LagrangeNewtonCotes

# LagrangeNewtonCotes - Inadequacy of Newton-Cotes...

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(3) (2) (1) Inadequacy of Newton-Cotes Integration The example below shows that Newton-Cotes Integration is not a very reliable method for approximating an integral. As the number of interpolation points grows large, there is no guarantee that the estimate gets closer to the integral. Romberg Integration converges to the integral as n grows large. So, it is a more reliable method. Below is the integral that we try to estimate using Newton-Cotes. As you will see, the estimate is not very good and becomes worse as the number of interpolation points increases. First we calculate the correct integral with a decimal representation and show the graph of the function. K 4 4 1 1 C x 2 d x 2 arctan 4 evalf % 2.651635328 with Student Calculus1 : f d x / 1 1 C x 2 x / 1 x 2 C 1 /

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(4) K 10 K 5 0 5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Now we give two Newton-Cotes estimates. We do this by determining the Lagrange interpolating polynomial and integrating it. This is what the Newton-Cotes does, indirectly. We also give a graph of the Lagrange polynomials to show why the integral is not a very good estimate. The calculations are for n = 8 and n = 16. with CurveFitting : PolynomialInterpolation K 4, K 3, K 2, K 1, 0, 1, 2, 3, 4 , 1 1 C 4 2 , 1 1 C 3 2 , 1 1 C 2 2 , 1 1 C 1 2 , 1 1 , 1 1 C 1 2 , 1 1 C 2 2 , 1 1 C 3 2 , 1 1 C 4 2 , x 1 1700 x 8 K 31 1700 x 6 C 76 425

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