LagrangeNewtonCotes - Inadequacy of Newton-Cotes...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
(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 /
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
(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
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern