【正文】 d u r e VD V w r iting t h is a s VY = w i thD V o r V D f r o m V ( w h ich is n o w k n o w n ) so lv e f o r c o lum n v e cto r Y u sing a sim p le r e cu r siYY????????????11111,2/ , 1 1/ono f th e f o r m = w ith in itial co n d ition n o w ca n so lv e f o r u sing t h e r e cu r sion w ith in itial co n d ition ii i i j jjpi i i j i jjip p pY V Y p iYY d V i pYd??????????????? ? ? ? ????? 1 1 ca lcula tio n p r o ce e d s b a ck w a r d s f r o m d o w n t o i p i? ? ?18 Cholesky Deposition Example 1121 2231 32 3341 42 43 4414112 2 21 13 3 31 1 32 2441 0 0 01 0 0101 co n t in u in g t h e e x a m p le w e so lv e f o r Y f ir st so lv in g f o r w e g e t YV YVV YV V V YYYYY V YY V Y V YYV??????????? ? ? ? ??? ? ? ? ??? ? ? ? ???? ? ? ? ??? ? ? ? ?? ? ? ????????? ? ???41 1 42 2 43 31 1 1 1 121 31 412 2 2 2 232 423 3 3 3 3434 4 4 4 41 / 0 0 0 /10 1 / 0 0 /010 0 1 / 0 /0 0 10 0 0 1 / /0 0 0 1 n e x t so lv e f o r f r o m e q u a t io nY V Y V Yd Y Y dVVVd Y Y dVVd Y Y dVd Y Y d????????? ? ? ? ? ? ? ???? ? ? ? ? ? ? ???? ? ? ? ? ? ? ?????? ? ? ? ? ? ???? ? ? ? ? ? ????? ? ? ? ? ? ? ? ?4 4 43 3 3 43 42 2 2 32 3 42 41 1 1 21 2 31 3 41 4//// g iv in g t h e r e su lts co m p le t in g t h e so lu t io nYdY d VY d V VY d V V V???? ? ?? ? ? ???????? ? ?? ? ? ??19 Covariance Method Minimum Error 111121( 0 , 0) ( 0 , )( 0 , 0)( 0 , 0)( 0 , 0) / t h e m in im u m m e a n sq u a r e d e r r o r ca n b e w r itt e n in t h e f o r m sin ce ca n w r ite t h is a s t h is c o m p u tpn n k nktntttnnpn k kkEkY D VE Y D YYd? ? ?? ? ???????????????????????a t io n f o r ca n b e u se d f o r a ll v a lu e s o f L P o r d e r f r o m1 t o ca n u n d e r st a n d h o w L P o r d e r r e d u ce s m e a n sq u a r e d e r r o rnEp ?20 Solution to Autocorrelation Method 1( ) ( ) 1pk n nkR i k R i i p???? ? ? ??? t h e m a tr i x e q u a tion f o r t h e a u to cor r e l a tion m e th o d is si n ce m a tr i x is T o e p l i tz ( sy m m e tr i c w i t h a l l e l e m e n ts a l o n g e a ch d i a g o n a l e q u a l )th e r e a r e se v e r a l so l u tion m e t( 0 )11 ( 1 )1()( ) ( 1 ) ( 1( 0)( ) ( ) / 1jiiiijiiii i ij j i i jERk R i R i j E i pkk??? ? ?????????????? ? ? ? ?????????h o d s, incl u d i n g t h e L e v i n son m e th o d , b u t w e w i l l u se th eD u r b i n r e cur si v e p r o ced u r e w h i ch g o e s a s f o l l o w s: )( ) 2 ( 1 )()()11( 1 )1 , 2 , ... ,11 , 2 , 3 , ... , 1iiipjjitjjiE k Eipjppj????? ? ?????? ? ?? so l v e e q u a tion s r e cur si v e l y f o r w i th t h e f i n a l so l u tion a l l p r e d i cto r s o f o r d e r s a r e f o u n d in D u r b i n 39。/ 。( , ) ( ) ( )() t h e r e a r e n o n z e r o t e r m s in t h e co m p u t a tio n o f f o r e a ch v a lue o f a n d ca n e a sil y sh o w t h a t w h e r e is t h e sh o r t tim e a u to co r r e latnnnni p k pN i k i kiki k f i k R i kR i k??? ? ? ?? ? ?? ? ? ????10()( ) ( ) ( )ion o f e v a lua te d a t w h e r e nNkn n nms m i kR k s m s m k???????13 Autocorrelation Method 14 Covariance Method 120()() t h e r e is a se con d b a si c a p p r o a ch to d e f i n i n g t h e sp e e ch seg m e n t a n d t h e li m i ts o n t h e su m s, n a m e l y f i x th e int e r v a l o v e r w h i chth e m e a n squ a r e d e r r o r is co m p u te d , g i v i ng nNnnmsmE e m??????21011011( ) ( )( , ) ( ) ( ) , 1 , 0(