scipy.integrate.romberg(function, a, b, args=(), tol=1.48e-08, rtol=1.48e-08, show=False, divmax=10, vec_func=False)[source]

Romberg integration of a callable function or method.

Returns the integral of function (a function of one variable) over the interval (a, b).

If show is 1, the triangular array of the intermediate results will be printed. If vec_func is True (default is False), then function is assumed to support vector arguments.


function : callable

Function to be integrated.

a : float

Lower limit of integration.

b : float

Upper limit of integration.


results : float

Result of the integration.

Other Parameters:

args : tuple, optional

Extra arguments to pass to function. Each element of args will be passed as a single argument to func. Default is to pass no extra arguments.

tol, rtol : float, optional

The desired absolute and relative tolerances. Defaults are 1.48e-8.

show : bool, optional

Whether to print the results. Default is False.

divmax : int, optional

Maximum order of extrapolation. Default is 10.

vec_func : bool, optional

Whether func handles arrays as arguments (i.e whether it is a “vector” function). Default is False.

See also

Fixed-order Gaussian quadrature.
Adaptive quadrature using QUADPACK.
Double integrals.
Triple integrals.
Integrators for sampled data.
Integrators for sampled data.
Cumulative integration for sampled data.
ODE integrator.
ODE integrator.


[R45]‘Romberg’s method’


Integrate a gaussian from 0 to 1 and compare to the error function.

>>> from scipy import integrate
>>> from scipy.special import erf
>>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2)
>>> result = integrate.romberg(gaussian, 0, 1, show=True)
Romberg integration of <function vfunc at ...> from [0, 1]
Steps  StepSize  Results
    1  1.000000  0.385872
    2  0.500000  0.412631  0.421551
    4  0.250000  0.419184  0.421368  0.421356
    8  0.125000  0.420810  0.421352  0.421350  0.421350
   16  0.062500  0.421215  0.421350  0.421350  0.421350  0.421350
   32  0.031250  0.421317  0.421350  0.421350  0.421350  0.421350  0.421350

The final result is 0.421350396475 after 33 function evaluations.

>>> print("%g %g" % (2*result, erf(1)))
0.842701 0.842701