import numpy as np import numpy.typing as npt import math import warnings from collections import namedtuple from collections.abc import Callable from scipy.special import roots_legendre from scipy.special import gammaln, logsumexp from scipy._lib._util import _rng_spawn from scipy._lib._array_api import _asarray, array_namespace, xp_broadcast_promote __all__ = ['fixed_quad', 'romb', 'trapezoid', 'simpson', 'cumulative_trapezoid', 'newton_cotes', 'qmc_quad', 'cumulative_simpson'] 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. The default is the last axis. 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.]) """ xp = array_namespace(y) y = _asarray(y, xp=xp, subok=True) # Cannot just use the broadcasted arrays that are returned # because trapezoid does not follow normal broadcasting rules # cf. https://github.com/scipy/scipy/pull/21524#issuecomment-2354105942 result_dtype = xp_broadcast_promote(y, force_floating=True, xp=xp)[0].dtype nd = y.ndim slice1 = [slice(None)]*nd slice2 = [slice(None)]*nd slice1[axis] = slice(1, None) slice2[axis] = slice(None, -1) if x is None: d = dx else: x = _asarray(x, xp=xp, subok=True) if x.ndim == 1: d = x[1:] - x[:-1] # make d broadcastable to y slice3 = [None] * nd slice3[axis] = slice(None) d = d[tuple(slice3)] else: # if x is n-D it should be broadcastable to y x = xp.broadcast_to(x, y.shape) d = x[tuple(slice1)] - x[tuple(slice2)] try: ret = xp.sum( d * (y[tuple(slice1)] + y[tuple(slice2)]) / 2.0, axis=axis, dtype=result_dtype ) except ValueError: # Operations didn't work, cast to ndarray d = xp.asarray(d) y = xp.asarray(y) ret = xp.sum( d * (y[tuple(slice1)] + y[tuple(slice2)]) / 2.0, axis=axis, dtype=result_dtype ) return ret 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 romb : integrators for sampled data simpson : integrators for sampled data cumulative_trapezoid : cumulative integration for sampled data 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 tupleset(t, i, value): l = list(t) l[i] = value return tuple(l) 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. 0 or None are the only values accepted. Default is None, which means `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 cumulative_simpson : cumulative integration using Simpson's 1/3 rule quad : adaptive quadrature using QUADPACK fixed_quad : fixed-order Gaussian quadrature dblquad : double integrals tplquad : triple integrals romb : integrators for sampled data 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 y.shape[axis] == 0: raise ValueError("At least one point is required along `axis`.") 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 initial != 0: raise ValueError("`initial` must be `None` or `0`.") 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 def simpson(y, x=None, *, dx=1.0, axis=-1): """ Integrate y(x) using samples along the given axis and the composite Simpson's rule. If x is None, spacing of dx is assumed. 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. Returns ------- float The estimated integral computed with the composite Simpson's rule. See Also -------- quad : adaptive quadrature using QUADPACK fixed_quad : fixed-order Gaussian quadrature dblquad : double integrals tplquad : triple integrals romb : integrators for sampled data cumulative_trapezoid : cumulative integration for sampled data cumulative_simpson : cumulative integration using Simpson's 1/3 rule 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=x) 40.5 >>> y = np.power(x, 3) >>> integrate.simpson(y, x=x) 1640.5 >>> integrate.quad(lambda x: x**3, 0, 9)[0] 1640.25 """ y = np.asarray(y) nd = len(y.shape) N = y.shape[axis] last_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.") if N % 2 == 0: val = 0.0 result = 0.0 slice_all = (slice(None),) * nd 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]) else: # 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.asarray([dx, dx], dtype=np.float64) 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] result += val else: result = _basic_simpson(y, 0, N-2, x, dx, axis) if returnshape: x = x.reshape(saveshape) return result def _cumulatively_sum_simpson_integrals( y: np.ndarray, dx: np.ndarray, integration_func: Callable[[np.ndarray, np.ndarray], np.ndarray], ) -> np.ndarray: """Calculate cumulative sum of Simpson integrals. Takes as input the integration function to be used. The integration_func is assumed to return the cumulative sum using composite Simpson's rule. Assumes the axis of summation is -1. """ sub_integrals_h1 = integration_func(y, dx) sub_integrals_h2 = integration_func(y[..., ::-1], dx[..., ::-1])[..., ::-1] shape = list(sub_integrals_h1.shape) shape[-1] += 1 sub_integrals = np.empty(shape) sub_integrals[..., :-1:2] = sub_integrals_h1[..., ::2] sub_integrals[..., 1::2] = sub_integrals_h2[..., ::2] # Integral over last subinterval can only be calculated from # formula for h2 sub_integrals[..., -1] = sub_integrals_h2[..., -1] res = np.cumsum(sub_integrals, axis=-1) return res def _cumulative_simpson_equal_intervals(y: np.ndarray, dx: np.ndarray) -> np.ndarray: """Calculate the Simpson integrals for all h1 intervals assuming equal interval widths. The function can also be used to calculate the integral for all h2 intervals by reversing the inputs, `y` and `dx`. """ d = dx[..., :-1] f1 = y[..., :-2] f2 = y[..., 1:-1] f3 = y[..., 2:] # Calculate integral over the subintervals (eqn (10) of Reference [2]) return d / 3 * (5 * f1 / 4 + 2 * f2 - f3 / 4) def _cumulative_simpson_unequal_intervals(y: np.ndarray, dx: np.ndarray) -> np.ndarray: """Calculate the Simpson integrals for all h1 intervals assuming unequal interval widths. The function can also be used to calculate the integral for all h2 intervals by reversing the inputs, `y` and `dx`. """ x21 = dx[..., :-1] x32 = dx[..., 1:] f1 = y[..., :-2] f2 = y[..., 1:-1] f3 = y[..., 2:] x31 = x21 + x32 x21_x31 = x21/x31 x21_x32 = x21/x32 x21x21_x31x32 = x21_x31 * x21_x32 # Calculate integral over the subintervals (eqn (8) of Reference [2]) coeff1 = 3 - x21_x31 coeff2 = 3 + x21x21_x31x32 + x21_x31 coeff3 = -x21x21_x31x32 return x21/6 * (coeff1*f1 + coeff2*f2 + coeff3*f3) def _ensure_float_array(arr: npt.ArrayLike) -> np.ndarray: arr = np.asarray(arr) if np.issubdtype(arr.dtype, np.integer): arr = arr.astype(float, copy=False) return arr def cumulative_simpson(y, *, x=None, dx=1.0, axis=-1, initial=None): r""" Cumulatively integrate y(x) using the composite Simpson's 1/3 rule. The integral of the samples at every point is calculated by assuming a quadratic relationship between each point and the two adjacent points. Parameters ---------- y : array_like Values to integrate. Requires at least one point along `axis`. If two or fewer points are provided along `axis`, Simpson's integration is not possible and the result is calculated with `cumulative_trapezoid`. x : array_like, optional The coordinate to integrate along. Must have the same shape as `y` or must be 1D with the same length as `y` along `axis`. `x` must also be strictly increasing along `axis`. If `x` is None (default), integration is performed using spacing `dx` between consecutive elements in `y`. dx : scalar or array_like, optional Spacing between elements of `y`. Only used if `x` is None. Can either be a float, or an array with the same shape as `y`, but of length one along `axis`. Default is 1.0. axis : int, optional Specifies the axis to integrate along. Default is -1 (last axis). initial : scalar or array_like, optional If given, insert this value at the beginning of the returned result, and add it to the rest of the result. 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. Can either be a float, or an array with the same shape as `y`, but of length one along `axis`. 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 cumulative_trapezoid : cumulative integration using the composite trapezoidal rule simpson : integrator for sampled data using the Composite Simpson's Rule Notes ----- .. versionadded:: 1.12.0 The composite Simpson's 1/3 method can be used to approximate the definite integral of a sampled input function :math:`y(x)` [1]_. The method assumes a quadratic relationship over the interval containing any three consecutive sampled points. Consider three consecutive points: :math:`(x_1, y_1), (x_2, y_2), (x_3, y_3)`. Assuming a quadratic relationship over the three points, the integral over the subinterval between :math:`x_1` and :math:`x_2` is given by formula (8) of [2]_: .. math:: \int_{x_1}^{x_2} y(x) dx\ &= \frac{x_2-x_1}{6}\left[\ \left\{3-\frac{x_2-x_1}{x_3-x_1}\right\} y_1 + \ \left\{3 + \frac{(x_2-x_1)^2}{(x_3-x_2)(x_3-x_1)} + \ \frac{x_2-x_1}{x_3-x_1}\right\} y_2\\ - \frac{(x_2-x_1)^2}{(x_3-x_2)(x_3-x_1)} y_3\right] The integral between :math:`x_2` and :math:`x_3` is given by swapping appearances of :math:`x_1` and :math:`x_3`. The integral is estimated separately for each subinterval and then cumulatively summed to obtain the final result. For samples that are equally spaced, the result is exact if the function is a polynomial of order three or less [1]_ and the number of subintervals is even. Otherwise, the integral is exact for polynomials of order two or less. References ---------- .. [1] Wikipedia page: https://en.wikipedia.org/wiki/Simpson's_rule .. [2] 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 >>> import matplotlib.pyplot as plt >>> x = np.linspace(-2, 2, num=20) >>> y = x**2 >>> y_int = integrate.cumulative_simpson(y, x=x, initial=0) >>> fig, ax = plt.subplots() >>> ax.plot(x, y_int, 'ro', x, x**3/3 - (x[0])**3/3, 'b-') >>> ax.grid() >>> plt.show() The output of `cumulative_simpson` is similar to that of iteratively calling `simpson` with successively higher upper limits of integration, but not identical. >>> def cumulative_simpson_reference(y, x): ... return np.asarray([integrate.simpson(y[:i], x=x[:i]) ... for i in range(2, len(y) + 1)]) >>> >>> rng = np.random.default_rng(354673834679465) >>> x, y = rng.random(size=(2, 10)) >>> x.sort() >>> >>> res = integrate.cumulative_simpson(y, x=x) >>> ref = cumulative_simpson_reference(y, x) >>> equal = np.abs(res - ref) < 1e-15 >>> equal # not equal when `simpson` has even number of subintervals array([False, True, False, True, False, True, False, True, True]) This is expected: because `cumulative_simpson` has access to more information than `simpson`, it can typically produce more accurate estimates of the underlying integral over subintervals. """ y = _ensure_float_array(y) # validate `axis` and standardize to work along the last axis original_y = y original_shape = y.shape try: y = np.swapaxes(y, axis, -1) except IndexError as e: message = f"`axis={axis}` is not valid for `y` with `y.ndim={y.ndim}`." raise ValueError(message) from e if y.shape[-1] < 3: res = cumulative_trapezoid(original_y, x, dx=dx, axis=axis, initial=None) res = np.swapaxes(res, axis, -1) elif x is not None: x = _ensure_float_array(x) message = ("If given, shape of `x` must be the same as `y` or 1-D with " "the same length as `y` along `axis`.") if not (x.shape == original_shape or (x.ndim == 1 and len(x) == original_shape[axis])): raise ValueError(message) x = np.broadcast_to(x, y.shape) if x.ndim == 1 else np.swapaxes(x, axis, -1) dx = np.diff(x, axis=-1) if np.any(dx <= 0): raise ValueError("Input x must be strictly increasing.") res = _cumulatively_sum_simpson_integrals( y, dx, _cumulative_simpson_unequal_intervals ) else: dx = _ensure_float_array(dx) final_dx_shape = tupleset(original_shape, axis, original_shape[axis] - 1) alt_input_dx_shape = tupleset(original_shape, axis, 1) message = ("If provided, `dx` must either be a scalar or have the same " "shape as `y` but with only 1 point along `axis`.") if not (dx.ndim == 0 or dx.shape == alt_input_dx_shape): raise ValueError(message) dx = np.broadcast_to(dx, final_dx_shape) dx = np.swapaxes(dx, axis, -1) res = _cumulatively_sum_simpson_integrals( y, dx, _cumulative_simpson_equal_intervals ) if initial is not None: initial = _ensure_float_array(initial) alt_initial_input_shape = tupleset(original_shape, axis, 1) message = ("If provided, `initial` must either be a scalar or have the " "same shape as `y` but with only 1 point along `axis`.") if not (initial.ndim == 0 or initial.shape == alt_initial_input_shape): raise ValueError(message) initial = np.broadcast_to(initial, alt_initial_input_shape) initial = np.swapaxes(initial, axis, -1) res += initial res = np.concatenate((initial, res), axis=-1) res = np.swapaxes(res, -1, axis) return res 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 fixed_quad : fixed-order Gaussian quadrature dblquad : double integrals tplquad : triple integrals simpson : integrators for sampled data cumulative_trapezoid : cumulative integration for sampled data 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)] # 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)