HW 3 solution

HW 3 solution - EGM 4313 S08 Homework Solution#3 1 Problem...

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EGM 4313 S08 Homework Solution #3 1/14 1. Problem Set 19.3, p.808, #3 (Quadratic interpolation) Calculate the Lagrange polynomial ( ) 2 p x for the 4D-values of the Gamma function [(24), App. 3.1] ( ) ( ) ( ) 1.00 1.0000, 1.02 0.9888, 1.04 0.9784 Γ = Γ = Γ = , and from it approximations of ( ) x Γ for x = 1.005, 1.010, 1.015, 1.025, 1.030, 1.035. Solution : Second-degree Lagrange polynomial ( ) 2 p x is given by ( ) ( ) ( ) ( ) 2 0 0 1 1 2 2 p x L x f L x f L x f = + + Where ( ) ( )( ) ( )( ) 1 2 0 0 1 0 2 x x x x L x x x x x = , ( ) ( )( ) ( )( ) 0 2 1 1 0 1 2 x x x x L x x x x x = , ( ) ( )( ) ( )( ) 0 1 2 2 0 2 1 x x x x L x x x x x = with 0 1 2 0 1 2 1.00, 1.02, 1.04, 1.0000, 0.9888, 0.9784 x x x f f f = = = = = = . Using the above values, we obtain ( ) 2 2 1.0000 2.5800 2.5800 p x x x = + Therefore, ( ) x Γ with 4D values for each x is ( ) 1.005 0.9971 Γ = , ( ) 1.010 0.9943 Γ = , ( ) 1.015 0.9915 Γ = ( ) 1.025 0.9861 Γ = , ( ) 1.030 0.9835 Γ = , ( ) 1.035 0.9809 Γ =

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