Transcript Numerical Differentiation Chapter 23

Chapter 23
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Numerical Differentiation
Chapter 23
• Notion of numerical differentiation has been
introduced in Chapter 4. In this chapter more
accurate formulas that retain more terms will
be developed.
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High Accuracy Differentiation
Formulas
• High-accuracy divided-difference formulas can be
generated by including additional terms from the
Taylor series expansion.
f ( xi ) 2
h 
2
f ( xi 1 )  f ( xi ) f ( xi )
f ( xi) 

h  O(h 2 )
h
2
f ( xi  2 )  2 f ( xi 1 )  f ( xi )
f ( xi ) 
 O ( h)
2
h
f ( xi 1 )  f ( xi ) f ( xi  2 )  2 f ( xi 1 )  f ( xi )
2
f ( xi) 

h

O
(
h
)
2
h
2h
 f ( xi  2 )  4 f ( xi 1 )  3 f ( xi )
f ( xi) 
 O(h 2 )
2h
f ( xi 1 )  f ( xi )  f ( xi )h 
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• Inclusion of the 2nd derivative term has
improved the accuracy to O(h2).
• Similar improved versions can be developed
for the backward and centered formulas as
well as for the approximations of the higher
derivatives.
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Richardson Extrapolation
• There are two ways to improve derivative
estimates when employing finite divided
differences:
– Decrease the step size, or
– Use a higher-order formula that employs more
points.
• A third approach, based on Richardson
extrapolation, uses two derivative estimates t
compute a third, more accurate approximation.
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I  I (h2 ) 
[ I (h2 )  I (h1 )]
2
(h1 / h2 )  1
h2  h1 / 2
4
1
I  I (h2 )  I (h1 )]
3
3
4
1
D  D (h2 )  D (h1 )]
3
3
• For centered difference approximations with
O(h2). The application of this formula yield a
new derivative estimate of O(h4).
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Derivatives of Unequally Spaced Data
• Data from experiments or field studies are often
collected at unequal intervals. One way to handle
such data is to fit a second-order Lagrange
interpolating polynomial.
f ( x)  f ( xi 1 )
f ( xi 1 )
2 x  xi  xi 1
2 x  xi 1  xi 1
 f ( xi )

xi 1  xi xi 1  xi 1 
xi  xi 1 xi  xi 1 
2 x  xi 1  xi
xi 1  xi 1 xi 1  xi 
x is the value at which you want to estimate the
derivative.
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Derivatives and Integrals for Data with
Errors
Figure 23.5
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