Numerical Integration

1. Numerical Integration • Basic Numerical Integration • We want to find integration of functions of various forms of the equation known as the Newton Cotes integration formulas. • Trapezoidal Rule • Simpson’s Rule • 1/3 Rule • 3/8 Rule • Midpoint • Gaussian Quadrature

2. Basic Numerical Integration • Weighted sum of function values f(x) x x0 x1 xn-1 xn

3. Numerical Integration Idea is to do integral in small parts, like the way you first learned integration - a summation Numerical methods just try to make it faster and more accurate

4. Numerical Integration • Newton-Cotes Closed Formulae -- Use both end points • Trapezoidal Rule : Linear • Simpson’s 1/3-Rule : Quadratic • Simpson’s 3/8-Rule : Cubic • Boole’s Rule : Fourth-order • Newton-Cotes Open Formulae -- Use only interior points • midpoint rule

5. Trapezoid Rule • Straight-line approximation f(x) L(x) x x0 x1

6. Trapezoid Rule • Lagrange interpolation

7. Trapezoid Rule • Integrate to obtain the rule

8. Example:Trapezoid Rule Evaluate the integral • Exact solution • Trapezoidal Rule

9. Simpson’s 1/3-Rule Approximate the function by a parabola L(x) f(x) x x0 h x1 h x2

10. Simpson’s 1/3-Rule

11. Simpson’s 1/3-Rule Integrate the Lagrange interpolation

12. Simpson’s 3/8-Rule Approximate by a cubic polynomial L(x) f(x) x x0 h x1 h x2 h x3

13. Simpson’s 3/8-Rule

14. Example: Simpson’s Rules Evaluate the integral • Simpson’s 1/3-Rule • Simpson’s 3/8-Rule

15. Midpoint Rule Newton-Cotes Open Formula f(x) x a xm b

16. Two-point Newton-Cotes Open Formula Approximate by a straight line f(x) x x0 h x1 h x2 h x3

17. Three-point Newton-Cotes Open Formula Approximate by a parabola f(x) x x0 h x1 h x2 h x3 h x4

18. Better Numerical Integration • Composite integration • Composite Trapezoidal Rule • Composite Simpson’s Rule • Richardson Extrapolation • Romberg integration

19. Two segments Three segments Four segments Many segments Apply trapezoid rule to multiple segments over integration limits

20. Composite Trapezoid Rule f(x) x x0 h x1 h x2 h x3 h x4

21. Composite Trapezoid Rule Evaluate the integral

22. Composite Trapezoid Example

23. Composite Trapezoid Rule with Unequal Segments Evaluate the integral • h1 = 2, h2 = 1, h3 = 0.5, h4 = 0.5

24. Composite Simpson’s Rule Piecewise Quadratic approximations f(x) …... x x0 h x1 h x2 h x3 h x4 xn-2 xn-1 xn

25. Composite Simpson’s Rule Multiple applications of Simpson’s rule

26. Composite Simpson’s Rule Evaluate the integral • n = 2, h = 2 • n = 4, h = 1

27. Composite Simpson’s Example

28. Composite Simpson’s Rule with Unequal Segments Evaluate the integral • h1 = 1.5, h2 = 0.5

29. Richardson Extrapolation Use trapezoidal rule as an example • subintervals: n = 2j = 1, 2, 4, 8, 16, ….

30. Richardson Extrapolation For trapezoidal rule • kth level of extrapolation

31. Romberg Integration Accelerated Trapezoid Rule

32. Romberg Integration Accelerated Trapezoid Rule

33. Romberg Integration Example

34. Gaussian Quadratures • Newton-Cotes Formulae • use evenly-spaced functional values • Gaussian Quadratures • select functional values at non-uniformly distributed points to achieve higher accuracy • change of variables so that the interval of integration is [-1,1] • Gauss-Legendre formulae

35. Gaussian Quadrature on [-1, 1] • Choose (c1, c2, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3 x1 x2 -1 1

36. Gaussian Quadrature on [-1, 1] Exact integral forf = x0, x1, x2, x3 • Four equations for four unknowns

37. Gaussian Quadrature on [-1, 1] • Choose (c1, c2, c3, x1, x2,x3) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5 x1 x2 x3 -1 1

38. Gaussian Quadrature on [-1, 1]

39. Gaussian Quadrature on [-1, 1] Exact integral forf = x0, x1, x2, x3, x4, x5

40. Gaussian Quadrature on [a, b] Coordinate transformation from [a,b] to [-1,1] a t1 t2 b

41. Example: Gaussian Quadrature Evaluate Coordinate transformation Two-point formula

42. Example: Gaussian Quadrature Three-point formula Four-point formula