set22

# set22 - \$ Set 22 Rational Interpolation Part 2 Kyle A...

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a39 a38 a36 a37 Set 22: Rational Interpolation – Part 2 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1

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a39 a38 a36 a37 Inverse Differences Definition 22.1. Given points ( x i , f i ) the inverse differences are defined as φ ( x i , x j ) = ( x i x j ) f i f j φ ( x i , x j , x k ) = ( x j x k ) φ ( x i , x j ) φ ( x i , x k ) φ ( x i , . . . , x s , x m , x n ) = ( x m x n ) φ ( x i , . . . , x s , x m ) φ ( x i , . . . , x s , x n ) Note. Values of can result. 2
a39 a38 a36 a37 Continued Fraction and Inverse Differences r nn ( x ) = p n ( x ) q n ( x ) , r nn ( x i ) = f i , 0 i 2 n r 00 ( x ) = f 0 , r 10 ( x ) = f 0 + ( x x 0 ) φ ( x 0 , x 1 ) r 11 ( x ) = f 0 + ( x x 0 ) φ ( x 0 , x 1 ) + ( x x 1 ) φ ( x 0 , x 1 , x 2 ) r 21 ( x ) = f 0 + ( x x 0 ) φ ( x 0 , x 1 ) + ( x x 1 ) φ ( x 0 , x 1 , x 2 ) + ( x x 2 ) φ ( x 0 , x 1 , x 2 , x 3 ) 3

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Continued Fraction and Inverse Differences r 22 ( x ) = f 0 + ( x - x 0 ) φ ( x 0 , x 1 ) + ( x - x 1 ) φ ( x 0 , x 1 , x 2 ) + ( x - x 2 ) φ ( x 0 , x 1 , x 2 , x 3 ) + ( x - x 3 ) φ ( x 0 , x 1 , x 2 , x 3 , x 4 ) 4
a39 a38 a36 a37 General Form Given the inverse differences we have: r nn ( x ) = f 0 + ( x - x 0 ) φ ( x 0 , x 1 ) + ( x - x 1 ) φ ( x 0 , x 1 , x 2 ) + ( x - x 2 ) φ ( x 0 , x 1 , x 2 , x 3 ) + · · · . . . + ( x - x 2 n - 1 ) φ ( x 0 , . . . , x 2 n ) 5

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a39 a38 a36 a37 Continued Fraction and Inverse Differences The expression can be evaluated in a Horner’s rule-like fashion, e.g., r 21 ( x ) : initialize τ = 0 τ = τ + φ ( x 0 , x 1 , x 2 , x 3 , x 4 ) τ = ( x x 3 ) τ τ = τ + φ ( x 0 , x 1 , x 2 , x 3 ) τ = ( x x 2 ) τ τ = τ + φ ( x 0 , x 1 , x 2 ) τ = ( x x 1 ) τ τ = τ + φ ( x 0 , x 1 ) τ = ( x x 0 ) τ r 21 ( x ) = f 0 + τ 6
a39 a38 a36 a37 Inverse Differences i x i f i φ ( x 0 , x i ) φ ( x 0 , x 1 , x i ) φ ( x 0 , x 1 , x 2 , x i ) 0 0 0 1 1 -1 1 2 2 -2/3 -3 1 / 2 3 3 9 1/3 3/2 1 / 2 Given this data we can build up to r 21 ( x ) . 7

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a39 a38 a36 a37 Continued Fraction and Inverse Differences r 10 ( x ) = 0 + ( x 0) ( 1) = x r 10 (0) = 0 , r 10 (1) = 1 , r 10 (2) = 2 negationslash = 2 3 , r 10 (3) = 3 negationslash = 9 8
a39 a38 a36 a37 Continued Fraction and Inverse Differences r 11 ( x ) = 0 + ( x 0) ( 1) + ( x 1) ( 1 / 2) = x 2 x + 1 r 11 (0) = 0 , r 11 (1) = 1 , r 11 (2) = 2 3 , r 11 (3) = 3 5 negationslash = 9 9

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a39 a38 a36 a37 Continued Fraction and Inverse Differences r 21 ( x ) = 0 + ( x 0) ( 1) + ( x 1) ( 1 / 2) + ( x 2) (1 / 2) = 4 x 2 9 x 2 x + 7 r 21 (0) = 0 , r 21 (1) = 1 , r 21 (2) = 2 3 , r 21 (3) = 9 10
a39 a38 a36 a37 Consistency Check 1 x 0 x 2 0 f 0 f 0 x 0 1 x 1 x 2 1 f 1 f 1 x 1 1 x 2 x 2 2 f 2 f 2 x 2 1 x 3 x 2 3 f 3 f 3 x 3 α 0 α 1 α 2 β 0 β 1 = 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 2 4 2 / 3 4 / 3 1 3 9 9 27

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