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

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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= , 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 px is given by () ( ) ( ) ( ) 20 0 1 1 2 2 px Lxf Lxf Lxf =+ + Where 12 0 0102 x xxx Lx x −− = , ( )( ) 02 1 1012 x xx x x = , ( )( ) 01 2 2021 x x x = with 0120 1 2 1.00, 1.02, 1.04, 1.0000, 0.9888, 0.9784 f f f ==== = = . Using the above values, we obtain 2 2 1.0000 2.5800 2.5800 x x =− + Therefore, ( ) x Γ with 4D values for each x is 1.005 0.9971 , ( ) 1.010 0.9943 , ( ) 1.015 0.9915 1.025 0.9861 , ( ) 1.030 0.9835 , ( ) 1.035 0.9809
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EGM 4313 S08 Homework Solution #3 2/14 2. Set 19.3, problem # 9, page 809 (Cubic Lagrange interpolation) Calculate and sketch or graph L 0 , L 1 , L 2 , L 3 for x=0,1,2,3 on common axes. Find p 3 (x) for the data (0, 1) (1, 0.765198) (2, 0.223891) (3, 0.260052) [values of the Bessel function J 0 (x)]. Find p 3 for x=0.5, 1.5, 2.5 by interpolation. Solutions: ) 3 )( 2 )( 1 ( 6 1 ) )( )( ( ) )( )( ( 3 0 2 0 1 0 3 2 1 0 = = x x x x x x x x x x x x x x x L ) 3 )( 2 )( ( 2 1 ) )( )( ( ) )( )( ( 3 1 2 1 0 1 3 2 0 1 = = x x x x x x x x x x x x x x x L ) 3 )( 1 )( ( 2 1 ) )( )( ( ) )( )( ( 3 2 1 2 0 2 3 1 0 2 = = x x x x x x x x x x x x x x x L ) 2 )( 1 )( ( 6 1 ) )( )( ( ) )( )( ( 2 3 1 3 0 3 2 1 0 3 = = x x x x x x x x x x x x x x x L
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EGM 4313 S08 Homework Solution #3 3/14 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 00 . 511 . 522 . 533 . 5 L0 L1 L2 L3 3 3 2 2 1 1 0 0 3 ) ( ) ( ) ( ) ( ) ( f x L f x L f x L f x L x p + + + = x 0 0.5 1 1.5 2 2.5 3 p3(x) 1 0.943654 0.765198 0.510116 0.223891 0.04799 0.26005 Appendix I L 0 L 1 L 2 L 3 0 1 0 0 0 0.1 0.8265 0.2755 0.1305 0.0285 0.2 0.672 0.504 0.224 0.048 0.3 0.5355 0.6885 0.2835 0.0595 0.4 0.416 0.832 0.312 0.064 0.5 0.3125 0.9375 0.3125 0.0625 0.6 0.224 1.008 0.288 0.056 0.7 0.1495 1.0465 0.2415 0.0455 0.8 0.088 1.056 0.176 0.032 0.9 0.0385 1.0395 0.0945 0.0165
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EGM 4313 S08 Homework Solution #3 4/14 1 3.7E 17 1 1.1E 16 1.85E 17 1.1 0.0285 0.9405 0.1045 0.0165 1.2 0.048 0.864 0.216 0.032 1.3 0.0595 0.7735 0.3315
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HW 3 solution - EGM 4313 S08 Homework Solution #3 1....

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