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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