from __future__ import annotations
from typing import TYPE_CHECKING, Callable, Any, cast
import numpy as np
import math
import warnings
from collections import namedtuple
from scipy.special import roots_legendre
from scipy.special import gammaln, logsumexp
from scipy._lib._util import _rng_spawn
__all__ = ['fixed_quad', 'quadrature', 'romberg', 'romb',
'trapezoid', 'trapz', 'simps', 'simpson',
'cumulative_trapezoid', 'cumtrapz', 'newton_cotes',
'qmc_quad', 'AccuracyWarning']
def trapezoid(y, x=None, dx=1.0, axis=-1):
r"""
Integrate along the given axis using the composite trapezoidal rule.
If `x` is provided, the integration happens in sequence along its
elements - they are not sorted.
Integrate `y` (`x`) along each 1d slice on the given axis, compute
:math:`\int y(x) dx`.
When `x` is specified, this integrates along the parametric curve,
computing :math:`\int_t y(t) dt =
\int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt`.
Parameters
----------
y : array_like
Input array to integrate.
x : array_like, optional
The sample points corresponding to the `y` values. If `x` is None,
the sample points are assumed to be evenly spaced `dx` apart. The
default is None.
dx : scalar, optional
The spacing between sample points when `x` is None. The default is 1.
axis : int, optional
The axis along which to integrate.
Returns
-------
trapezoid : float or ndarray
Definite integral of `y` = n-dimensional array as approximated along
a single axis by the trapezoidal rule. If `y` is a 1-dimensional array,
then the result is a float. If `n` is greater than 1, then the result
is an `n`-1 dimensional array.
See Also
--------
cumulative_trapezoid, simpson, romb
Notes
-----
Image [2]_ illustrates trapezoidal rule -- y-axis locations of points
will be taken from `y` array, by default x-axis distances between
points will be 1.0, alternatively they can be provided with `x` array
or with `dx` scalar. Return value will be equal to combined area under
the red lines.
References
----------
.. [1] Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule
.. [2] Illustration image:
https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png
Examples
--------
Use the trapezoidal rule on evenly spaced points:
>>> import numpy as np
>>> from scipy import integrate
>>> integrate.trapezoid([1, 2, 3])
4.0
The spacing between sample points can be selected by either the
``x`` or ``dx`` arguments:
>>> integrate.trapezoid([1, 2, 3], x=[4, 6, 8])
8.0
>>> integrate.trapezoid([1, 2, 3], dx=2)
8.0
Using a decreasing ``x`` corresponds to integrating in reverse:
>>> integrate.trapezoid([1, 2, 3], x=[8, 6, 4])
-8.0
More generally ``x`` is used to integrate along a parametric curve. We can
estimate the integral :math:`\int_0^1 x^2 = 1/3` using:
>>> x = np.linspace(0, 1, num=50)
>>> y = x**2
>>> integrate.trapezoid(y, x)
0.33340274885464394
Or estimate the area of a circle, noting we repeat the sample which closes
the curve:
>>> theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True)
>>> integrate.trapezoid(np.cos(theta), x=np.sin(theta))
3.141571941375841
``trapezoid`` can be applied along a specified axis to do multiple
computations in one call:
>>> a = np.arange(6).reshape(2, 3)
>>> a
array([[0, 1, 2],
[3, 4, 5]])
>>> integrate.trapezoid(a, axis=0)
array([1.5, 2.5, 3.5])
>>> integrate.trapezoid(a, axis=1)
array([2., 8.])
"""
# Future-proofing, in case NumPy moves from trapz to trapezoid for the same
# reasons as SciPy
if hasattr(np, 'trapezoid'):
return np.trapezoid(y, x=x, dx=dx, axis=axis)
else:
return np.trapz(y, x=x, dx=dx, axis=axis)
# Note: alias kept for backwards compatibility. Rename was done
# because trapz is a slur in colloquial English (see gh-12924).
def trapz(y, x=None, dx=1.0, axis=-1):
"""An alias of `trapezoid`.
`trapz` is kept for backwards compatibility. For new code, prefer
`trapezoid` instead.
"""
return trapezoid(y, x=x, dx=dx, axis=axis)
class AccuracyWarning(Warning):
pass
if TYPE_CHECKING:
# workaround for mypy function attributes see:
# https://github.com/python/mypy/issues/2087#issuecomment-462726600
from typing import Protocol
class CacheAttributes(Protocol):
cache: dict[int, tuple[Any, Any]]
else:
CacheAttributes = Callable
def cache_decorator(func: Callable) -> CacheAttributes:
return cast(CacheAttributes, func)
@cache_decorator
def _cached_roots_legendre(n):
"""
Cache roots_legendre results to speed up calls of the fixed_quad
function.
"""
if n in _cached_roots_legendre.cache:
return _cached_roots_legendre.cache[n]
_cached_roots_legendre.cache[n] = roots_legendre(n)
return _cached_roots_legendre.cache[n]
_cached_roots_legendre.cache = dict()
def fixed_quad(func, a, b, args=(), n=5):
"""
Compute a definite integral using fixed-order Gaussian quadrature.
Integrate `func` from `a` to `b` using Gaussian quadrature of
order `n`.
Parameters
----------
func : callable
A Python function or method to integrate (must accept vector inputs).
If integrating a vector-valued function, the returned array must have
shape ``(..., len(x))``.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function, if any.
n : int, optional
Order of quadrature integration. Default is 5.
Returns
-------
val : float
Gaussian quadrature approximation to the integral
none : None
Statically returned value of None
See Also
--------
quad : adaptive quadrature using QUADPACK
dblquad : double integrals
tplquad : triple integrals
romberg : adaptive Romberg quadrature
quadrature : adaptive Gaussian quadrature
romb : integrators for sampled data
simpson : integrators for sampled data
cumulative_trapezoid : cumulative integration for sampled data
ode : ODE integrator
odeint : ODE integrator
Examples
--------
>>> from scipy import integrate
>>> import numpy as np
>>> f = lambda x: x**8
>>> integrate.fixed_quad(f, 0.0, 1.0, n=4)
(0.1110884353741496, None)
>>> integrate.fixed_quad(f, 0.0, 1.0, n=5)
(0.11111111111111102, None)
>>> print(1/9.0) # analytical result
0.1111111111111111
>>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=4)
(0.9999999771971152, None)
>>> integrate.fixed_quad(np.cos, 0.0, np.pi/2, n=5)
(1.000000000039565, None)
>>> np.sin(np.pi/2)-np.sin(0) # analytical result
1.0
"""
x, w = _cached_roots_legendre(n)
x = np.real(x)
if np.isinf(a) or np.isinf(b):
raise ValueError("Gaussian quadrature is only available for "
"finite limits.")
y = (b-a)*(x+1)/2.0 + a
return (b-a)/2.0 * np.sum(w*func(y, *args), axis=-1), None
def vectorize1(func, args=(), vec_func=False):
"""Vectorize the call to a function.
This is an internal utility function used by `romberg` and
`quadrature` to create a vectorized version of a function.
If `vec_func` is True, the function `func` is assumed to take vector
arguments.
Parameters
----------
func : callable
User defined function.
args : tuple, optional
Extra arguments for the function.
vec_func : bool, optional
True if the function func takes vector arguments.
Returns
-------
vfunc : callable
A function that will take a vector argument and return the
result.
"""
if vec_func:
def vfunc(x):
return func(x, *args)
else:
def vfunc(x):
if np.isscalar(x):
return func(x, *args)
x = np.asarray(x)
# call with first point to get output type
y0 = func(x[0], *args)
n = len(x)
dtype = getattr(y0, 'dtype', type(y0))
output = np.empty((n,), dtype=dtype)
output[0] = y0
for i in range(1, n):
output[i] = func(x[i], *args)
return output
return vfunc
def quadrature(func, a, b, args=(), tol=1.49e-8, rtol=1.49e-8, maxiter=50,
vec_func=True, miniter=1):
"""
Compute a definite integral using fixed-tolerance Gaussian quadrature.
Integrate `func` from `a` to `b` using Gaussian quadrature
with absolute tolerance `tol`.
Parameters
----------
func : function
A Python function or method to integrate.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function.
tol, rtol : float, optional
Iteration stops when error between last two iterates is less than
`tol` OR the relative change is less than `rtol`.
maxiter : int, optional
Maximum order of Gaussian quadrature.
vec_func : bool, optional
True or False if func handles arrays as arguments (is
a "vector" function). Default is True.
miniter : int, optional
Minimum order of Gaussian quadrature.
Returns
-------
val : float
Gaussian quadrature approximation (within tolerance) to integral.
err : float
Difference between last two estimates of the integral.
See Also
--------
romberg : adaptive Romberg quadrature
fixed_quad : fixed-order Gaussian quadrature
quad : adaptive quadrature using QUADPACK
dblquad : double integrals
tplquad : triple integrals
romb : integrator for sampled data
simpson : integrator for sampled data
cumulative_trapezoid : cumulative integration for sampled data
ode : ODE integrator
odeint : ODE integrator
Examples
--------
>>> from scipy import integrate
>>> import numpy as np
>>> f = lambda x: x**8
>>> integrate.quadrature(f, 0.0, 1.0)
(0.11111111111111106, 4.163336342344337e-17)
>>> print(1/9.0) # analytical result
0.1111111111111111
>>> integrate.quadrature(np.cos, 0.0, np.pi/2)
(0.9999999999999536, 3.9611425250996035e-11)
>>> np.sin(np.pi/2)-np.sin(0) # analytical result
1.0
"""
if not isinstance(args, tuple):
args = (args,)
vfunc = vectorize1(func, args, vec_func=vec_func)
val = np.inf
err = np.inf
maxiter = max(miniter+1, maxiter)
for n in range(miniter, maxiter+1):
newval = fixed_quad(vfunc, a, b, (), n)[0]
err = abs(newval-val)
val = newval
if err < tol or err < rtol*abs(val):
break
else:
warnings.warn(
"maxiter (%d) exceeded. Latest difference = %e" % (maxiter, err),
AccuracyWarning)
return val, err
def tupleset(t, i, value):
l = list(t)
l[i] = value
return tuple(l)
# Note: alias kept for backwards compatibility. Rename was done
# because cumtrapz is a slur in colloquial English (see gh-12924).
def cumtrapz(y, x=None, dx=1.0, axis=-1, initial=None):
"""An alias of `cumulative_trapezoid`.
`cumtrapz` is kept for backwards compatibility. For new code, prefer
`cumulative_trapezoid` instead.
"""
return cumulative_trapezoid(y, x=x, dx=dx, axis=axis, initial=initial)
[docs]
def cumulative_trapezoid(y, x=None, dx=1.0, axis=-1, initial=None):
"""
Cumulatively integrate y(x) using the composite trapezoidal rule.
Parameters
----------
y : array_like
Values to integrate.
x : array_like, optional
The coordinate to integrate along. If None (default), use spacing `dx`
between consecutive elements in `y`.
dx : float, optional
Spacing between elements of `y`. Only used if `x` is None.
axis : int, optional
Specifies the axis to cumulate. Default is -1 (last axis).
initial : scalar, optional
If given, insert this value at the beginning of the returned result.
Typically this value should be 0. Default is None, which means no
value at ``x[0]`` is returned and `res` has one element less than `y`
along the axis of integration.
Returns
-------
res : ndarray
The result of cumulative integration of `y` along `axis`.
If `initial` is None, the shape is such that the axis of integration
has one less value than `y`. If `initial` is given, the shape is equal
to that of `y`.
See Also
--------
numpy.cumsum, numpy.cumprod
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
romb : integrators for sampled data
ode : ODE integrators
odeint : ODE integrators
Examples
--------
>>> from scipy import integrate
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-2, 2, num=20)
>>> y = x
>>> y_int = integrate.cumulative_trapezoid(y, x, initial=0)
>>> plt.plot(x, y_int, 'ro', x, y[0] + 0.5 * x**2, 'b-')
>>> plt.show()
"""
y = np.asarray(y)
if x is None:
d = dx
else:
x = np.asarray(x)
if x.ndim == 1:
d = np.diff(x)
# reshape to correct shape
shape = [1] * y.ndim
shape[axis] = -1
d = d.reshape(shape)
elif len(x.shape) != len(y.shape):
raise ValueError("If given, shape of x must be 1-D or the "
"same as y.")
else:
d = np.diff(x, axis=axis)
if d.shape[axis] != y.shape[axis] - 1:
raise ValueError("If given, length of x along axis must be the "
"same as y.")
nd = len(y.shape)
slice1 = tupleset((slice(None),)*nd, axis, slice(1, None))
slice2 = tupleset((slice(None),)*nd, axis, slice(None, -1))
res = np.cumsum(d * (y[slice1] + y[slice2]) / 2.0, axis=axis)
if initial is not None:
if not np.isscalar(initial):
raise ValueError("`initial` parameter should be a scalar.")
shape = list(res.shape)
shape[axis] = 1
res = np.concatenate([np.full(shape, initial, dtype=res.dtype), res],
axis=axis)
return res
def _basic_simpson(y, start, stop, x, dx, axis):
nd = len(y.shape)
if start is None:
start = 0
step = 2
slice_all = (slice(None),)*nd
slice0 = tupleset(slice_all, axis, slice(start, stop, step))
slice1 = tupleset(slice_all, axis, slice(start+1, stop+1, step))
slice2 = tupleset(slice_all, axis, slice(start+2, stop+2, step))
if x is None: # Even-spaced Simpson's rule.
result = np.sum(y[slice0] + 4.0*y[slice1] + y[slice2], axis=axis)
result *= dx / 3.0
else:
# Account for possibly different spacings.
# Simpson's rule changes a bit.
h = np.diff(x, axis=axis)
sl0 = tupleset(slice_all, axis, slice(start, stop, step))
sl1 = tupleset(slice_all, axis, slice(start+1, stop+1, step))
h0 = h[sl0].astype(float, copy=False)
h1 = h[sl1].astype(float, copy=False)
hsum = h0 + h1
hprod = h0 * h1
h0divh1 = np.true_divide(h0, h1, out=np.zeros_like(h0), where=h1 != 0)
tmp = hsum/6.0 * (y[slice0] *
(2.0 - np.true_divide(1.0, h0divh1,
out=np.zeros_like(h0divh1),
where=h0divh1 != 0)) +
y[slice1] * (hsum *
np.true_divide(hsum, hprod,
out=np.zeros_like(hsum),
where=hprod != 0)) +
y[slice2] * (2.0 - h0divh1))
result = np.sum(tmp, axis=axis)
return result
# Note: alias kept for backwards compatibility. simps was renamed to simpson
# because the former is a slur in colloquial English (see gh-12924).
def simps(y, x=None, dx=1.0, axis=-1, even=None):
"""An alias of `simpson`.
`simps` is kept for backwards compatibility. For new code, prefer
`simpson` instead.
"""
return simpson(y, x=x, dx=dx, axis=axis, even=even)
def simpson(y, x=None, dx=1.0, axis=-1, even=None):
"""
Integrate y(x) using samples along the given axis and the composite
Simpson's rule. If x is None, spacing of dx is assumed.
If there are an even number of samples, N, then there are an odd
number of intervals (N-1), but Simpson's rule requires an even number
of intervals. The parameter 'even' controls how this is handled.
Parameters
----------
y : array_like
Array to be integrated.
x : array_like, optional
If given, the points at which `y` is sampled.
dx : float, optional
Spacing of integration points along axis of `x`. Only used when
`x` is None. Default is 1.
axis : int, optional
Axis along which to integrate. Default is the last axis.
even : {None, 'simpson', 'avg', 'first', 'last'}, optional
'avg' : Average two results:
1) use the first N-2 intervals with
a trapezoidal rule on the last interval and
2) use the last
N-2 intervals with a trapezoidal rule on the first interval.
'first' : Use Simpson's rule for the first N-2 intervals with
a trapezoidal rule on the last interval.
'last' : Use Simpson's rule for the last N-2 intervals with a
trapezoidal rule on the first interval.
None : equivalent to 'simpson' (default)
'simpson' : Use Simpson's rule for the first N-2 intervals with the
addition of a 3-point parabolic segment for the last
interval using equations outlined by Cartwright [1]_.
If the axis to be integrated over only has two points then
the integration falls back to a trapezoidal integration.
.. versionadded:: 1.11.0
.. versionchanged:: 1.11.0
The newly added 'simpson' option is now the default as it is more
accurate in most situations.
.. deprecated:: 1.11.0
Parameter `even` is deprecated and will be removed in SciPy
1.13.0. After this time the behaviour for an even number of
points will follow that of `even='simpson'`.
Returns
-------
float
The estimated integral computed with the composite Simpson's rule.
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
romb : integrators for sampled data
cumulative_trapezoid : cumulative integration for sampled data
ode : ODE integrators
odeint : ODE integrators
Notes
-----
For an odd number of samples that are equally spaced the result is
exact if the function is a polynomial of order 3 or less. If
the samples are not equally spaced, then the result is exact only
if the function is a polynomial of order 2 or less.
References
----------
.. [1] Cartwright, Kenneth V. Simpson's Rule Cumulative Integration with
MS Excel and Irregularly-spaced Data. Journal of Mathematical
Sciences and Mathematics Education. 12 (2): 1-9
Examples
--------
>>> from scipy import integrate
>>> import numpy as np
>>> x = np.arange(0, 10)
>>> y = np.arange(0, 10)
>>> integrate.simpson(y, x)
40.5
>>> y = np.power(x, 3)
>>> integrate.simpson(y, x)
1640.5
>>> integrate.quad(lambda x: x**3, 0, 9)[0]
1640.25
>>> integrate.simpson(y, x, even='first')
1644.5
"""
y = np.asarray(y)
nd = len(y.shape)
N = y.shape[axis]
last_dx = dx
first_dx = dx
returnshape = 0
if x is not None:
x = np.asarray(x)
if len(x.shape) == 1:
shapex = [1] * nd
shapex[axis] = x.shape[0]
saveshape = x.shape
returnshape = 1
x = x.reshape(tuple(shapex))
elif len(x.shape) != len(y.shape):
raise ValueError("If given, shape of x must be 1-D or the "
"same as y.")
if x.shape[axis] != N:
raise ValueError("If given, length of x along axis must be the "
"same as y.")
# even keyword parameter is deprecated
if even is not None:
warnings.warn(
"The 'even' keyword is deprecated as of SciPy 1.11.0 and will be "
"removed in SciPy 1.13.0",
DeprecationWarning, stacklevel=2
)
if N % 2 == 0:
val = 0.0
result = 0.0
slice_all = (slice(None),) * nd
# default is 'simpson'
even = even if even is not None else "simpson"
if even not in ['avg', 'last', 'first', 'simpson']:
raise ValueError(
"Parameter 'even' must be 'simpson', "
"'avg', 'last', or 'first'."
)
if N == 2:
# need at least 3 points in integration axis to form parabolic
# segment. If there are two points then any of 'avg', 'first',
# 'last' should give the same result.
slice1 = tupleset(slice_all, axis, -1)
slice2 = tupleset(slice_all, axis, -2)
if x is not None:
last_dx = x[slice1] - x[slice2]
val += 0.5 * last_dx * (y[slice1] + y[slice2])
# calculation is finished. Set `even` to None to skip other
# scenarios
even = None
if even == 'simpson':
# use Simpson's rule on first intervals
result = _basic_simpson(y, 0, N-3, x, dx, axis)
slice1 = tupleset(slice_all, axis, -1)
slice2 = tupleset(slice_all, axis, -2)
slice3 = tupleset(slice_all, axis, -3)
h = np.asfarray([dx, dx])
if x is not None:
# grab the last two spacings from the appropriate axis
hm2 = tupleset(slice_all, axis, slice(-2, -1, 1))
hm1 = tupleset(slice_all, axis, slice(-1, None, 1))
diffs = np.float64(np.diff(x, axis=axis))
h = [np.squeeze(diffs[hm2], axis=axis),
np.squeeze(diffs[hm1], axis=axis)]
# This is the correction for the last interval according to
# Cartwright.
# However, I used the equations given at
# https://en.wikipedia.org/wiki/Simpson%27s_rule#Composite_Simpson's_rule_for_irregularly_spaced_data
# A footnote on Wikipedia says:
# Cartwright 2017, Equation 8. The equation in Cartwright is
# calculating the first interval whereas the equations in the
# Wikipedia article are adjusting for the last integral. If the
# proper algebraic substitutions are made, the equation results in
# the values shown.
num = 2 * h[1] ** 2 + 3 * h[0] * h[1]
den = 6 * (h[1] + h[0])
alpha = np.true_divide(
num,
den,
out=np.zeros_like(den),
where=den != 0
)
num = h[1] ** 2 + 3.0 * h[0] * h[1]
den = 6 * h[0]
beta = np.true_divide(
num,
den,
out=np.zeros_like(den),
where=den != 0
)
num = 1 * h[1] ** 3
den = 6 * h[0] * (h[0] + h[1])
eta = np.true_divide(
num,
den,
out=np.zeros_like(den),
where=den != 0
)
result += alpha*y[slice1] + beta*y[slice2] - eta*y[slice3]
# The following code (down to result=result+val) can be removed
# once the 'even' keyword is removed.
# Compute using Simpson's rule on first intervals
if even in ['avg', 'first']:
slice1 = tupleset(slice_all, axis, -1)
slice2 = tupleset(slice_all, 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(slice_all, axis, 0)
slice2 = tupleset(slice_all, 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
>>> import numpy as np
>>> 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 # may vary
"""
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, "=" * len(title), sep="\n", end="\n")
for i in range(k+1):
for j in range(i+1):
print(formstr % R[(i, j)], end=" ")
print()
print("=" * len(title))
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
>>> import numpy as np
>>> 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.
Notes
-----
Normally, the Newton-Cotes rules are used on smaller integration
regions and a composite rule is used to return the total integral.
Examples
--------
Compute the integral of sin(x) in [0, :math:`\pi`]:
>>> from scipy.integrate import newton_cotes
>>> import numpy as np
>>> 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
"""
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
def _qmc_quad_iv(func, a, b, n_points, n_estimates, qrng, log):
# lazy import to avoid issues with partially-initialized submodule
if not hasattr(qmc_quad, 'qmc'):
from scipy import stats
qmc_quad.stats = stats
else:
stats = qmc_quad.stats
if not callable(func):
message = "`func` must be callable."
raise TypeError(message)
# a, b will be modified, so copy. Oh well if it's copied twice.
a = np.atleast_1d(a).copy()
b = np.atleast_1d(b).copy()
a, b = np.broadcast_arrays(a, b)
dim = a.shape[0]
try:
func((a + b) / 2)
except Exception as e:
message = ("`func` must evaluate the integrand at points within "
"the integration range; e.g. `func( (a + b) / 2)` "
"must return the integrand at the centroid of the "
"integration volume.")
raise ValueError(message) from e
try:
func(np.array([a, b]).T)
vfunc = func
except Exception as e:
message = ("Exception encountered when attempting vectorized call to "
f"`func`: {e}. For better performance, `func` should "
"accept two-dimensional array `x` with shape `(len(a), "
"n_points)` and return an array of the integrand value at "
"each of the `n_points.")
warnings.warn(message, stacklevel=3)
def vfunc(x):
return np.apply_along_axis(func, axis=-1, arr=x)
n_points_int = np.int64(n_points)
if n_points != n_points_int:
message = "`n_points` must be an integer."
raise TypeError(message)
n_estimates_int = np.int64(n_estimates)
if n_estimates != n_estimates_int:
message = "`n_estimates` must be an integer."
raise TypeError(message)
if qrng is None:
qrng = stats.qmc.Halton(dim)
elif not isinstance(qrng, stats.qmc.QMCEngine):
message = "`qrng` must be an instance of scipy.stats.qmc.QMCEngine."
raise TypeError(message)
if qrng.d != a.shape[0]:
message = ("`qrng` must be initialized with dimensionality equal to "
"the number of variables in `a`, i.e., "
"`qrng.random().shape[-1]` must equal `a.shape[0]`.")
raise ValueError(message)
rng_seed = getattr(qrng, 'rng_seed', None)
rng = stats._qmc.check_random_state(rng_seed)
if log not in {True, False}:
message = "`log` must be boolean (`True` or `False`)."
raise TypeError(message)
return (vfunc, a, b, n_points_int, n_estimates_int, qrng, rng, log, stats)
QMCQuadResult = namedtuple('QMCQuadResult', ['integral', 'standard_error'])
def qmc_quad(func, a, b, *, n_estimates=8, n_points=1024, qrng=None,
log=False):
"""
Compute an integral in N-dimensions using Quasi-Monte Carlo quadrature.
Parameters
----------
func : callable
The integrand. Must accept a single argument ``x``, an array which
specifies the point(s) at which to evaluate the scalar-valued
integrand, and return the value(s) of the integrand.
For efficiency, the function should be vectorized to accept an array of
shape ``(d, n_points)``, where ``d`` is the number of variables (i.e.
the dimensionality of the function domain) and `n_points` is the number
of quadrature points, and return an array of shape ``(n_points,)``,
the integrand at each quadrature point.
a, b : array-like
One-dimensional arrays specifying the lower and upper integration
limits, respectively, of each of the ``d`` variables.
n_estimates, n_points : int, optional
`n_estimates` (default: 8) statistically independent QMC samples, each
of `n_points` (default: 1024) points, will be generated by `qrng`.
The total number of points at which the integrand `func` will be
evaluated is ``n_points * n_estimates``. See Notes for details.
qrng : `~scipy.stats.qmc.QMCEngine`, optional
An instance of the QMCEngine from which to sample QMC points.
The QMCEngine must be initialized to a number of dimensions ``d``
corresponding with the number of variables ``x1, ..., xd`` passed to
`func`.
The provided QMCEngine is used to produce the first integral estimate.
If `n_estimates` is greater than one, additional QMCEngines are
spawned from the first (with scrambling enabled, if it is an option.)
If a QMCEngine is not provided, the default `scipy.stats.qmc.Halton`
will be initialized with the number of dimensions determine from
the length of `a`.
log : boolean, default: False
When set to True, `func` returns the log of the integrand, and
the result object contains the log of the integral.
Returns
-------
result : object
A result object with attributes:
integral : float
The estimate of the integral.
standard_error :
The error estimate. See Notes for interpretation.
Notes
-----
Values of the integrand at each of the `n_points` points of a QMC sample
are used to produce an estimate of the integral. This estimate is drawn
from a population of possible estimates of the integral, the value of
which we obtain depends on the particular points at which the integral
was evaluated. We perform this process `n_estimates` times, each time
evaluating the integrand at different scrambled QMC points, effectively
drawing i.i.d. random samples from the population of integral estimates.
The sample mean :math:`m` of these integral estimates is an
unbiased estimator of the true value of the integral, and the standard
error of the mean :math:`s` of these estimates may be used to generate
confidence intervals using the t distribution with ``n_estimates - 1``
degrees of freedom. Perhaps counter-intuitively, increasing `n_points`
while keeping the total number of function evaluation points
``n_points * n_estimates`` fixed tends to reduce the actual error, whereas
increasing `n_estimates` tends to decrease the error estimate.
Examples
--------
QMC quadrature is particularly useful for computing integrals in higher
dimensions. An example integrand is the probability density function
of a multivariate normal distribution.
>>> import numpy as np
>>> from scipy import stats
>>> dim = 8
>>> mean = np.zeros(dim)
>>> cov = np.eye(dim)
>>> def func(x):
... # `multivariate_normal` expects the _last_ axis to correspond with
... # the dimensionality of the space, so `x` must be transposed
... return stats.multivariate_normal.pdf(x.T, mean, cov)
To compute the integral over the unit hypercube:
>>> from scipy.integrate import qmc_quad
>>> a = np.zeros(dim)
>>> b = np.ones(dim)
>>> rng = np.random.default_rng()
>>> qrng = stats.qmc.Halton(d=dim, seed=rng)
>>> n_estimates = 8
>>> res = qmc_quad(func, a, b, n_estimates=n_estimates, qrng=qrng)
>>> res.integral, res.standard_error
(0.00018429555666024108, 1.0389431116001344e-07)
A two-sided, 99% confidence interval for the integral may be estimated
as:
>>> t = stats.t(df=n_estimates-1, loc=res.integral,
... scale=res.standard_error)
>>> t.interval(0.99)
(0.0001839319802536469, 0.00018465913306683527)
Indeed, the value reported by `scipy.stats.multivariate_normal` is
within this range.
>>> stats.multivariate_normal.cdf(b, mean, cov, lower_limit=a)
0.00018430867675187443
"""
args = _qmc_quad_iv(func, a, b, n_points, n_estimates, qrng, log)
func, a, b, n_points, n_estimates, qrng, rng, log, stats = args
def sum_product(integrands, dA, log=False):
if log:
return logsumexp(integrands) + np.log(dA)
else:
return np.sum(integrands * dA)
def mean(estimates, log=False):
if log:
return logsumexp(estimates) - np.log(n_estimates)
else:
return np.mean(estimates)
def std(estimates, m=None, ddof=0, log=False):
m = m or mean(estimates, log)
if log:
estimates, m = np.broadcast_arrays(estimates, m)
temp = np.vstack((estimates, m + np.pi * 1j))
diff = logsumexp(temp, axis=0)
return np.real(0.5 * (logsumexp(2 * diff)
- np.log(n_estimates - ddof)))
else:
return np.std(estimates, ddof=ddof)
def sem(estimates, m=None, s=None, log=False):
m = m or mean(estimates, log)
s = s or std(estimates, m, ddof=1, log=log)
if log:
return s - 0.5*np.log(n_estimates)
else:
return s / np.sqrt(n_estimates)
# The sign of the integral depends on the order of the limits. Fix this by
# ensuring that lower bounds are indeed lower and setting sign of resulting
# integral manually
if np.any(a == b):
message = ("A lower limit was equal to an upper limit, so the value "
"of the integral is zero by definition.")
warnings.warn(message, stacklevel=2)
return QMCQuadResult(-np.inf if log else 0, 0)
i_swap = b < a
sign = (-1)**(i_swap.sum(axis=-1)) # odd # of swaps -> negative
a[i_swap], b[i_swap] = b[i_swap], a[i_swap]
A = np.prod(b - a)
dA = A / n_points
estimates = np.zeros(n_estimates)
rngs = _rng_spawn(qrng.rng, n_estimates)
for i in range(n_estimates):
# Generate integral estimate
sample = qrng.random(n_points)
# The rationale for transposing is that this allows users to easily
# unpack `x` into separate variables, if desired. This is consistent
# with the `xx` array passed into the `scipy.integrate.nquad` `func`.
x = stats.qmc.scale(sample, a, b).T # (n_dim, n_points)
integrands = func(x)
estimates[i] = sum_product(integrands, dA, log)
# Get a new, independently-scrambled QRNG for next time
qrng = type(qrng)(seed=rngs[i], **qrng._init_quad)
integral = mean(estimates, log)
standard_error = sem(estimates, m=integral, log=log)
integral = integral + np.pi*1j if (log and sign < 0) else integral*sign
return QMCQuadResult(integral, standard_error)