Numerical Methods

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Todays class

• Numerical Differentiation
• Finite Difference Methods

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Numerical Differentiation

• Finite Difference Methods
• Forward
• Backward
• Centered
• Error Magnitude
• O(h) for forward and backward
• O(h2) for centered

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Forward First Derivative

• Consider a function f(x) which can be expanded in
  a Taylor series in the neighborhood of a point x

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Forward First Derivative
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Backward First Derivative

• Consider a function f(x) which can be expanded in
  a Taylor series in the neighborhood of a point x

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Backward First Derivative
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Central First Derivative
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Central First Derivative
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Numerical Differentiation
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2nd-order Forward Difference
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High-Accuracy Differentiation
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Forward Finite-Divided Difference
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Backward Difference Scheme
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Backward Finite-Divided Difference
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Centered Difference Scheme
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Centered Divided Difference
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Basic Differentiation

• Example
• Find derivative at x0.5, h0.25
• True
• Forward

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Basic Differentiation

• Example
• Backward
• Centered

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High-Accuracy Differentiation

• Forward
• Backward
• Centered

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Summary

• Forward Divided Difference method uses the value
  of points in front of or at the point where the
  derivative is calculated.
• Backward Divided Difference method uses the value
  of points behind of or at the point where the
  derivative is calculated.

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Summary

• Centered Divided Difference uses the value of
  points both in front and behind of the point
  where the derivative is calculated.
• Centered method is usually more accurate than
  forward backward methods
• Accurate formulas use more points in the
  calculations.

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Richardson Extrapolation

• As with integration, use two approximations to
  arrive at a better approximation
• D is the true value but unknown and D(h1) is an
  approximation based on the step size h1. Reducing
  the step size to half, h2 h1/2, we obtained
  another approximation D(h2).
• By properly combining the two approximations,
  D(h1) D(h2), the error is reduced to O(h4).

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Richardson Extrapolation
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Richardson Extrapolation
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Richardson Extrapolation
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Richardsons Extrapolation

• Example
• h0.5
• h0.25
• Extrapolate

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Unevenly Spaced Data
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Unevenly Spaced Data
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Unevenly Spaced Data
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Unevenly Spaced Data
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Unevenly Spaced Data
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Next class

• Ordinary Differential Equations
• Read Chapter PT7, 25