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Numerical Methods

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Today s class Numerical Differentiation Finite Difference Methods Numerical Methods Lecture 14 Prof. Jinbo Bi CSE, UConn * Numerical Differentiation Finite ... – PowerPoint PPT presentation

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Title: Numerical Methods


1
Todays class
  • Numerical Differentiation
  • Finite Difference Methods

2
Numerical Differentiation
  • Finite Difference Methods
  • Forward
  • Backward
  • Centered
  • Error Magnitude
  • O(h) for forward and backward
  • O(h2) for centered

3
Forward First Derivative
  • Consider a function f(x) which can be expanded in
    a Taylor series in the neighborhood of a point x

4
Forward First Derivative
5
Backward First Derivative
  • Consider a function f(x) which can be expanded in
    a Taylor series in the neighborhood of a point x

6
Backward First Derivative
7
Central First Derivative
8
Central First Derivative
9
Numerical Differentiation
10
2nd-order Forward Difference
11
High-Accuracy Differentiation
12
Forward Finite-Divided Difference
13
Backward Difference Scheme
14
Backward Finite-Divided Difference
15
Centered Difference Scheme
16
Centered Divided Difference
17
Basic Differentiation
  • Example
  • Find derivative at x0.5, h0.25
  • True
  • Forward

18
Basic Differentiation
  • Example
  • Backward
  • Centered

19
High-Accuracy Differentiation
  • Forward
  • Backward
  • Centered

20
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.

21
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.

22
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).

23
Richardson Extrapolation
24
Richardson Extrapolation
25
Richardson Extrapolation
26
Richardsons Extrapolation
  • Example
  • h0.5
  • h0.25
  • Extrapolate

27
Unevenly Spaced Data
28
Unevenly Spaced Data
29
Unevenly Spaced Data
30
Unevenly Spaced Data
31
Unevenly Spaced Data
32
Next class
  • Ordinary Differential Equations
  • Read Chapter PT7, 25
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