Lecture31 - Quadratic Interpolation Quadratic interpolation...

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Quadratic interpolation – need three points Use the parabola This is the same as ( 29 ( 29 ( 29 ( 29 2 1 3 1 2 1 2 x x b x x b b x f + - + = = - - = + - = + + = 3 3 2 3 1 3 2 2 2 1 3 1 2 1 1 3 2 1 2 b a x b x b b a x x b x b b a a x a a x f ) ( Quadratic Interpolation where
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To get the coefficients b ’s 1) Set x = x 1 , get b 1 = f ( x 1 ) 2) Use b 1 and set x = x 2 to get b 2 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 2 1 1 1 3 1 1 2 1 1 2 x f b x x x x b x x b b x f = = - - + - + = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 2 2 2 2 1 2 3 1 2 2 1 2 2 x x x f x f b x f x x x x b x x b x f x f - - = = - - + - + = Quadratic Interpolation ( 29 ( 29 ( 29 ( 29 2 1 3 1 2 1 2 x x x x b x x b b x f - - + - + =
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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 3 3 3 2 3 1 3 3 1 3 1 2 1 2 1 3 2 x x b x f x x x x b x x x x x f x f x f x f - - = = - - + - - - + = b is a constant (0th order) b gives slope (finite difference) b gives curvature (difference of finite differences) Quadratic Interpolation 3) Use b 1 and b 2 , and set x = x 3 to get b 3
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Example: interpolate e 2 using e 1 , e 3 , and e 5 ( 29 ( 29 ( 29 41 148 x f 5 x 086 20 x f 3 x 7183 2 x f 1 x 3 3 2 2 1 1 = = = = = = . . . 870 13 1 5 3 5 e e b ; 6836 8 b ; 7183 2 b 3 5 3 2 1 . . . = - - - - = = = = ( 29 ( 29 ( 29 ( 29 4681 2 3 2 1 2 870 13 1 2 6836 8 7183 2 2 f 2 . * . * . . - = - - + - + = 3891 7 e Exact 2 . = Hand Calculation Example
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Example: interpolate e 2 using e 1 , e 1.5 , and e 2.5
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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Lecture31 - Quadratic Interpolation Quadratic interpolation...

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