 0 3 \ w 9 1 0 \ t e s t r o z w u k l r l i n . m a t l i n e 1 0
d z i e l e n i e l e w e : x = A \ b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 4 . 1 1 3 2 8 e + 0 0 1 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 4 . 1 1 3 2 8 e + 0 0 1 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 6 . 9 9 9 3 2 e + 0 0 1
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 6 . 9 9 9 3 2 e + 0 0 1
r e s i d u a : m a x | r e s ( i ) | = 8 . 8 8 1 7 8 e - 0 1 6 d l a i = 7
W a r n i n g : M a t r i x i s c l o s e t o s i n g u l a r o r b a d l y s c a l e d .
R e s u l t s m a y b e i n a c c u r a t e . R C O N D = 9 . 8 6 4 9 5 9 e - 0 1 9 .
> I n E : \ 0 1 0 2 \ d y d a k t y k a \ 0 2 3 - m a t - o b l i c z e n i o w a \ 2 0 0 3 \ w 9 1 0 \ t e s t r o z w u k l r l i n . m a t l i n e 2 1
m a c i e r z o d w r o t n a : x = i n v ( A ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 2 . 8 6 2 6 4 e + 0 0 1 d l a i = 6
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 2 . 8 6 2 6 4 e + 0 0 1 d l a i = 6
r o z w : n o r m ( x - x 0 ) = 4 . 4 0 7 0 3 e + 0 0 1
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 4 . 4 0 7 0 3 e + 0 0 1
r e s i d u a : m a x | r e s ( i ) | = 1 . 4 1 7 1 1 e - 0 0 1 d l a i = 1 3
m a c i e r z o d w r o t n a d o m . H i l b e r t a : x = i n v h i l b ( n ) * b
r o z w : m a x | x ( i ) - x 0 ( i ) | = 3 . 0 5 6 4 7 e + 0 0 1 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 3 . 0 5 6 4 7 e + 0 0 1 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 3 . 2 9 7 1 0 e + 0 0 1
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 3 . 2 9 7 1 0 e + 0 0 1
r e s i d u a : m a x | r e s ( i ) | = 3 . 4 9 0 2 1 e + 0 0 0 d l a i = 1 3
w a r i a n t = 1
r o z w : m a x | x ( i ) - x 0 ( i ) | = 3 . 8 8 8 9 4 e + 0 0 1 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 3 . 8 8 8 9 4 e + 0 0 1 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 6 . 6 2 9 6 2 e + 0 0 1
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 6 . 6 2 9 6 2 e + 0 0 1
9
A d a m S z u s t a l e w i c z M a t e m a t y k a o b l i c z e n i o w a 2 0 0 2 / 2 0 0 3 w . 1 0 1 6 . 1 2 . 0 2
r e s i d u a : m a x | r e s ( i ) | = 1 . 1 1 0 2 2 e - 0 1 5 d l a i = 1 1
w a r i a n t = 2
r o z w : m a x | x ( i ) - x 0 ( i ) | = 5 . 6 4 7 1 7 e + 0 0 0 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 5 . 6 4 7 1 7 e + 0 0 0 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 9 . 4 4 8 4 4 e + 0 0 0
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 9 . 4 4 8 4 4 e + 0 0 0
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 8
w a r i a n t = 3
r o z w : m a x | x ( i ) - x 0 ( i ) | = 2 . 0 5 4 8 6 e + 0 0 0 d l a i = 4
r o z w : m a x | x ( i ) - x 0 ( i ) | / | x 0 ( i ) | = 2 . 0 5 4 8 6 e + 0 0 0 d l a i = 4
r o z w : n o r m ( x - x 0 ) = 3 . 5 1 8 7 3 e + 0 0 0
r o z w : n o r m ( ( x - x 0 ) . / x 0 ) = 3 . 5 1 8 7 3 e + 0 0 0
r e s i d u a : m a x | r e s ( i ) | = 4 . 4 4 0 8 9 e - 0 1 6 d l a i = 1 3
4 . U w a r u n k o w a n i e z a d a n i a r o z w i a z y w a n i a u k Ba d u r ó w n a D l i n i o w y c h
R o z w i a z u j e m y u k Ba d r ó w n a D
A x = b
z m a c i e r z a n i e o s o b l i w a A . Z a b u r z a m y p r a w a s t r o n e b ’! b + ´b ; w t e d y z m i a n i e u l e g a r o z w i a z a n i e d o k Ba d n e :
A ( x + ´x ) = ( b + ´b ) .
M o ze m y n a p i s a :
A ´x = ´b c z y l i ´x = A - 1 ´b .
D l a z g o d n y c h n o r m s p e Bn i o n e s a n i e r ó w n o [c i :
| | ´x | | d" | | A - 1 | | | | ´b | | o r a z | | b | | d" | | A | | | | x | | .
S t a d o t r z y m u j e m y n i e r ó w n o [
| | ´x | | | | ´b | |
d" | | A | | | | A - 1 | | .
| | x | | | | b | |
D E F . W i e l k o [
c o n d ( A ) = | | A | | | | A - 1 | |
n a z y w a m y w s k a zn i k i e m u w a r u n k o w a n i a z a d a n i a r o z w i a z y w a n i a u k Ba d u r ó w n a D l i n i o w y c h A x = b .
5 . Z a d a n i a
1 . W y z n a c z y r o z w i a z a n i a p o n i zs z y c h u k Ba d ó w r ó w n a D l i n i o w y c h . W y j a [n i t a k d u ze r ó zn i c e p o m i e d z y
r o z w i a z a n i a m i .
1 0 0 0 x 1 + 2 0 0 1 x 2 = 4 0 0 3 1 0 0 0 x 1 + 1 9 9 9 x 2 = 4 0 0 3
o r a z
x 1 + 2 x 2 = 4 x 1 + 2 x 2 = 4
2 . S t o s u j a c a r y t m e t y k e z 3 c y f r a m i z n a c z a c y m i r o z w i a z a p o d a n y n i ze j u k Ba d r ó w n a D n a s t e p u j a c y m i a l g o -
r y t m a m i :
( a ) e l i m i n u j a c z d r u g i e g o r ó w n a n i a x 1 ,
( b ) e l i m i n u j a c n i e w i a d o m a x 1 z p i e r w s z e g o r ó w n a n i a .
. 1 0 0 1 0 - 3 x 1 + . 1 0 0 1 0 1 x 2 = . 1 0 0 1 0 1
. 1 0 0 1 0 1 x 1 + . 1 0 0 1 0 1 x 2 = . 2 0 0 1 0 1
3 . Z a p r o g r a m o w a r o z w i a z y w a n i e u k Ba d u r ó w n a D l i n i o w y c h z a p o m o c a w z o r ó w C r a m e r a ( d l a c h e t n y c h :
p r o p o n u j e o p r a c o w a w e r s j e r e k u r e n c y j n a ) i p o r ó w n a w y n i k i z w y n i k a m i z a m i e s z c z o n y m i w
m a t e r i a Ba c h . . .
* * *
1 0
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