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Chapter 19 Numerical Differentiation

. . . . . . . . . xi-3 xi-2 xi-1 xi xi 1 xi 2 xi 3. . . . . Evenly distributed points along the x-axis. . . . . . x1 x2 x3 . Unevenly distributed points along the x-axis. Distance between two neighboring points is the same, i.e. h..

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Chapter 19 Numerical Differentiation

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    1. Chapter 19 Numerical Differentiation Estimate the derivatives (slope, curvature, etc.) of a function by using the function values at only a set of discrete points Ordinary differential equation (ODE) Partial differential equation (PDE) Represent the function by Taylor polynomials or Lagrange interpolation Evaluate the derivatives of the interpolation polynomial at selected (unevenly distributed) nodal points

    4. Forward difference

    5. Forward difference

    6. Backward difference

    7. Centered difference

    8. First Derivatives Forward difference Backward difference Central difference

    9. Truncation Errors Uniform grid spacing

    10. Example: First Derivatives Use forward and backward difference approximations to estimate the first derivative of at x = 0.5 with h = 0.5 and 0.25 (exact sol. = -0.9125) Forward Difference Backward Difference

    11. Example: First Derivative Use central difference approximation to estimate the first derivative of at x = 0.5 with h = 0.5 and 0.25 (exact sol. = -0.9125) Central Difference

    12. Second-Derivatives Taylor-series expansion Uniform grid spacing Second-order accurate O(h2)

    13. Centered Finite-Divided Differences

    14. Forward Finite-divided differences

    15. Backward finite-divided differences

    16. First Derivatives 3 -point Forward difference 3 -point Backward difference

    17. Example: First Derivatives Use forward and backward difference approximations of O(h2) to estimate the first derivative of at x = 0.5 with h = 0.25 (exact sol. = -0.9125) Forward Difference Backward Difference

    18. Higher Derivatives All second-order accurate O(h2) More nodal points are needed for higher derivatives Higher order formula may be derived

    19. 19.3 Richardson Extrapolation

    20. Example of using Richardson Extrapolation Central Difference Scheme

    21. Ex19.2: Richardson Extrapolation Use central difference approximation to estimate the first derivative of at x = 0.5 with h = 0.5 and 0.25 (exact sol. = -0.9125)

    22. General Three-Point Formula Lagrange interpolation polynomial for unequally spaced data

    23. Lagrange Interpolation 1st-order Lagrange polynomial Second-order Lagrange polynomial

    24. Lagrange Interpolation Third-order Lagrange polynomial

    25. Lagrange Interpolation

    26. General Three-Point Formula Lagrange interpolation polynomial for unequally spaced data First derivative

    27. Second Derivative First Derivative for unequally spaced data Second Derivative for unequally spaced data

    28. Differentiation of Noisy Data

    29. MATLAB’s Methods Derivatives are sensitive to the noise Use least square fit before taking derivatives p = polyfit(x, y, n) - coefficients of Pn(x) polyfit(p, x) - evaluation of Pn(x) polyder(p) - differentiation

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