romberg(function, a, b, args=(), tol=1.48e-08, rtol=1.48e-08, show=False, divmax=10, vec_func=False)

Returns the integral of :None:None:`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 :None:None:`vec_func` is True (default is False), then :None:None:`function` is assumed to support vector arguments.

Other Parameters

args : tuple, optional

Extra arguments to pass to function. Each element of :None:None:`args` will be passed as a single argument to :None:None:`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 :None:None:`func` handles arrays as arguments (i.e., whether it is a "vector" function). Default is False.

Parameters

function : callable

Function to be integrated.

a : float

Lower limit of integration.

b : float

Upper limit of integration.

Returns

results : float

Result of the integration.

Romberg integration of a callable function or method.

See Also

cumulative_trapezoid

Cumulative integration for sampled data.

dblquad

Double integrals.

fixed_quad

Fixed-order Gaussian quadrature.

ode

ODE integrator.

odeint

ODE integrator.

quad

Adaptive quadrature using QUADPACK.

romb

Integrators for sampled data.

simpson

Integrators for sampled data.

tplquad

Triple integrals.

Examples

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

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.integrate._quadrature.simpson scipy.integrate._quadrature.quadrature scipy.integrate._quadrature.romb scipy.integrate._quadrature.cumulative_trapezoid scipy.integrate._quadrature.fixed_quad scipy.integrate._quadrature.vectorize1 scipy.integrate._quadrature.romberg

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