Romberg Method

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1/10/2010 http://numericalmethods.eng.usf.edu 1

Romberg Rule of Integration

Major: All Engineering Majors

Authors: Autar Kaw, Charlie Barker

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates
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Romberg Rule ofIntegration

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http://numericalmethods.eng.usf.edu3

Basis of Romberg RuleIntegration

=b

a

dx)x(fI

The process of measuringthe 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


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What is The Romberg Rule?

Romberg Integration is an extrapolation formula of

the Trapezoidal Rule for integration. It provides abetter approximation of the integral by reducing theTrue Error.


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Error in Multiple Segment

Trapezoidal RuleThe true error in a multiple segment Trapezoidal

Rule with n segments for an integral

Is given by

=b

a

dx)x(fI

( )

( )

n

f

nabE

n

i

i

t

=

= 12

3

12

where for each i, is a point somewhere in thedomain , .

i( )[ ]iha,hia ++ 1


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Trapezoidal RuleThe term can be viewed as an( )

n

fn

ii

=

1

approximate average value of in .( )xf [ ]b,a

This leads us to say that the true error, Et

previously defined can be approximated as

2

1

nEt


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Trapezoidal Rule

The true error gets approximately quartered asthe number of segments is doubled. Thisinformation is used to get a better approximationof the integral, and is the basis of Richardsons

extrapolation.



Richardsons Extrapolation for

Trapezoidal Rule

The true error, in the n-segment Trapezoidal ruleis estimated as

tE

2nCEt

where Cis an approximate constantofproportionality. Since

nt ITVE =

Where TV = true value and = approx. valuenI



Richardsons Extrapolation for

Trapezoidal Rule

From the previous development, it can be shownthat

( ) nITVnC

222

when the segment size is doubled and that

32

2nn

n

IIITV

+

which is Richardsons Extrapolation.



Example 1The vertical distance covered by a rocket from 8 to 30seconds is given by

=

30

8

892100140000

1400002000 dtt.

tlnx

a) Use Richardsons rule to find the distance covered.Use the 2-segment and 4-segment Trapezoidalrule results given in Table 1.

b) Find the true error, Et for part (a).c) Find the absolute relative true error, for part (a).a



Solutiona) mI 112662 = mI 111134 =

Using Richardsons extrapolation formulafor Trapezoidal rule

3

2

2

nnn

IIITV

+

and choosing n=2,

3

24

4

IIITV +

3

112661111311113 +=

m11062=



Solution (cont.)b) The exact value of the above integral is

=

30

8 892100140000

140000

2000 dtt.tlnx

m11061=

Hence

ValueeApproximatValueTrueEt =

1106211061=

m1=



Solution (cont.)c) The absolute relative true error t would then be

10011061

1106211061

=t

%.009040=

Table 2 shows the Richardsons extrapolationresults using 1, 2, 4, 8 segments. Results arecompared with those of Trapezoidal rule.



Solution (cont.)Table 2: The values obtained using Richardsonsextrapolation formula for Trapezoidal rule for

=30

8

892100140000

1400002000 dtt.t

lnx

n TrapezoidalRule

for TrapezoidalRule

RichardsonsExtrapolation

for RichardsonsExtrapolation

1248

11868112661111311074

7.2961.8540.46550.1165

--110651106211061

--0.036160.0090410.0000

Table 2: Richardsons Extrapola tion Values

t t



Romberg IntegrationRomberg integration is same as Richardsonsextrapolation formula as given previously. However,Romberg used a recursive algorithm for theextrapolation. Recall

3

2

2

nnn

IIITV

+

This can alternately be written as

( )3

2

22

nnnRn

IIII

+=

1412

2

2

+=

nnn

III



Note that the variable TVis replaced by as the

value obtained using Richardsons extrapolation formula.Note also that the sign is replaced by = sign.

( )Rn

I2

Romberg Integration

Hence the estimate of the true value now is

( ) 42

ChITVRn

+

Where Ch4 is an approximation of the true error.



Romberg IntegrationDetermine another integral value with further halvingthe step size (doubling the number of segments),

( ) 324

44

nnnRn

II

II

+=

It follows from the two previous expressionsthat the true value TV can be written as

( ) ( ) ( )15

24

4

RnRn

Rn

IIITV +

( ) ( )

1413

24

4

+=

RnRnn

III



Romberg Integration

214

1

11111

+=

+

+ k,IIII kj,kj,k

j,kj,k

The index krepresents the order of extrapolation.

k=1 represents the values obtained from the regularTrapezoidal rule, k=2 represents values obtained using thetrue estimate as O(h2). The indexjrepresents the more andless accurate estimate of the integral.

A general expression for Romberg integration can bewritten as



Example 2

The vertical distance covered by a rocket from

8=t to 30=t seconds is given by

=

30

8

892100140000

1400002000 dtt.

tlnx

Use Rombergs rule to find the distance covered. Usethe 1, 2, 4, and 8-segment Trapezoidal rule results asgiven in the Table 1.



SolutionFrom Table 1, the needed values from originalTrapezoidal rule are

1186811 =,I 1126621 =,I

1111331 =,I 1107441 =,I

where the above four values correspond to using 1, 2,4 and 8 segment Trapezoidal rule, respectively.


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Solution (cont.)To get the first order extrapolation values,

11065

3118681126611266

3

1,12,1

2,11,2

=

+=

+=

IIII

Similarly,

11062

3

112661111311113

3

2,13,1

3,12,2

=

+=

+= IIII

11061

3

111131107411074

3

3,14,1

4,13,2

=

+=

+= IIII


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Solution (cont.)For the second order extrapolation values,

11062

15

110651106211062

15

1,22,2

2,21,3

=

+=

+=

IIII

Similarly,

11061

15

110621106111061

15

2,23,2

3,22,3

=

+=

+=

IIII


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Solution (cont.)For the third order extrapolation values,

63

1323

2314

,,

,,

IIII

+=

63

110621106111061

+=

m11061

=Table 3 shows these increased correct values in a treegraph.


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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


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Additional ResourcesFor all resources on this topic such as digital audiovisuallectures, primers, textbook chapters, multiple-choicetests, worksheets in MATLAB, MATHEMATICA, MathCad

and MAPLE, blogs, related physical problems, pleasevisit

http://numericalmethods.eng.usf.edu/topics/romberg_

method.html
http://numericalmethods.eng.usf.edu/topics/romberg_method.htmlhttp://numericalmethods.eng.usf.edu/topics/romberg_method.htmlhttp://numericalmethods.eng.usf.edu/topics/romberg_method.htmlhttp://numericalmethods.eng.usf.edu/topics/romberg_method.html


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