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Lecture32

# Lecture32 - Lagrange Interpolating Polynomials Give the...

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Lagrange Interpolating Polynomials Give the same result as the Newton’s polynomials, but different approach ) ( ) )( ( ) )( ( ) ( ) )( ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( n k 1 i i 1 i i 2 i 1 i n 1 i 1 i 2 1 i i i n i j 1 j j i j i i n 1 i i n n 2 2 1 1 1 n x x x x x x x x x x x x x x x x x x x x x P x P x x x x x L x f x L x f x L x f x L x f x L x f - - - - - - - - - - = = - - = = + + + = + - + - = = - ij j i j i i i i i i i x L 0 x L i j 1 x P x P x L i ; j Note δ = = = = = ) ( ) ( ; ) ( ) ( ) ( :

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Lagrange Interpolation 1st-order Lagrange polynomial Second-order Lagrange polynomial Third-order Lagrange polynomial ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 1 2 1 2 2 2 1 1 1 x f x x x x x f x x x x x f x L x f L x f - - + - - = + = ) ( ) )( ( ) )( ( ) ( ) )( ( ) )( ( ) ( ) )( ( ) )( ( ) ( 3 2 3 1 3 2 1 2 3 2 1 2 3 1 1 3 1 2 1 3 2 2 x f x x x x x x x x x f x x x x x x x x x f x x x x x x x x x f - - - - + - - - - + - - - - = ) ( ) )( )( ( ) )( )( ( ) ( ) )( )( ( ) )( )( ( ) ( ) )( )( ( ) )( )( ( ) ( ) )( )( ( ) )( )( ( ) ( 4 3 4 2 4 1 4 3 2 1 3 4 3 2 3 1 3 4 2 1 2 4 2 3 2 1 2 4 3 1 1 4 1 3 1 2 1 4 3 2 4 x f x x x x x x x x x x x x x f x x x x x x x x x x x x x f x x x x x x x x x x x x x f x x x x x x x x x x x x x f - - - - - - + - - - - - - + - - - - - - + - - - - - - =
Linear Lagrange Interpolation

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Quadratic Lagrange Interpolation L 2 ( x ) f ( x 2 ) L 3 ( x ) f ( x 3 ) x 1 x 2 x 3
Cubic Lagrange Interpolation -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 2 3 4 L0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 2 3 4 L1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 2 3 4 L2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -1 0 1 2 3 4 L3 L2 L3 L4 L1

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f ( x ) = e x Interpolation at [0 4] First-order Lagrange interpolation
f ( x ) = e x , Interpolation at [0 1 4] Second-order Lagrange interpolation f ( x )

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f ( x ) = e x , Interpolation at [0 1 4 3] Third-order Lagrange interpolation
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