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Numerical Integration

Numerical Integration. Dr. Asaf Varol. Numerical Integration. Numerical integration is a primary tool used for definite integrals that cannot be solved analytically. A numerical integration rule has the form

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Numerical Integration

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  1. Numerical Integration Dr. Asaf Varol

  2. Numerical Integration • Numerical integration is a primary tool used for definite integrals that cannot be solved analytically. A numerical integration rule has the form • we investigate several basic quadrature formulas that use function values at equally spaced points; these methods are known as Newton-Cotes formulas. There are two types of Newton –Cotes formulas, depending on whether or not the function values at the ends of the interval of integration are used. The trapezoid and Simpson rules are examples of “closed” formulas, in which the endpoint values are used. The midpoint rule is the simplest example of an “open” formula, in which the endpoints are not used [2].

  3. Newton-Cotes Closed FormulasTrapezoid Rule • One of the simplest ways to approximate the area under a curve is to approximate the curve by a straight line. The trapezoid rule approximates the curve by the straight line that passes through the points [a, f(a) and b, f(b)], the two ends of the interval of interest. We have x0=a, x1=b, and h=b-a, and then

  4. Example

  5. Matlab Program

  6. Diagram

  7. Newton-Cotes Closed FormulasSimpson’s Rule

  8. Newton-Cotes Closed FormulasSimpson’s Rule (Cont’d)

  9. Matlab Program

  10. Diagram

  11. Newton-Cotes Closed FormulasMidpoint Rule

  12. Figure • Figure given on the right side compares the actual value of the area with that found by using the midpoint rule. The area given by the integral S (hatched) and the approximation using the midpoint rule (shaded) [2].

  13. Matlab Program

  14. Diagram

  15. Gaussian quadrature

  16. Gaussian quadrature

  17. Example

  18. End of Chapter 5

  19. References • Celik, Ismail, B., “Introductory Numerical Methods for Engineering Applications”, Ararat Books & Publishing, LCC., Morgantown, 2001 • Fausett, Laurene, V. “Numerical Methods, Algorithms and Applications”, Prentice Hall, 2003 by Pearson Education, Inc., Upper Saddle River, NJ 07458 • Rao, Singiresu, S., “Applied Numerical Methods for Engineers and Scientists, 2002 Prentice Hall, Upper Saddle River, NJ 07458 • Mathews, John, H.; Fink, Kurtis, D., “Numerical Methods Using MATLAB” Fourth Edition, 2004 Prentice Hall, Upper Saddle River, NJ 07458 • Varol, A., “Sayisal Analiz (Numerical Analysis), in Turkish, Course notes, Firat University, 2001

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