Numerical Integration: Romberg's Method

endpoint a

0

endpoint b

2π

subdivisions

3

2

4

2

5

2

function

x-sin(5x)-3



2

x

+x+1cos(x)

Romberg's method table

function f graphic

\*SubsuperscriptBox\(∫\),[{0}]\)\),[{2, , π}]\)\)\)\*TagBox[FormBox[RowBox[{x, -, RowBox[{sin, (, RowBox[{5, , x}], )}], -, 3}], TraditionalForm], TraditionalForm, Rule[Editable, True]]\)x

integral actual value: 0.889653

n	h	R(n,0)	R(n,1)	R(n,2)	R(n,3)	R(n,4)	R(n,5)
0	2π	0.889653
1	π	0.889653	0.889653
2	π 2	0.889653	0.889653	0.889653
3	π 4	0.889653	0.889653	0.889653	0.889653
4	π 8	0.889653	0.889653	0.889653	0.889653	0.889653
5	π 16	0.889653	0.889653	0.889653	0.889653	0.889653	0.889653

Romberg's method is a powerful numerical integration technique that uses refinements of the extended trapezoidal rule to reduce error in definite integrals.

Details

Each element of the column

j

is calculated from the elements of the preceding column,

j-1

, using the expression

R(i,j)=

j

4

R(i,j-1)-R(i-1,j-1)

j

4

-1

.

The elements of the first column are calculated using the trapezoidal rule with

n

2

subdivisions.

External Links

Romberg Integration (Wolfram MathWorld)

Trapezoidal Rule (Wolfram MathWorld)

Permanent Citation

Eugenio Bravo Sevilla

"Numerical Integration: Romberg's Method"
http://demonstrations.wolfram.com/NumericalIntegrationRombergsMethod/
Wolfram Demonstrations Project
Published: March 7, 2011