mtl_gen_int_ppt_romberg.ppt 6 6 6

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1. 02/27/25 http://numericalmethods.eng.usf.edu 1
Romberg Rule of
Integration
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.ed
u
Transforming Numerical Methods Education for STEM
Undergraduates

2. Romberg Rule of
Integration

http://numericalmethods.eng.usf.e
du

3. http://numericalmethods.eng.usf.edu3
Basis of Romberg Rule
Integration


b
a
dx)x(fI
The process of measuring
the area under a curve.
Where:
f(x) is the integrand
a= lower limit of integration
b= upper limit of integration
f(x)
a b
y
x

b
a
dx)x(f

4. http://numericalmethods.eng.usf.edu4
What is The Romberg Rule?
Romberg Integration is an extrapolation formula
of the Trapezoidal Rule for integration. It provides
a better approximation of the integral by reducing
the True Error.

5. http://numericalmethods.eng.usf.edu5
Error in Multiple Segment
Trapezoidal Rule
The true error in a multiple segment Trapezoidal
Rule with n segments for an integral
Is given by


b
a
dx)x(fI


n
f
n
ab
E
n
i
i
t





1
2
3
12
where for each i, is a point somewhere in the
domain , .
i

 iha,hia 1

6. http://numericalmethods.eng.usf.edu6
Error in Multiple Segment
Trapezoidal Rule
The term can be viewed as an 
n
f
n
i
i


1
approximate average value of in .xf b,a
This leads us to say that the true error, E
t

previously defined can be approximated as
2
1
n
E
t


7. http://numericalmethods.eng.usf.edu7
Error in Multiple Segment
Trapezoidal Rule
Table 1 shows the
results obtained for the
integral using multiple
segment Trapezoidal
rule for
n Value E
t
1 11868 807 7.296 ---
2 11266 205 1.854 5.343
3 11153 91.4 0.8265 1.019
4 11113 51.5 0.46550.3594
5 11094 33.0 0.29810.1669
6 11084 22.9 0.20700.09082
7 11078 16.8 0.15210.05482
8 11074 12.9 0.11650.03560
 














30
8
89
2100140000
140000
2000 dtt.
t
lnx
Table 1: Multiple Segment Trapezoidal Rule Values
%
t
 %
a


8. http://numericalmethods.eng.usf.edu8
Error in Multiple Segment
Trapezoidal Rule
The true error gets approximately quartered
as the number of segments is doubled. This
information is used to get a better
approximation of the integral, and is the basis
of Richardson’s extrapolation.

9. http://numericalmethods.eng.usf.edu9
Richardson’s Extrapolation for
Trapezoidal Rule
The true error, in the n-segment Trapezoidal
rule is estimated as
t
E
2
n
C
E
t
where C is an approximate constant of
proportionality. Since
nt ITVE 
Where TV = true value and = approx. value
n
I

10. http://numericalmethods.eng.usf.edu10
Richardson’s Extrapolation for
Trapezoidal Rule
From the previous development, it can be
shown that

n
ITV
n
C
22
2

when the segment size is doubled and that
3
2
2
nn
n
II
ITV


which is Richardson’s Extrapolation.

11. http://numericalmethods.eng.usf.edu11
Example 1
The vertical distance covered by a rocket from 8 to
30 seconds is given by
 














30
8
89
2100140000
140000
2000 dtt.
t
lnx
a)Use Richardson’s rule to find the distance
covered.
Use the 2-segment and 4-segment Trapezoidal
rule results given in Table 1.
b) Find the true error, E
t
for part (a).
c) Find the absolute relative true error, for part
(a).
a

12. http://numericalmethods.eng.usf.edu12
Solution
a) mI11266
2 mI11113
4
Using Richardson’s extrapolation
formula for Trapezoidal rule
3
2
2
nn
n
II
ITV


and choosing n=2,
3
24
4
II
ITV


3
1126611113
11113


m11062

13. http://numericalmethods.eng.usf.edu13
Solution (cont.)
b)The exact value of the above integral is
 














30
8
89
2100140000
140000
2000 dtt.
t
lnx
m11061
Hence
ValueeApproximatValueTrueE
t 
1106211061
m1

14. http://numericalmethods.eng.usf.edu14
Solution (cont.)
c)The absolute relative true error twould then be
100
11061
1106211061



t
%.009040
Table 2 shows the Richardson’s extrapolation
results using 1, 2, 4, 8 segments. Results are
compared with those of Trapezoidal rule.

15. http://numericalmethods.eng.usf.edu15
Solution (cont.)
Table 2: The values obtained using Richardson’s
extrapolation formula for Trapezoidal rule for
 














30
8
89
2100140000
140000
2000 dtt.
t
lnx
n Trapezoidal
Rule
for Trapezoidal
Rule
Richardson’s
Extrapolation
for Richardson’s
Extrapolation
1
2
4
8
11868
11266
11113
11074
7.296
1.854
0.4655
0.1165
--
11065
11062
11061
--
0.03616
0.009041
0.0000
Table 2: Richardson’s Extrapolation Values
t

t


16. http://numericalmethods.eng.usf.edu16
Romberg Integration
Romberg integration is same as Richardson’s
extrapolation formula as given previously.
However, Romberg used a recursive algorithm for
the extrapolation. Recall
3
2
2
nn
n
II
ITV


This can alternately be written as

3
2
22
nn
nRn
II
II


14
12
2
2




nn
n
II
I

17. http://numericalmethods.eng.usf.edu17
Note that the variable TV is replaced by as the
value obtained using Richardson’s extrapolation formula.
Note also that the sign is replaced by = sign.

Rn
I
2

Romberg Integration
Hence the estimate of the true value now is

4
2 ChITV
Rn
Where Ch
4
is an approximation of the true
error.

18. http://numericalmethods.eng.usf.edu18
Romberg Integration
Determine another integral value with further
halving the step size (doubling the number of
segments),

3
24
44
nn
nRn
II
II


It follows from the two previous expressions
that the true value TV can be written as


15
24
4
RnRn
Rn
II
ITV



14
13
24
4




RnRn
n
II
I

19. http://numericalmethods.eng.usf.edu19
Romberg Integration
2
14
1
111
11 





 k,
II
II
k
j,kj,k
j,kj,k
The index k represents the order of
extrapolation.
k=1 represents the values obtained from the regular
Trapezoidal rule, k=2 represents values obtained using
the true estimate as O(h
2
). The index j represents the
more and less accurate estimate of the integral.
A general expression for Romberg integration can
be written as

20. http://numericalmethods.eng.usf.edu20
Example 2
The vertical distance covered by a rocket from
8t to30t seconds is given by
 














30
8
89
2100140000
140000
2000 dtt.
t
lnx
Use Romberg’s rule to find the distance covered.
Use the 1, 2, 4, and 8-segment Trapezoidal rule
results as given in the Table 1.

21. http://numericalmethods.eng.usf.edu21
Solution
From Table 1, the needed values from original
Trapezoidal rule are
11868
11

,
I 11266
21

,
I
11113
31

,
I
11074
41

,
I
where the above four values correspond to using 1,
2, 4 and 8 segment Trapezoidal rule, respectively.

22. http://numericalmethods.eng.usf.edu22
Solution (cont.)
To get the first order extrapolation values,
11065
3
1186811266
11266
3
1,12,1
2,11,2





II
II
Similarly,
11062
3
1126611113
11113
3
2,13,1
3,12,2





II
II
11061
3
1111311074
11074
3
3,14,1
4,13,2





II
II

23. http://numericalmethods.eng.usf.edu23
Solution (cont.)
For the second order extrapolation values,
11062
15
1106511062
11062
15
1,22,2
2,21,3





II
II
Similarly,
11061
15
1106211061
11061
15
2,23,2
3,22,3





II
II

24. http://numericalmethods.eng.usf.edu24
Solution (cont.)
For the third order extrapolation values,
63
1323
2314
,,
,,
II
II


63
1106211061
11061


m11061
Table 3 shows these increased correct values in a tree
graph.

25. http://numericalmethods.eng.usf.edu25
Solution (cont.)
11868
1126
11113
11074
11065
11062
11061
11062
11061
11061
1-segment
2-segment
4-segment
8-segment
First Order Second OrderThird Order
Table 3: Improved estimates of the integral value using Romberg Integration

26. Additional Resources
For all resources on this topic such as digital
audiovisual lectures, primers, textbook chapters,
multiple-choice tests, worksheets in MATLAB,
MATHEMATICA, MathCad and MAPLE, blogs, related
physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/romber
g_method.html

27. THE END
http://numericalmethods.eng.usf.edu