3_Newton_s_Difference_Method.pdf

# 2 1 1 1 2 1 2 2 1 1 2 1 2 2 x x x x x f x f x x x f x

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0 2 0 1 0 1 1 2 1 2 0 2 0 1 1 2 0 1 2 2 ) ( ) ( ) ( ) ( ] , [ ] , [ ] , , [ x x x x x f x f x x x f x f x x x x f x x f x x x f b

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15 General Form Given ) 1 ( n data points,     n n n n y x y x y x y x , , , , ...... , , , , 1 1 1 1 0 0 as ) )...( )( ( .... ) ( ) ( 1 1 0 0 1 0 n n n x x x x x x b x x b b x f where ] [ 0 0 x f b ] , [ 0 1 1 x x f b ] , , [ 0 1 2 2 x x x f b ] , .... , , [ 0 2 1 1 x x x f b n n n ] , .... , , [ 0 1 x x x f b n n n
16 General form The third order polynomial, given ), , ( 0 0 y x ), , ( 1 1 y x ), , ( 2 2 y x and ), , ( 3 3 y x is ) )( )( ]( , , , [ ) )( ]( , , [ ) ]( , [ ] [ ) ( 2 1 0 0 1 2 3 1 0 0 1 2 0 0 1 0 3 x x x x x x x x x x f x x x x x x x f x x x x f x f x f 0 b 0 x ) ( 0 x f 1 b ] , [ 0 1 x x f 2 b 1 x ) ( 1 x f ] , , [ 0 1 2 x x x f 3 b ] , [ 1 2 x x f ] , , , [ 0 1 2 3 x x x x f 2 x ) ( 2 x f ] , , [ 1 2 3 x x x f ] , [ 2 3 x x f 3 x ) ( 3 x f

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17 Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for cubic interpolation. Table. Velocity as a function of time Figure. Velocity vs. time data for the rocket example 0 0 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 ) s ( t ) m/s ( ) ( t v