~ Numerical Differentiation and Integration ~
Integration of Equations
Chapter 22
Credit: Prof. Lale Yurttas, Chemical Eng., Texas A&M University
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1
Romberg Integration
Successive application of the trapezoidal rule to attain efficient numerical integrals of functions.
Richardson’s Extrapolation: In numerical analysis, Richardson extrapolation is a
sequence acceleration method, used to improve the rate of convergence of a sequence.
Here we use two estimates of an integral to compute a third and more accurate approximation.
I  I ( h)  E ( h)
h  (b  a) / n
n  (b  a ) / h
I (h1 )  E (h1 )  I (h2 )  E (h2 ) I = exact value of integral E(h) = the truncation error
I(h): trapezoidal rule (n segments, step size h)
ba 2
2
E 
h f   O ( h )
12
(assume f  is constant for different step sizes)
2
1
2
2
E (h1 ) h

E (h2 ) h

I (h1 )  E (h2 )h1 / h2 
2
 h1 
E (h1 )  E (h2 ) 
 h2 
2
I (h2 )  I (h1 )
 I (h2 )  E (h2 )  E (h2 ) 
h1 / h2 2 1
I  I ( h2 )  E ( h2 )
1
I (h2 )  I (h1 )
I  I ( h2 ) 
2
h1 / h2   1
Improved estimate of the integral.
It is shown that the error of this
estimate is O(h4). Trapezoidal rule
had an error estimate of O(h2).
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1
I (h2 )  I (h1 )
I  I (h2 ) 
2
h1 / h2   1
If (h2  h1 / 2) 
I  I (h2 ) 
1
4
1


I
(
h
)

I
(
h
)

I
(
h
)

I (h1 )
2
1
2
2
2 1
3
3
Example
f(x) = 0.2 +25x – 200x2 + 675x3 – 900x4 + 400x5
I (True Integral value) = 1.6405
Evaluate the integral of
from a=0 to b=0.8.
Segments
h
Integral
εtr%
Segments 1 & 2 combined to give :
4
1
(1.0688)  (0.1728)  1.3675
3
3
E t  1.6405  1.3675  0.273 ( t  16.6%)
1
0.8
0.1728
89.5
2
0.4
1.0688
34.9
I
Segments 2 & 4 combined to give :
4
0.2
1.4848
9.5
In each case, two estimates with
error O(h2) are combined to give a
third estimate with error O(h4)
4
1
(1.4848)  (1.0688)  1.6234
3
3
E t  1.6405  1.6234  0.0171 ( t  1%)
I
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3
In Example 22.1, we computed two improved estimates of O(h4). These two
estimates can, in turn, be combined to yield an even better value with error O(h6).
For the special case where the original trapezoidal estimates are based on
successive halving of the step size, the equation used for O(h6) accuracy is:
where Im and Il are more and less accurate estimates
16
1
I
15
Im 
15
Il
Similarly, two O(h6) estimates can be combined to compute an I that is O(h8).
64
1
I
Im  Il
63
63
4
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The Romberg Integration Algorithm
I j ,k 
4 k 1 I j 1,k 1  I j ,k 1
4 k 1  1
k=1 refers to trapezoidal rule, hence O(h2) accuracy.
k=2 refers to O(h4) and
k=3  O(h6)
Index j is used to distinguish between the more (j+1) and the less (j) accurate estimates.
5
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