assignment_2[1]

assignment_2[1] - Lagrange and Hermite interpolating p n n...

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Math 400 Assignment #2 -- Interpolation ( 29 ( 29 { } 1 2 1 1 For Lagrange interpolation on the nodes of the data , n n n i i i x x x x x f x - = < < < < ( 29 ( 29 ( 29 ( 29 ( 29 ( 29( 29 ( 29 ( 29 ( 29( 29 ( 29 , 1 1 1 1 , 1 1 1 the interpolating polynomial is , where n i n i i i i n n i i i i i i i n p x f x L x x x x x x x x x L x x x x x x x x x = - + - + = - - - - = - - - - ( 29 ( 29 ( 29 ( 29( 29 ( 29( 29 ( 29( 29 ( 29 ( 29( 29 ( 29 ( 29 1 1 2 1 2 1 ,1 1 2 1 2 1 3 1 2 1 3 1 , Prove that 1 and state the result for . n n n n i x x x x x x x x x x x x L x x x x x x x x x x x x x L x - - - - - - - = + + + + - - - - - - ( 29 ( 29 ( 29 { } 1 2 1 1 Generalize Hermite interpolation on the nodes for data , , , . This involves finding explicit formulas for appropriate degree polynomials such that the interpolat n n n i i i i i x x x x x f x f x f x - = < < < < ′′ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 ing polynomial can be written n n n i i i i i i i i i p x f x A x f x B x f x C x = = = ′′ = + + 1. 2. ( 29 ( 29 { } ( 29 ( 29 1 2 1 1 10 10 2 1 1 1 1 7 2 3 5 11 For the nodes where 0 1 6 3 2 12 3 4 6 12 produce data , , , for 1.6 sin 3 . Using this data find the
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Unformatted text preview: Lagrange and Hermite interpolating p n n x i i i i x x x x x x x f x f x f x e x π--= < < < < = < < < < < < < < < = ′ = ⋯ ( 29 olynomials as well as piecewise linear and cubic spline interpolants. Using R or C write programs to compute the interpolating polynomials, and produce graphs, each of which shows the graph of f x and one interpolant. f 3. ( 29 ( 29 1 2 1 1 10 10,6 1 1 1 7 2 3 5 11 Using the nodes where 1 6 3 2 12 3 4 6 12 create a graph of and, on a separate plot a graph of the polynomial n n x x x x x x L x x-< < < < = < < < < < < < < < = = ⋯ ( 29( 29 ( 29( 29 1 2 9 10 . x x x x x x x x----⋯ 4. Due: April 6, 2010...
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