mtl_gen_int_ppt_romberg.ppt 6 6 6

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

Romberg Rule of
Integration
http://numericalmethods.eng.usf.e
du

http://numericalmethods.eng.usf.edu
3
Basis of Romberg Rule
Integration


b
a
dx
)
x
(
f
I
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
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
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
(
f
I
   
n
f
n
a
b
E
n
i
i
t






 1
2
3
12
where for each i, is a point somewhere in the
domain , .
i

 
 
ih
a
,
h
i
a 

 1

6
Trapezoidal Rule
The term can be viewed as an
 
n
f
n
i
i





1
approximate average value of in .
 
x
f 
  
b
,
a
This leads us to say that the true error, Et
previously defined can be approximated as
2
1
n
Et 


7
Trapezoidal Rule
Table 1 shows the
results obtained for the
integral using multiple
segment Trapezoidal
rule for
n Value Et
1 11868 807 7.296 ---
2 11266 205 1.854 5.343
3 11153 91.4 0.8265 1.019
4 11113 51.5 0.4655 0.3594
5 11094 33.0 0.2981 0.1669
6 11084 22.9 0.2070 0.09082
7 11078 16.8 0.1521 0.05482
8 11074 12.9 0.1165 0.03560
 














30
8
8
9
2100
140000
140000
2000 dt
t
.
t
ln
x
Table 1: Multiple Segment Trapezoidal Rule Values
%
t
 %
a


8
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
Richardson’s Extrapolation for
Trapezoidal Rule
The true error, in the n-segment Trapezoidal
rule is estimated as
t
E
2
n
C
Et 
where C is an approximate constant of
proportionality. Since
n
t I
TV
E 

Where TV = true value and = approx. value
n
I

10
Richardson’s Extrapolation for
Trapezoidal Rule
From the previous development, it can be
shown that
  n
I
TV
n
C
2
2
2


when the segment size is doubled and that
3
2
2
n
n
n
I
I
I
TV



which is Richardson’s Extrapolation.

11
Example 1
The vertical distance covered by a rocket from 8 to
30 seconds is given by
 














30
8
8
9
2100
140000
140000
2000 dt
t
.
t
ln
x
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, Et for part (a).
c) Find the absolute relative true error, for part
a


12
Solution
a) m
I 11266
2  m
I 11113
4 
Using Richardson’s extrapolation
formula for Trapezoidal rule
3
2
2
n
n
n
I
I
I
TV



and choosing n=2,
3
2
4
4
I
I
I
TV



3
11266
11113
11113



m
11062


13
Solution (cont.)
b)The exact value of the above integral is
 














30
8
8
9
2100
140000
140000
2000 dt
t
.
t
ln
x
m
11061

Hence
Value
e
Approximat
Value
True
Et 

11062
11061

m
1



14
Solution (cont.)
c) The absolute relative true error t
 would then be
100
11061
11062
11061



t
%
.00904
0

Table 2 shows the Richardson’s extrapolation
results using 1, 2, 4, 8 segments. Results are
compared with those of Trapezoidal rule.

15
Solution (cont.)
Table 2: The values obtained using Richardson’s
extrapolation formula for Trapezoidal rule for
 














30
8
8
9
2100
140000
140000
2000 dt
t
.
t
ln
x
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
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
n
n
n
I
I
I
TV



This can alternately be written as
 
3
2
2
2
n
n
n
R
n
I
I
I
I



1
4 1
2
2
2



 
n
n
n
I
I
I

17
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.
 R
n
I2

Romberg Integration
Hence the estimate of the true value now is
  4
2 Ch
I
TV R
n 

Where Ch4
is an approximation of the true
error.

18
Romberg Integration
Determine another integral value with further
halving the step size (doubling the number of
segments),
 
3
2
4
4
4
n
n
n
R
n
I
I
I
I



It follows from the two previous expressions
that the true value TV can be written as
 
   
15
2
4
4
R
n
R
n
R
n
I
I
I
TV



   
1
4 1
3
2
4
4



 
R
n
R
n
n
I
I
I

19
Romberg Integration
2
1
4 1
1
1
1
1
1 



 




 k
,
I
I
I
I k
j
,
k
j
,
k
j
,
k
j
,
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(h2
). The index j represents the
more and less accurate estimate of the integral.
A general expression for Romberg integration can
be written as

20
Example 2
The vertical distance covered by a rocket from
8

t to 30

t seconds is given by
 














30
8
8
9
2100
140000
140000
2000 dt
t
.
t
ln
x
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
Solution
From Table 1, the needed values from original
Trapezoidal rule are
11868
1
1 
,
I 11266
2
1 
,
I
11113
3
1 
,
I 11074
4
1 
,
I
where the above four values correspond to using 1,
2, 4 and 8 segment Trapezoidal rule, respectively.

22
Solution (cont.)
To get the first order extrapolation values,
11065
3
11868
11266
11266
3
1
,
1
2
,
1
2
,
1
1
,
2







I
I
I
I
Similarly,
11062
3
11266
11113
11113
3
2
,
1
3
,
1
3
,
1
2
,
2







I
I
I
I
11061
3
11113
11074
11074
3
3
,
1
4
,
1
4
,
1
3
,
2







I
I
I
I

23
Solution (cont.)
For the second order extrapolation values,
11062
15
11065
11062
11062
15
1
,
2
2
,
2
2
,
2
1
,
3







I
I
I
I
Similarly,
11061
15
11062
11061
11061
15
2
,
2
3
,
2
3
,
2
2
,
3







I
I
I
I

24
Solution (cont.)
For the third order extrapolation values,
63
1
3
2
3
2
3
1
4
,
,
,
,
I
I
I
I



63
11062
11061
11061



m
11061

Table 3 shows these increased correct values in a tree
graph.

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

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

THE END

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