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spline_400_2010 - A Presentation on Interpolation Using...

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A Presentation on Interpolation Using Piecewise Linear and Cubic Spline Functions for Math 400 - Spring 2010 Bruce Cohen [email protected] http://www.cgl.ucsf.edu/home/bic David Sklar [email protected]
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Write a formula for a piecewise linear function that interpolates five given data points ( 29 1,1 ( 29 2,2 ( 29 3,2.5 ( 29 4,1.5 ( 29 5,2 0 1 2 3 4 5 6 ( 29 1 2 11 2 1 1 2 2 1 2 1 2 3 3 4 4 5 x if x x if x p x x if x x if x + = - + - Connecting the Dots
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Write a formula for a piecewise linear function that interpolates n given data points ( 29 2 1 1 2 2 1 1 2 2 1 2 1 3 2 2 3 3 2 2 3 3 2 3 2 1 1 1 1 1 1 n n n n n n n n n n n n y y y x y x x if x x x x x x x y y y x y x x if x x x x x x x p x y y y x y x x if x x x x x x x - - - - - - - - + - - - - + - - = - - + - - M M ( 29 1 1 , x y ( 29 2 2 , x y ( 29 3 3 , x y ( 29 1 1 , n n x y - - ( 29 , n n x y 1 x 2 x 3 x 1 n x - n x
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Interpolation using a Linear Spline Basis The “linear spline function” approach involves carefully choosing a set of “basis functions” such that the interpolating function can be written as a simple linear combination: 1 2 , , , n Ψ Ψ Ψ K f ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 2 1 n n n i i i p x y x y x y x y x = = Ψ + Ψ + + Ψ = Ψ L 2 i x - 1 i x - i x 1 i x + 2 i x + ( 29 2 2 , i i x y - - ( 29 1 1 , i i x y - - ( 29 , i i x y ( 29 1 1 , i i x y + + ( 29 2 2 , i i x y + + ( 29 ,1 i x i Ψ 1 , for all n x x x ( 29 1 if 0 if i j ij i j x i j δ = Ψ = = On the data points we have
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( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 2 1 n n n i i i p x y x y x y x y x = = Ψ + Ψ + + Ψ = Ψ L 2 i x - 1 i x - i x 1 i x + 2 i x + ( 29 2 2 , i i x y - - ( 29 1 1 , i i x y - - ( 29 , i i x y ( 29 1 1 , i i x y + + ( 29 2 2 , i i x y + + ( 29 ,1 i x i Ψ i i y Ψ 1 1 i i y - - Ψ 1 1 i i i i y y - - Ψ + Ψ A closer look at a linear combination of basis functions
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The linear spline basis functions can be constructed as sums of translations and horizontal scalings of two “elementary basis functions” ( 29 [ 29 [ 29 1 1 0,1 0 0,1 x if x x if x θ - = ( 29 [ 29 [ 29 2 0,1 0 0,1 x if x x if x θ = ( 29 1 2 1 i i i i x x x x x θ - - - Ψ = - 1 i x - i x 1 i x + 0 1 0 1 1 1 i i i x x x x θ + - + -
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A summary description of the linear spline basis ( 29 [ 29 [ 29 2 0,1 0 0,1 x if x x if x θ = ( 29 [ 29 [ 29 1 1 0,1 0 0,1 x if x x if x θ - = 0 1 0 1 1. Elementary basis functions – basically constructed on the unit interval 2. A set of nodes -- 1 2 n x x x < < < L 3. Spline basis functions – sums of (usually) two translated and scaled elementary basis functions 1 i x - i x 1 i x + ( 29 1 2 1 1 1 i i i i i i i x x x x x x x x x θ θ - - + - - Ψ = + - - interior: 2, , 1 i n = - K endpoints: 1 and i n = ( 29 1 1 1 2 1 x x x x x θ - Ψ = - ( 29 1 2 1 n n n n x x x x x θ - - - Ψ = - 1 x 2 x n x 1 n x -
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Points from sin(x)
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Points from sin(x) Linear Interpolation
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Points from sin(x) Linear Interpolation & sin(x)
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Points from ( 29 2 2 1 2 x g x e π - =
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Points from g(x) Linear Interpolation
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Points from g(x) Linear Interpolation & g(x)
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