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AM2032 JAYANTA MUKHERJEE

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AM2032 JAYANTA MUKHERJEE

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    1. AM2032 JAYANTA MUKHERJEE

    2. AM2032 JAYANTA MUKHERJEE

    3. AM2032 JAYANTA MUKHERJEE Numerical Integration

    4. AM2032 JAYANTA MUKHERJEE What Will We Learn? Trapezoidal Rule Simpson’s Rule 1/3 Rule 3/8 Rule Midpoint Gaussian Quadrature

    5. AM2032 JAYANTA MUKHERJEE Basic Numerical Integration

    6. AM2032 JAYANTA MUKHERJEE

    7. AM2032 JAYANTA MUKHERJEE

    8. AM2032 JAYANTA MUKHERJEE 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

    9. AM2032 JAYANTA MUKHERJEE Numerical Integration Newton-Cotes Open Formulae – Use only interior points midpoint rule

    10. AM2032 JAYANTA MUKHERJEE Trapezoid Rule Straight-line approximation (n=1)

    11. AM2032 JAYANTA MUKHERJEE Example:Trapezoid Rule Evaluate the integral Exact solution Trapezoidal Rule

    12. AM2032 JAYANTA MUKHERJEE Simpson’s 1/3-Rule Approximate the function by a parabola (n=2)

    13. AM2032 JAYANTA MUKHERJEE Simpson’s 3/8-Rule Approximate by a cubic polynomial (n=3)

    14. AM2032 JAYANTA MUKHERJEE Example: Simpson’s Rules Evaluate the Integral Simpson’s 1/3-Rule Simpson’s 3/8-Rule

    15. AM2032 JAYANTA MUKHERJEE Midpoint Rule Newton-Cotes Open Formula

    16. AM2032 JAYANTA MUKHERJEE Two-point Newton-Cotes Open Formula Approximate by a straight line

    17. AM2032 JAYANTA MUKHERJEE Three-Point Newton-Cotes Open Formula Approximate by a parabola

    18. AM2032 JAYANTA MUKHERJEE Better Numerical Integration Composite integration Composite Trapezoidal Rule Composite Simpson’s Rule Romberg integration

    19. AM2032 JAYANTA MUKHERJEE

    20. AM2032 JAYANTA MUKHERJEE Composite Trapezoid Rule

    21. AM2032 JAYANTA MUKHERJEE Composite Trapezoid Rule Evaluate the integral

    22. AM2032 JAYANTA MUKHERJEE Composite Trapezoid Rule with Unequal Segments Evaluate the integral h1 = 2, h2 = 1, h3 = 0.5, h4 = 0.5

    23. AM2032 JAYANTA MUKHERJEE Composite Simpson’s Rule

    24. AM2032 JAYANTA MUKHERJEE Composite Simpson’s Rule Multiple applications of Simpson’s rule

    25. AM2032 JAYANTA MUKHERJEE Composite Simpson’s Rule Evaluate the integral n = 2, h = 2 n = 4, h = 1

    26. AM2032 JAYANTA MUKHERJEE Composite Simpson’s Rule with Unequal Segments Evaluate the integral h1 = 1.5, h2 = 0.5

    27. AM2032 JAYANTA MUKHERJEE Romberg Integration

    28. AM2032 JAYANTA MUKHERJEE Romberg Integration

    29. AM2032 JAYANTA MUKHERJEE 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]

    30. AM2032 JAYANTA MUKHERJEE Gaussian Quadrature on [-1, 1] Choose (c1, c2, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3

    31. AM2032 JAYANTA MUKHERJEE Gaussian Quadrature on [-1, 1] Exact integral for f = x0, x1, x2, x3 Four equations for four unknowns

    32. AM2032 JAYANTA MUKHERJEE 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

    33. AM2032 JAYANTA MUKHERJEE Gaussian Quadrature on [-1, 1]

    34. AM2032 JAYANTA MUKHERJEE Gaussian Quadrature on [-1, 1] Exact integral for f = x0, x1, x2, x3, x4, x5

    35. AM2032 JAYANTA MUKHERJEE Gaussian Quadrature on [a, b] Coordinate transformation from [a,b] to [-1,1]

    36. AM2032 JAYANTA MUKHERJEE Example: Gaussian Quadrature Evaluate Coordinate transformation Two-point formula

    37. AM2032 JAYANTA MUKHERJEE Example: Gaussian Quadrature Three-point formula Four-point formula

    38. AM2032 JAYANTA MUKHERJEE

    39. AM2032 JAYANTA MUKHERJEE

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