today’s class numerical differentiation finite difference methods numerical methods lecture 14...

Today’s class

• Numerical Differentiation• Finite Difference Methods

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Numerical Differentiation

• Finite Difference Methods• Forward• Backward• Centered

• Error Magnitude• O(h) for forward and backward• O(h2) for centered

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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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

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Forward First Derivative

Numerical MethodLecture 14

Prof. Jinbo BiCSE, UConn

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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

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Backward First Derivative

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Central First Derivative

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Central First Derivative

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Numerical Differentiation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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2nd-order Forward Difference

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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High-Accuracy Differentiation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Forward Finite-Divided Difference

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Backward Difference Scheme

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Backward Finite-Divided Difference

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Centered Difference Scheme

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Centered Divided Difference

• Example:

• Find derivative at x=0.5, h=0.25• True

• Forward

Basic Differentiation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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• Example:• Backward

• Centered

Basic Differentiation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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• Forward

• Backward

• Centered

High-Accuracy Differentiation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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• 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.

Summary

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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• 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.

Summary

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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• 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).

Richardson Extrapolation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Richardson Extrapolation

2)( hhE

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Richardson Extrapolation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Richardson Extrapolation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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• Example:• h=0.5

• h=0.25

• Extrapolate

Richardson’s Extrapolation

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Unevenly Spaced Data

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Unevenly Spaced Data

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Unevenly Spaced Data

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Unevenly Spaced Data

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Unevenly Spaced Data

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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Next class

• Ordinary Differential Equations• Read Chapter PT7, 25

Numerical MethodsLecture 14

Prof. Jinbo BiCSE, UConn

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