Numerical differentiation
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Numerical differentiation

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Numerical differentiation Numerical differentiation Presentation Transcript

  • NUMERICAL DIFFERENTIATION
    • We like to estimate the value of f '(x) for a given function f(x).
    • The derivative represents the rate of change of a dependent variable with respect to an independent variable.
    • The difference approximation is
    • If x is allowed to approach zero, the difference becomes a derivative:
  • f (Xi + ∆X)
    ∆y
    f(Xi)
    X
    ∆Xi
    Xi + ∆X
    ∆x
  • Y
    f (Xi + ∆X)
    ∆y
    f(Xi)
    ∆Xi
    Xi + ∆X
    X
    ∆x
  • Y
    f’(Xi)
    Xi
    X
    • The Taylor series expansion of f(x) about xi is
    • From this:
    • This formula is called the first forward divided difference formula and the error is of order O(h).
    • Or equivalently, the Taylor series expansion of f(x) about xi can be written as
    • From this:
    • This formula is called the first backward divided difference formula and the error is of order O(h).
    • A third way to approximate the first derivative is to subtract the backward from the forward Taylor series expansions:
    • This yields to
    • This formula is called the centered divided difference formula and the error is of order O(h2).
  • FORWARD
    True derivate
    Y
    Approximation
    h
    X
    Xi
    Xi+1
  • BACKWARD
    True derivate
    Y
    Approximation
    h
    X
    Xi-1
    Xi
  • CENTERED
    True derivate
    Y
    Approximation
    2h
    Xi+1
    Xi-1
    X
  • THANK YOU!
    SOURCE: Dr Muhammad Al-Salamah, Industrial Engineering KFUPM