Lecture 32 - Newtons Interpolating Polynomials General form...

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General form for Newton’s interpolating polynomials Bracketed functions are divided differences ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 x x x x x x x x b x x x x b x x b b x f 1 n 3 2 1 n 2 1 3 1 2 1 1 n - - - - - - + + + - - + - + = ( 29 [ ] [ ] [ ] 1 2 1 n n n 1 2 3 3 1 2 2 1 1 x x x x f b x x x f b x x f b x f b , , , , , , , - = = = = Newton’s Interpolating Polynomials
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First divided difference Second divided difference The difference of two divided differences [ ] ( 29 ( 29 j i j i j i x x x f x f x x f - - = , [ ] [ ] [ ] k i k j j i k j i x x x x f x x f x x x f - - = , , , , Newton’s Divided Differences
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The n th divided difference [ ] [ ] [ ] 1 n 1 2 2 n 1 n 2 3 1 n n 1 2 1 n n x x x x x x f x x x x f x x x x f - - = - - - - , , , , , , , , , , , , Procedure: 1. Evaluate all first-order divided differences; save f ( x 1 ) for b 1 2. Evaluate second-order from firsts; save f [ x 2 , x 1 ] for b 2 3. Continue to n th -order, saving needed ones Newton’s Divided Differences
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Newton Interpolation [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 5 1 2 3 4 2 3 4 5 1 2 3 4 5 5 1 4 1 2 3 2 3 4 1 2 3 4 4 1 3 1 2 2 3 1 2 3 3 1 2 1 2 1 2 2 1 1 1 1 n 1 n 2 1 3 1 2 1 1 n x x x x x x f x x x x f x x x x x f b x x x x x f x x x f x x x x f b x x x x f x x f x x x f b x x x f x f x x f b x f x f b x x x x b x x x x b x x b b x f - - = = - - = = - - = = - - = = = = - - + + - - + - + = - - , , , , , , , , , , , , , , , , , , , , , , ) ( ) ( ) ( ) )( ( ) ( ) ( No need to solve a system of simultaneous equations i
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[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ) ( , ) ( , , , ) ( , , , , , , ) ( , , , , , , , , , , ) ( , , , , , , , , , , ) ( , , , , , , , ) ( 6 6 5 6 5 5 4 5 6 4 5 4 4 3 4 5 6 3 4 5 3 4 3 3 2 3 4 5 6 2 3 4 5 2 3 4 2 3 2 2 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1 1 i 4 i i 3 i i 1 i 2 i i 1 i i i i x f x 6 x x f x f x 5 x x x f x x f x f x 4 x x x x f x x x f x x f x f x 3 x x x x x f x x x x f x x x f x x f x f x 2 x x x x x f x x x x f x x x f x x f x f x 1 x x f x x f x x x f x x f x f y x i + + + + + = Newton Interpolation [ ] [ ] [ ] 1 n 1 2 2 n 1 n 1 3 1 n n 1 2 1 n n n 1 n 1 n 2
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Fall '09 term at University of Florida.

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Lecture 32 - Newtons Interpolating Polynomials General form...

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