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1008 lines
32 KiB
Python
1008 lines
32 KiB
Python
import functools
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import numpy as np
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import math
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import types
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import warnings
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# trapezoid is a public function for scipy.integrate,
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# even though it's actually a NumPy function.
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from numpy import trapz as trapezoid
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from scipy.special import roots_legendre
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from scipy.special import gammaln
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__all__ = ['fixed_quad', 'quadrature', 'romberg', 'romb',
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'trapezoid', 'trapz', 'simps', 'simpson',
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'cumulative_trapezoid', 'cumtrapz', 'newton_cotes',
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'AccuracyWarning']
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# Make See Also linking for our local copy work properly
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def _copy_func(f):
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"""Based on http://stackoverflow.com/a/6528148/190597 (Glenn Maynard)"""
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g = types.FunctionType(f.__code__, f.__globals__, name=f.__name__,
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argdefs=f.__defaults__, closure=f.__closure__)
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g = functools.update_wrapper(g, f)
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g.__kwdefaults__ = f.__kwdefaults__
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return g
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trapezoid = _copy_func(trapezoid)
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if trapezoid.__doc__:
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trapezoid.__doc__ = trapezoid.__doc__.replace(
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'sum, cumsum', 'numpy.cumsum')
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# Note: alias kept for backwards compatibility. Rename was done
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# because trapz is a slur in colloquial English (see gh-12924).
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def trapz(y, x=None, dx=1.0, axis=-1):
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"""`An alias of `trapezoid`.
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`trapz` is kept for backwards compatibility. For new code, prefer
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`trapezoid` instead.
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"""
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return trapezoid(y, x=x, dx=dx, axis=axis)
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class AccuracyWarning(Warning):
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pass
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def _cached_roots_legendre(n):
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"""
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Cache roots_legendre results to speed up calls of the fixed_quad
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function.
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"""
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if n in _cached_roots_legendre.cache:
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return _cached_roots_legendre.cache[n]
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_cached_roots_legendre.cache[n] = roots_legendre(n)
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return _cached_roots_legendre.cache[n]
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_cached_roots_legendre.cache = dict()
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def fixed_quad(func, a, b, args=(), n=5):
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"""
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Compute a definite integral using fixed-order Gaussian quadrature.
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Integrate `func` from `a` to `b` using Gaussian quadrature of
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order `n`.
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Parameters
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----------
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func : callable
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A Python function or method to integrate (must accept vector inputs).
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If integrating a vector-valued function, the returned array must have
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shape ``(..., len(x))``.
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a : float
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Lower limit of integration.
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b : float
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Upper limit of integration.
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args : tuple, optional
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Extra arguments to pass to function, if any.
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n : int, optional
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Order of quadrature integration. Default is 5.
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Returns
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-------
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val : float
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Gaussian quadrature approximation to the integral
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none : None
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Statically returned value of None
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See Also
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--------
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quad : adaptive quadrature using QUADPACK
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dblquad : double integrals
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tplquad : triple integrals
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romberg : adaptive Romberg quadrature
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quadrature : adaptive Gaussian quadrature
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romb : integrators for sampled data
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simpson : integrators for sampled data
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cumulative_trapezoid : cumulative integration for sampled data
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ode : ODE integrator
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odeint : ODE integrator
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Examples
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--------
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>>> from scipy import integrate
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>>> f = lambda x: x**8
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>>> integrate.fixed_quad(f, 0.0, 1.0, n=4)
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(0.1110884353741496, None)
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>>> integrate.fixed_quad(f, 0.0, 1.0, n=5)
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(0.11111111111111102, None)
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>>> print(1/9.0) # analytical result
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0.1111111111111111
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>>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=4)
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(0.9999999771971152, None)
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>>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=5)
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(1.000000000039565, None)
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>>> np.sin(np.pi/2)-np.sin(0) # analytical result
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1.0
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"""
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x, w = _cached_roots_legendre(n)
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x = np.real(x)
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if np.isinf(a) or np.isinf(b):
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raise ValueError("Gaussian quadrature is only available for "
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"finite limits.")
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y = (b-a)*(x+1)/2.0 + a
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return (b-a)/2.0 * np.sum(w*func(y, *args), axis=-1), None
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def vectorize1(func, args=(), vec_func=False):
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"""Vectorize the call to a function.
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This is an internal utility function used by `romberg` and
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`quadrature` to create a vectorized version of a function.
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If `vec_func` is True, the function `func` is assumed to take vector
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arguments.
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Parameters
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----------
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func : callable
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User defined function.
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args : tuple, optional
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Extra arguments for the function.
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vec_func : bool, optional
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True if the function func takes vector arguments.
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Returns
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-------
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vfunc : callable
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A function that will take a vector argument and return the
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result.
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"""
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if vec_func:
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def vfunc(x):
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return func(x, *args)
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else:
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def vfunc(x):
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if np.isscalar(x):
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return func(x, *args)
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x = np.asarray(x)
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# call with first point to get output type
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y0 = func(x[0], *args)
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n = len(x)
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dtype = getattr(y0, 'dtype', type(y0))
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output = np.empty((n,), dtype=dtype)
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output[0] = y0
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for i in range(1, n):
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output[i] = func(x[i], *args)
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return output
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return vfunc
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def quadrature(func, a, b, args=(), tol=1.49e-8, rtol=1.49e-8, maxiter=50,
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vec_func=True, miniter=1):
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"""
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Compute a definite integral using fixed-tolerance Gaussian quadrature.
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Integrate `func` from `a` to `b` using Gaussian quadrature
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with absolute tolerance `tol`.
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Parameters
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----------
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func : function
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A Python function or method to integrate.
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a : float
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Lower limit of integration.
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b : float
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Upper limit of integration.
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args : tuple, optional
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Extra arguments to pass to function.
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tol, rtol : float, optional
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Iteration stops when error between last two iterates is less than
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`tol` OR the relative change is less than `rtol`.
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maxiter : int, optional
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Maximum order of Gaussian quadrature.
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vec_func : bool, optional
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True or False if func handles arrays as arguments (is
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a "vector" function). Default is True.
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miniter : int, optional
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Minimum order of Gaussian quadrature.
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Returns
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-------
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val : float
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Gaussian quadrature approximation (within tolerance) to integral.
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err : float
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Difference between last two estimates of the integral.
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See also
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--------
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romberg: adaptive Romberg quadrature
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fixed_quad: fixed-order Gaussian quadrature
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quad: adaptive quadrature using QUADPACK
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dblquad: double integrals
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tplquad: triple integrals
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romb: integrator for sampled data
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simpson: integrator for sampled data
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cumulative_trapezoid: cumulative integration for sampled data
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ode: ODE integrator
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odeint: ODE integrator
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Examples
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--------
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>>> from scipy import integrate
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>>> f = lambda x: x**8
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>>> integrate.quadrature(f, 0.0, 1.0)
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(0.11111111111111106, 4.163336342344337e-17)
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>>> print(1/9.0) # analytical result
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0.1111111111111111
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>>> integrate.quadrature(np.cos, 0.0, np.pi/2)
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(0.9999999999999536, 3.9611425250996035e-11)
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>>> np.sin(np.pi/2)-np.sin(0) # analytical result
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1.0
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"""
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if not isinstance(args, tuple):
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args = (args,)
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vfunc = vectorize1(func, args, vec_func=vec_func)
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val = np.inf
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err = np.inf
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maxiter = max(miniter+1, maxiter)
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for n in range(miniter, maxiter+1):
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newval = fixed_quad(vfunc, a, b, (), n)[0]
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err = abs(newval-val)
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val = newval
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if err < tol or err < rtol*abs(val):
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break
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else:
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warnings.warn(
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"maxiter (%d) exceeded. Latest difference = %e" % (maxiter, err),
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AccuracyWarning)
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return val, err
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def tupleset(t, i, value):
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l = list(t)
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l[i] = value
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return tuple(l)
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# Note: alias kept for backwards compatibility. Rename was done
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# because cumtrapz is a slur in colloquial English (see gh-12924).
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def cumtrapz(y, x=None, dx=1.0, axis=-1, initial=None):
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"""`An alias of `cumulative_trapezoid`.
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`cumtrapz` is kept for backwards compatibility. For new code, prefer
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`cumulative_trapezoid` instead.
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"""
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return cumulative_trapezoid(y, x=x, dx=dx, axis=axis, initial=initial)
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def cumulative_trapezoid(y, x=None, dx=1.0, axis=-1, initial=None):
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"""
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Cumulatively integrate y(x) using the composite trapezoidal rule.
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Parameters
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----------
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y : array_like
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Values to integrate.
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x : array_like, optional
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The coordinate to integrate along. If None (default), use spacing `dx`
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between consecutive elements in `y`.
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dx : float, optional
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Spacing between elements of `y`. Only used if `x` is None.
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axis : int, optional
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Specifies the axis to cumulate. Default is -1 (last axis).
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initial : scalar, optional
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If given, insert this value at the beginning of the returned result.
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Typically this value should be 0. Default is None, which means no
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value at ``x[0]`` is returned and `res` has one element less than `y`
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along the axis of integration.
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Returns
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-------
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res : ndarray
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The result of cumulative integration of `y` along `axis`.
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If `initial` is None, the shape is such that the axis of integration
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has one less value than `y`. If `initial` is given, the shape is equal
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to that of `y`.
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See Also
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--------
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numpy.cumsum, numpy.cumprod
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quad: adaptive quadrature using QUADPACK
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romberg: adaptive Romberg quadrature
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quadrature: adaptive Gaussian quadrature
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fixed_quad: fixed-order Gaussian quadrature
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dblquad: double integrals
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tplquad: triple integrals
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romb: integrators for sampled data
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ode: ODE integrators
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odeint: ODE integrators
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Examples
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--------
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>>> from scipy import integrate
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>>> import matplotlib.pyplot as plt
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>>> x = np.linspace(-2, 2, num=20)
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>>> y = x
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>>> y_int = integrate.cumulative_trapezoid(y, x, initial=0)
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>>> plt.plot(x, y_int, 'ro', x, y[0] + 0.5 * x**2, 'b-')
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>>> plt.show()
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"""
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y = np.asarray(y)
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if x is None:
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d = dx
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else:
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x = np.asarray(x)
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if x.ndim == 1:
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d = np.diff(x)
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# reshape to correct shape
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shape = [1] * y.ndim
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shape[axis] = -1
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d = d.reshape(shape)
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elif len(x.shape) != len(y.shape):
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raise ValueError("If given, shape of x must be 1-D or the "
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"same as y.")
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else:
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d = np.diff(x, axis=axis)
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if d.shape[axis] != y.shape[axis] - 1:
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raise ValueError("If given, length of x along axis must be the "
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"same as y.")
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nd = len(y.shape)
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slice1 = tupleset((slice(None),)*nd, axis, slice(1, None))
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slice2 = tupleset((slice(None),)*nd, axis, slice(None, -1))
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res = np.cumsum(d * (y[slice1] + y[slice2]) / 2.0, axis=axis)
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if initial is not None:
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if not np.isscalar(initial):
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raise ValueError("`initial` parameter should be a scalar.")
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shape = list(res.shape)
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shape[axis] = 1
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res = np.concatenate([np.full(shape, initial, dtype=res.dtype), res],
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axis=axis)
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return res
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def _basic_simpson(y, start, stop, x, dx, axis):
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nd = len(y.shape)
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if start is None:
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start = 0
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step = 2
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slice_all = (slice(None),)*nd
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slice0 = tupleset(slice_all, axis, slice(start, stop, step))
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slice1 = tupleset(slice_all, axis, slice(start+1, stop+1, step))
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slice2 = tupleset(slice_all, axis, slice(start+2, stop+2, step))
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if x is None: # Even-spaced Simpson's rule.
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result = np.sum(dx/3.0 * (y[slice0]+4*y[slice1]+y[slice2]),
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axis=axis)
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else:
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# Account for possibly different spacings.
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# Simpson's rule changes a bit.
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h = np.diff(x, axis=axis)
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sl0 = tupleset(slice_all, axis, slice(start, stop, step))
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sl1 = tupleset(slice_all, axis, slice(start+1, stop+1, step))
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h0 = h[sl0]
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h1 = h[sl1]
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hsum = h0 + h1
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hprod = h0 * h1
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h0divh1 = h0 / h1
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tmp = hsum/6.0 * (y[slice0]*(2-1.0/h0divh1) +
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y[slice1]*hsum*hsum/hprod +
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y[slice2]*(2-h0divh1))
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result = np.sum(tmp, axis=axis)
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return result
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# Note: alias kept for backwards compatibility. simps was renamed to simpson
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# because the former is a slur in colloquial English (see gh-12924).
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def simps(y, x=None, dx=1, axis=-1, even='avg'):
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"""`An alias of `simpson`.
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`simps` is kept for backwards compatibility. For new code, prefer
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`simpson` instead.
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"""
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return simpson(y, x=x, dx=dx, axis=axis, even=even)
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def simpson(y, x=None, dx=1, axis=-1, even='avg'):
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"""
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Integrate y(x) using samples along the given axis and the composite
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Simpson's rule. If x is None, spacing of dx is assumed.
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If there are an even number of samples, N, then there are an odd
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number of intervals (N-1), but Simpson's rule requires an even number
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of intervals. The parameter 'even' controls how this is handled.
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Parameters
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----------
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y : array_like
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Array to be integrated.
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x : array_like, optional
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If given, the points at which `y` is sampled.
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dx : int, optional
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Spacing of integration points along axis of `x`. Only used when
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`x` is None. Default is 1.
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axis : int, optional
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Axis along which to integrate. Default is the last axis.
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even : str {'avg', 'first', 'last'}, optional
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'avg' : Average two results:1) use the first N-2 intervals with
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a trapezoidal rule on the last interval and 2) use the last
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N-2 intervals with a trapezoidal rule on the first interval.
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'first' : Use Simpson's rule for the first N-2 intervals with
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a trapezoidal rule on the last interval.
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'last' : Use Simpson's rule for the last N-2 intervals with a
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trapezoidal rule on the first interval.
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See Also
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--------
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quad: adaptive quadrature using QUADPACK
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romberg: adaptive Romberg quadrature
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quadrature: adaptive Gaussian quadrature
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fixed_quad: fixed-order Gaussian quadrature
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dblquad: double integrals
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tplquad: triple integrals
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romb: integrators for sampled data
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cumulative_trapezoid: cumulative integration for sampled data
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ode: ODE integrators
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odeint: ODE integrators
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Notes
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-----
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For an odd number of samples that are equally spaced the result is
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exact if the function is a polynomial of order 3 or less. If
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the samples are not equally spaced, then the result is exact only
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if the function is a polynomial of order 2 or less.
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Examples
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--------
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>>> from scipy import integrate
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>>> x = np.arange(0, 10)
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>>> y = np.arange(0, 10)
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>>> integrate.simpson(y, x)
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40.5
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>>> y = np.power(x, 3)
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>>> integrate.simpson(y, x)
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1642.5
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>>> integrate.quad(lambda x: x**3, 0, 9)[0]
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1640.25
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>>> integrate.simpson(y, x, even='first')
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1644.5
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"""
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y = np.asarray(y)
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nd = len(y.shape)
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N = y.shape[axis]
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last_dx = dx
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first_dx = dx
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returnshape = 0
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if x is not None:
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x = np.asarray(x)
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if len(x.shape) == 1:
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shapex = [1] * nd
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shapex[axis] = x.shape[0]
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saveshape = x.shape
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returnshape = 1
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x = x.reshape(tuple(shapex))
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elif len(x.shape) != len(y.shape):
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raise ValueError("If given, shape of x must be 1-D or the "
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"same as y.")
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if x.shape[axis] != N:
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raise ValueError("If given, length of x along axis must be the "
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"same as y.")
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if N % 2 == 0:
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val = 0.0
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result = 0.0
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slice1 = (slice(None),)*nd
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slice2 = (slice(None),)*nd
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if even not in ['avg', 'last', 'first']:
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raise ValueError("Parameter 'even' must be "
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"'avg', 'last', or 'first'.")
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# Compute using Simpson's rule on first intervals
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if even in ['avg', 'first']:
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slice1 = tupleset(slice1, axis, -1)
|
|
slice2 = tupleset(slice2, axis, -2)
|
|
if x is not None:
|
|
last_dx = x[slice1] - x[slice2]
|
|
val += 0.5*last_dx*(y[slice1]+y[slice2])
|
|
result = _basic_simpson(y, 0, N-3, x, dx, axis)
|
|
# Compute using Simpson's rule on last set of intervals
|
|
if even in ['avg', 'last']:
|
|
slice1 = tupleset(slice1, axis, 0)
|
|
slice2 = tupleset(slice2, axis, 1)
|
|
if x is not None:
|
|
first_dx = x[tuple(slice2)] - x[tuple(slice1)]
|
|
val += 0.5*first_dx*(y[slice2]+y[slice1])
|
|
result += _basic_simpson(y, 1, N-2, x, dx, axis)
|
|
if even == 'avg':
|
|
val /= 2.0
|
|
result /= 2.0
|
|
result = result + val
|
|
else:
|
|
result = _basic_simpson(y, 0, N-2, x, dx, axis)
|
|
if returnshape:
|
|
x = x.reshape(saveshape)
|
|
return result
|
|
|
|
|
|
def romb(y, dx=1.0, axis=-1, show=False):
|
|
"""
|
|
Romberg integration using samples of a function.
|
|
|
|
Parameters
|
|
----------
|
|
y : array_like
|
|
A vector of ``2**k + 1`` equally-spaced samples of a function.
|
|
dx : float, optional
|
|
The sample spacing. Default is 1.
|
|
axis : int, optional
|
|
The axis along which to integrate. Default is -1 (last axis).
|
|
show : bool, optional
|
|
When `y` is a single 1-D array, then if this argument is True
|
|
print the table showing Richardson extrapolation from the
|
|
samples. Default is False.
|
|
|
|
Returns
|
|
-------
|
|
romb : ndarray
|
|
The integrated result for `axis`.
|
|
|
|
See also
|
|
--------
|
|
quad : adaptive quadrature using QUADPACK
|
|
romberg : adaptive Romberg quadrature
|
|
quadrature : adaptive Gaussian quadrature
|
|
fixed_quad : fixed-order Gaussian quadrature
|
|
dblquad : double integrals
|
|
tplquad : triple integrals
|
|
simpson : integrators for sampled data
|
|
cumulative_trapezoid : cumulative integration for sampled data
|
|
ode : ODE integrators
|
|
odeint : ODE integrators
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy import integrate
|
|
>>> x = np.arange(10, 14.25, 0.25)
|
|
>>> y = np.arange(3, 12)
|
|
|
|
>>> integrate.romb(y)
|
|
56.0
|
|
|
|
>>> y = np.sin(np.power(x, 2.5))
|
|
>>> integrate.romb(y)
|
|
-0.742561336672229
|
|
|
|
>>> integrate.romb(y, show=True)
|
|
Richardson Extrapolation Table for Romberg Integration
|
|
====================================================================
|
|
-0.81576
|
|
4.63862 6.45674
|
|
-1.10581 -3.02062 -3.65245
|
|
-2.57379 -3.06311 -3.06595 -3.05664
|
|
-1.34093 -0.92997 -0.78776 -0.75160 -0.74256
|
|
====================================================================
|
|
-0.742561336672229
|
|
"""
|
|
y = np.asarray(y)
|
|
nd = len(y.shape)
|
|
Nsamps = y.shape[axis]
|
|
Ninterv = Nsamps-1
|
|
n = 1
|
|
k = 0
|
|
while n < Ninterv:
|
|
n <<= 1
|
|
k += 1
|
|
if n != Ninterv:
|
|
raise ValueError("Number of samples must be one plus a "
|
|
"non-negative power of 2.")
|
|
|
|
R = {}
|
|
slice_all = (slice(None),) * nd
|
|
slice0 = tupleset(slice_all, axis, 0)
|
|
slicem1 = tupleset(slice_all, axis, -1)
|
|
h = Ninterv * np.asarray(dx, dtype=float)
|
|
R[(0, 0)] = (y[slice0] + y[slicem1])/2.0*h
|
|
slice_R = slice_all
|
|
start = stop = step = Ninterv
|
|
for i in range(1, k+1):
|
|
start >>= 1
|
|
slice_R = tupleset(slice_R, axis, slice(start, stop, step))
|
|
step >>= 1
|
|
R[(i, 0)] = 0.5*(R[(i-1, 0)] + h*y[slice_R].sum(axis=axis))
|
|
for j in range(1, i+1):
|
|
prev = R[(i, j-1)]
|
|
R[(i, j)] = prev + (prev-R[(i-1, j-1)]) / ((1 << (2*j))-1)
|
|
h /= 2.0
|
|
|
|
if show:
|
|
if not np.isscalar(R[(0, 0)]):
|
|
print("*** Printing table only supported for integrals" +
|
|
" of a single data set.")
|
|
else:
|
|
try:
|
|
precis = show[0]
|
|
except (TypeError, IndexError):
|
|
precis = 5
|
|
try:
|
|
width = show[1]
|
|
except (TypeError, IndexError):
|
|
width = 8
|
|
formstr = "%%%d.%df" % (width, precis)
|
|
|
|
title = "Richardson Extrapolation Table for Romberg Integration"
|
|
print("", title.center(68), "=" * 68, sep="\n", end="\n")
|
|
for i in range(k+1):
|
|
for j in range(i+1):
|
|
print(formstr % R[(i, j)], end=" ")
|
|
print()
|
|
print("=" * 68)
|
|
print()
|
|
|
|
return R[(k, k)]
|
|
|
|
# Romberg quadratures for numeric integration.
|
|
#
|
|
# Written by Scott M. Ransom <ransom@cfa.harvard.edu>
|
|
# last revision: 14 Nov 98
|
|
#
|
|
# Cosmetic changes by Konrad Hinsen <hinsen@cnrs-orleans.fr>
|
|
# last revision: 1999-7-21
|
|
#
|
|
# Adapted to SciPy by Travis Oliphant <oliphant.travis@ieee.org>
|
|
# last revision: Dec 2001
|
|
|
|
|
|
def _difftrap(function, interval, numtraps):
|
|
"""
|
|
Perform part of the trapezoidal rule to integrate a function.
|
|
Assume that we had called difftrap with all lower powers-of-2
|
|
starting with 1. Calling difftrap only returns the summation
|
|
of the new ordinates. It does _not_ multiply by the width
|
|
of the trapezoids. This must be performed by the caller.
|
|
'function' is the function to evaluate (must accept vector arguments).
|
|
'interval' is a sequence with lower and upper limits
|
|
of integration.
|
|
'numtraps' is the number of trapezoids to use (must be a
|
|
power-of-2).
|
|
"""
|
|
if numtraps <= 0:
|
|
raise ValueError("numtraps must be > 0 in difftrap().")
|
|
elif numtraps == 1:
|
|
return 0.5*(function(interval[0])+function(interval[1]))
|
|
else:
|
|
numtosum = numtraps/2
|
|
h = float(interval[1]-interval[0])/numtosum
|
|
lox = interval[0] + 0.5 * h
|
|
points = lox + h * np.arange(numtosum)
|
|
s = np.sum(function(points), axis=0)
|
|
return s
|
|
|
|
|
|
def _romberg_diff(b, c, k):
|
|
"""
|
|
Compute the differences for the Romberg quadrature corrections.
|
|
See Forman Acton's "Real Computing Made Real," p 143.
|
|
"""
|
|
tmp = 4.0**k
|
|
return (tmp * c - b)/(tmp - 1.0)
|
|
|
|
|
|
def _printresmat(function, interval, resmat):
|
|
# Print the Romberg result matrix.
|
|
i = j = 0
|
|
print('Romberg integration of', repr(function), end=' ')
|
|
print('from', interval)
|
|
print('')
|
|
print('%6s %9s %9s' % ('Steps', 'StepSize', 'Results'))
|
|
for i in range(len(resmat)):
|
|
print('%6d %9f' % (2**i, (interval[1]-interval[0])/(2.**i)), end=' ')
|
|
for j in range(i+1):
|
|
print('%9f' % (resmat[i][j]), end=' ')
|
|
print('')
|
|
print('')
|
|
print('The final result is', resmat[i][j], end=' ')
|
|
print('after', 2**(len(resmat)-1)+1, 'function evaluations.')
|
|
|
|
|
|
def romberg(function, a, b, args=(), tol=1.48e-8, rtol=1.48e-8, show=False,
|
|
divmax=10, vec_func=False):
|
|
"""
|
|
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.
|
|
|
|
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.
|
|
|
|
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_quad : Fixed-order Gaussian quadrature.
|
|
quad : Adaptive quadrature using QUADPACK.
|
|
dblquad : Double integrals.
|
|
tplquad : Triple integrals.
|
|
romb : Integrators for sampled data.
|
|
simpson : Integrators for sampled data.
|
|
cumulative_trapezoid : Cumulative integration for sampled data.
|
|
ode : ODE integrator.
|
|
odeint : ODE integrator.
|
|
|
|
References
|
|
----------
|
|
.. [1] 'Romberg's method' https://en.wikipedia.org/wiki/Romberg%27s_method
|
|
|
|
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
|
|
|
|
"""
|
|
if np.isinf(a) or np.isinf(b):
|
|
raise ValueError("Romberg integration only available "
|
|
"for finite limits.")
|
|
vfunc = vectorize1(function, args, vec_func=vec_func)
|
|
n = 1
|
|
interval = [a, b]
|
|
intrange = b - a
|
|
ordsum = _difftrap(vfunc, interval, n)
|
|
result = intrange * ordsum
|
|
resmat = [[result]]
|
|
err = np.inf
|
|
last_row = resmat[0]
|
|
for i in range(1, divmax+1):
|
|
n *= 2
|
|
ordsum += _difftrap(vfunc, interval, n)
|
|
row = [intrange * ordsum / n]
|
|
for k in range(i):
|
|
row.append(_romberg_diff(last_row[k], row[k], k+1))
|
|
result = row[i]
|
|
lastresult = last_row[i-1]
|
|
if show:
|
|
resmat.append(row)
|
|
err = abs(result - lastresult)
|
|
if err < tol or err < rtol * abs(result):
|
|
break
|
|
last_row = row
|
|
else:
|
|
warnings.warn(
|
|
"divmax (%d) exceeded. Latest difference = %e" % (divmax, err),
|
|
AccuracyWarning)
|
|
|
|
if show:
|
|
_printresmat(vfunc, interval, resmat)
|
|
return result
|
|
|
|
|
|
# Coefficients for Newton-Cotes quadrature
|
|
#
|
|
# These are the points being used
|
|
# to construct the local interpolating polynomial
|
|
# a are the weights for Newton-Cotes integration
|
|
# B is the error coefficient.
|
|
# error in these coefficients grows as N gets larger.
|
|
# or as samples are closer and closer together
|
|
|
|
# You can use maxima to find these rational coefficients
|
|
# for equally spaced data using the commands
|
|
# a(i,N) := integrate(product(r-j,j,0,i-1) * product(r-j,j,i+1,N),r,0,N) / ((N-i)! * i!) * (-1)^(N-i);
|
|
# Be(N) := N^(N+2)/(N+2)! * (N/(N+3) - sum((i/N)^(N+2)*a(i,N),i,0,N));
|
|
# Bo(N) := N^(N+1)/(N+1)! * (N/(N+2) - sum((i/N)^(N+1)*a(i,N),i,0,N));
|
|
# B(N) := (if (mod(N,2)=0) then Be(N) else Bo(N));
|
|
#
|
|
# pre-computed for equally-spaced weights
|
|
#
|
|
# num_a, den_a, int_a, num_B, den_B = _builtincoeffs[N]
|
|
#
|
|
# a = num_a*array(int_a)/den_a
|
|
# B = num_B*1.0 / den_B
|
|
#
|
|
# integrate(f(x),x,x_0,x_N) = dx*sum(a*f(x_i)) + B*(dx)^(2k+3) f^(2k+2)(x*)
|
|
# where k = N // 2
|
|
#
|
|
_builtincoeffs = {
|
|
1: (1,2,[1,1],-1,12),
|
|
2: (1,3,[1,4,1],-1,90),
|
|
3: (3,8,[1,3,3,1],-3,80),
|
|
4: (2,45,[7,32,12,32,7],-8,945),
|
|
5: (5,288,[19,75,50,50,75,19],-275,12096),
|
|
6: (1,140,[41,216,27,272,27,216,41],-9,1400),
|
|
7: (7,17280,[751,3577,1323,2989,2989,1323,3577,751],-8183,518400),
|
|
8: (4,14175,[989,5888,-928,10496,-4540,10496,-928,5888,989],
|
|
-2368,467775),
|
|
9: (9,89600,[2857,15741,1080,19344,5778,5778,19344,1080,
|
|
15741,2857], -4671, 394240),
|
|
10: (5,299376,[16067,106300,-48525,272400,-260550,427368,
|
|
-260550,272400,-48525,106300,16067],
|
|
-673175, 163459296),
|
|
11: (11,87091200,[2171465,13486539,-3237113, 25226685,-9595542,
|
|
15493566,15493566,-9595542,25226685,-3237113,
|
|
13486539,2171465], -2224234463, 237758976000),
|
|
12: (1, 5255250, [1364651,9903168,-7587864,35725120,-51491295,
|
|
87516288,-87797136,87516288,-51491295,35725120,
|
|
-7587864,9903168,1364651], -3012, 875875),
|
|
13: (13, 402361344000,[8181904909, 56280729661, -31268252574,
|
|
156074417954,-151659573325,206683437987,
|
|
-43111992612,-43111992612,206683437987,
|
|
-151659573325,156074417954,-31268252574,
|
|
56280729661,8181904909], -2639651053,
|
|
344881152000),
|
|
14: (7, 2501928000, [90241897,710986864,-770720657,3501442784,
|
|
-6625093363,12630121616,-16802270373,19534438464,
|
|
-16802270373,12630121616,-6625093363,3501442784,
|
|
-770720657,710986864,90241897], -3740727473,
|
|
1275983280000)
|
|
}
|
|
|
|
|
|
def newton_cotes(rn, equal=0):
|
|
r"""
|
|
Return weights and error coefficient for Newton-Cotes integration.
|
|
|
|
Suppose we have (N+1) samples of f at the positions
|
|
x_0, x_1, ..., x_N. Then an N-point Newton-Cotes formula for the
|
|
integral between x_0 and x_N is:
|
|
|
|
:math:`\int_{x_0}^{x_N} f(x)dx = \Delta x \sum_{i=0}^{N} a_i f(x_i)
|
|
+ B_N (\Delta x)^{N+2} f^{N+1} (\xi)`
|
|
|
|
where :math:`\xi \in [x_0,x_N]`
|
|
and :math:`\Delta x = \frac{x_N-x_0}{N}` is the average samples spacing.
|
|
|
|
If the samples are equally-spaced and N is even, then the error
|
|
term is :math:`B_N (\Delta x)^{N+3} f^{N+2}(\xi)`.
|
|
|
|
Parameters
|
|
----------
|
|
rn : int
|
|
The integer order for equally-spaced data or the relative positions of
|
|
the samples with the first sample at 0 and the last at N, where N+1 is
|
|
the length of `rn`. N is the order of the Newton-Cotes integration.
|
|
equal : int, optional
|
|
Set to 1 to enforce equally spaced data.
|
|
|
|
Returns
|
|
-------
|
|
an : ndarray
|
|
1-D array of weights to apply to the function at the provided sample
|
|
positions.
|
|
B : float
|
|
Error coefficient.
|
|
|
|
Examples
|
|
--------
|
|
Compute the integral of sin(x) in [0, :math:`\pi`]:
|
|
|
|
>>> from scipy.integrate import newton_cotes
|
|
>>> def f(x):
|
|
... return np.sin(x)
|
|
>>> a = 0
|
|
>>> b = np.pi
|
|
>>> exact = 2
|
|
>>> for N in [2, 4, 6, 8, 10]:
|
|
... x = np.linspace(a, b, N + 1)
|
|
... an, B = newton_cotes(N, 1)
|
|
... dx = (b - a) / N
|
|
... quad = dx * np.sum(an * f(x))
|
|
... error = abs(quad - exact)
|
|
... print('{:2d} {:10.9f} {:.5e}'.format(N, quad, error))
|
|
...
|
|
2 2.094395102 9.43951e-02
|
|
4 1.998570732 1.42927e-03
|
|
6 2.000017814 1.78136e-05
|
|
8 1.999999835 1.64725e-07
|
|
10 2.000000001 1.14677e-09
|
|
|
|
Notes
|
|
-----
|
|
Normally, the Newton-Cotes rules are used on smaller integration
|
|
regions and a composite rule is used to return the total integral.
|
|
|
|
"""
|
|
try:
|
|
N = len(rn)-1
|
|
if equal:
|
|
rn = np.arange(N+1)
|
|
elif np.all(np.diff(rn) == 1):
|
|
equal = 1
|
|
except Exception:
|
|
N = rn
|
|
rn = np.arange(N+1)
|
|
equal = 1
|
|
|
|
if equal and N in _builtincoeffs:
|
|
na, da, vi, nb, db = _builtincoeffs[N]
|
|
an = na * np.array(vi, dtype=float) / da
|
|
return an, float(nb)/db
|
|
|
|
if (rn[0] != 0) or (rn[-1] != N):
|
|
raise ValueError("The sample positions must start at 0"
|
|
" and end at N")
|
|
yi = rn / float(N)
|
|
ti = 2 * yi - 1
|
|
nvec = np.arange(N+1)
|
|
C = ti ** nvec[:, np.newaxis]
|
|
Cinv = np.linalg.inv(C)
|
|
# improve precision of result
|
|
for i in range(2):
|
|
Cinv = 2*Cinv - Cinv.dot(C).dot(Cinv)
|
|
vec = 2.0 / (nvec[::2]+1)
|
|
ai = Cinv[:, ::2].dot(vec) * (N / 2.)
|
|
|
|
if (N % 2 == 0) and equal:
|
|
BN = N/(N+3.)
|
|
power = N+2
|
|
else:
|
|
BN = N/(N+2.)
|
|
power = N+1
|
|
|
|
BN = BN - np.dot(yi**power, ai)
|
|
p1 = power+1
|
|
fac = power*math.log(N) - gammaln(p1)
|
|
fac = math.exp(fac)
|
|
return ai, BN*fac
|