Romberg Integration

Romberg Integration

romberg(f, a, b, maxit = 25, tol = 1e-12, ...)

Arguments

f

function to be integrated.

a, b

end points of the interval.

maxit

maximum number of iterations.

tol

requested tolerance.

...

variables to be passed to the function.

Details

Simple Romberg integration with an explicit Richardson method applied to a series of trapezoidal integrals. This scheme works best with smooth and non-oscillatory functions and needs the least number of function calls among all integration routines.

The function does not need to be vectorized.

Value

List of value, number or iterations, and relative error.

References

Chapra, S. C., and R. P. Canale (2006). Numerical Methods for Engineers. Fifth Edition, McGraw-Hill, New York.

Note

Using a trapezoid formula Romberg integration will use 2*(2^iter-1)+iter function calls. By remembering function values this could be reduced to 2^iter+1 calls.

See also

integrate, quadgr

Examples

romberg(sin, 0, pi, tol = 1e-15)    #  2                 , rel.error 1e-15
#> $value
#> [1] 2
#> 
#> $iter
#> [1] 6
#> 
#> $rel.error
#> [1] 0
#> 
romberg(exp, 0, 1,  tol = 1e-15)    #  1.718281828459044 , rel error 1e-15
#> $value
#> [1] 1.718282
#> 
#> $iter
#> [1] 7
#> 
#> $rel.error
#> [1] 0
#> 
                                    #  1.718281828459045 , i.e. exp(1) - 1

f <- function(x, p) sin(x) * cos(p*x)
romberg(f, 0, pi, p = 2)            #  2/3               , abs.err 1.5e-14
#> $value
#> [1] -0.6666667
#> 
#> $iter
#> [1] 7
#> 
#> $rel.error
#> [1] 5.995204e-15
#> 
# value: -0.6666667, iter: 7, rel.error: 1e-12