set22 - a39 a38 S e t 2 2 : R a t i o n a l I n t e r p o l...

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Unformatted text preview: a39 a38 S e t 2 2 : R a t i o n a l I n t e r p o l a t i o n P a r t 2 K y l e A . G a l l i v a n D e p a r t m e n t o f M a t h e m a t i c s F l o r i d a S t a t e U n i v e r s i t y F o u n d a t i o n s o f C o m p u t a t i o n a l M a t h 2 S p r i n g 2 1 1 1 a39 a38 I n v e r s e D i f f e r e n c e s D e fi n i t i o n 2 2 . 1 . G i v e n p o i n t s ( x i , f i ) t h e i n v e r s e d i f f e r e n c e s a r e d e fi n e d a s ( x i , x j ) = ( x i x j ) f i f j ( x i , x j , x k ) = ( x j x k ) ( x i , x j ) ( x i , x k ) ( x i , . . . , x s , x m , x n ) = ( x m x n ) ( x i , . . . , x s , x m ) ( x i , . . . , x s , x n ) N o t e . V a l u e s o f c a n r e s u l t . 2 a39 a38 C o n t i n u e d F r a c t i o n a n d I n v e r s e D i f f e r e n c e s r n n ( x ) = p n ( x ) q n ( x ) , r n n ( x i ) = f i , i 2 n r ( x ) = f , r 1 ( x ) = f + ( x x ) ( x , x 1 ) r 1 1 ( x ) = f + ( x x ) ( x , x 1 ) + ( x x 1 ) ( x , x 1 , x 2 ) r 2 1 ( x ) = f + ( x x ) ( x , x 1 ) + ( x x 1 ) ( x , x 1 , x 2 ) + ( x x 2 ) ( x , x 1 , x 2 , x 3 ) 3 C o n t i n u e d F r a c t i o n a n d I n v e r s e D i f f e r e n c e s r 2 2 ( x ) = f + ( x- x ) ( x , x 1 ) + ( x- x 1 ) ( x , x 1 , x 2 ) + ( x- x 2 ) ( x , x 1 , x 2 , x 3 ) + ( x- x 3 ) ( x , x 1 , x 2 , x 3 , x 4 ) 4 a39 a38 G e n e r a l F o r m G i v e n t h e i n v e r s e d i f f e r e n c e s w e h a v e : r n n ( x ) = f + ( x- x ) ( x , x 1 ) + ( x- x 1 ) ( x , x 1 , x 2 ) + ( x- x 2 ) ( x , x 1 , x 2 , x 3 ) + . . . + ( x- x 2 n- 1 ) ( x , . . . , x 2 n ) 5 a39 a38 C o n t i n u e d F r a c t i o n a n d I n v e r s e D i f f e r e n c e s T h e e x p r e s s i o n c a n b e e v a l u a t e d i n a H o r n e r s r u l e- l i k e f a s h i o n , e . g . , r 2 1 ( x ) : i n i t i a l i z e = = + ( x , x 1 , x 2 , x 3 , x 4 ) = ( x x 3 ) = + ( x , x 1 , x 2 , x 3 ) = ( x x 2 ) = + ( x , x 1 , x 2 ) = ( x x 1 ) = + ( x , x 1 ) = ( x x ) r 2 1 ( x ) = f + 6 a39 a38 I n v e r s e D i f f e r e n c e s i x i f i ( x , x i ) ( x , x 1 , x i ) ( x , x 1 , x 2 , x i ) 1 1- 1 1 2 2- 2 / 3- 3 1 / 2 3 3 9 1 / 3 3 / 2 1 / 2 G i v e n t h i s d a t a w e c a n b u i l d u p t o r 2 1 ( x ) . 7 a39 a38 C o n t i n u e d F r a c t i o n a n d I n v e r s e D i f f e r e n c e s r 1 ( x ) = + ( x ) ( 1 ) = x r 1 ( ) = , r 1 ( 1 ) = 1 , r 1 ( 2 ) = 2 negationslash = 2 3 , r 1 ( 3 ) = 3 negationslash = 9 8 a39 a38 C o n t i n u e d F r a c t i o n a n d I n v e r s e D i f f e r e n c e s r 1 1 ( x ) = + ( x ) ( 1 ) + ( x 1 ) ( 1 / 2 ) = x...
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set22 - a39 a38 S e t 2 2 : R a t i o n a l I n t e r p o l...

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