Romberg Integration Table Calculator
Enter f(x), the interval a to b, and a level count to build the Romberg table R(i,j) and show the best estimate R(n,n).
Input
Enter the integrand f(x), the integration interval, and a level count to build the Romberg table R(i,j). The trapezoidal rule with Richardson extrapolation gives a high-accuracy estimate of the definite integral.
Examples: sin(x), exp(-x^2), 1/(1+x^2). Use x as the variable; pi and e are available.
Number of table rows (1 to 16). More rows give higher accuracy.
Result
Best estimate R(5,5)
2
Definite integral over [0, 3.14159265]
Levels
6
Last-step delta
5.287326e-12
Romberg table R(i, j)
Row i uses 2 to the power i subintervals; column j is the extrapolation order. The bottom-right cell is the best estimate.
| i over j | j=0 | j=1 | j=2 | j=3 | j=4 | j=5 |
|---|---|---|---|---|---|---|
| 0 | 1.92367069e-16 | |||||
| 1 | 1.5707963268 | 2.0943951024 | ||||
| 2 | 1.8961188979 | 2.004559755 | 1.9985707318 | |||
| 3 | 1.9742316019 | 2.0002691699 | 1.9999831309 | 2.00000555 | ||
| 4 | 1.9935703438 | 2.000016591 | 1.9999997525 | 2.0000000163 | 1.9999999946 | |
| 5 | 1.998393361 | 2.0000010334 | 1.9999999962 | 2.0000000001 | 2 | 2 |
How it works
- The first column R(i,0) is the composite trapezoidal rule with 2 to the power i subintervals.
- Later columns use Richardson extrapolation R(i,j) equals R(i,j-1) plus (R(i,j-1) minus R(i-1,j-1)) divided by (4 to the power j minus 1).
- The table is lower-triangular (values only where j is at most i); the bottom-right entry R(n,n) is the best estimate.
- f(x) supports functions such as sin, cos, exp, log, sqrt and the constants pi and e.
- Functions that become non-finite at the endpoints or sample points (those with singularities) cannot be evaluated.
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Romberg Integration Table Calculator