from __future__ import division, print_function, absolute_import import numpy as np import math import warnings # trapz is a public function for scipy.integrate, # even though it's actually a numpy function. from numpy import trapz from scipy.special.orthogonal import p_roots from scipy.special import gammaln from scipy._lib.six import xrange __all__ = ['fixed_quad', 'quadrature', 'romberg', 'trapz', 'simps', 'romb', 'cumtrapz', 'newton_cotes'] class AccuracyWarning(Warning): pass def _cached_p_roots(n): """ Cache p_roots results to speed up calls of the fixed_quad function. """ if n in _cached_p_roots.cache: return _cached_p_roots.cache[n] _cached_p_roots.cache[n] = p_roots(n) return _cached_p_roots.cache[n] _cached_p_roots.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). 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 simps : integrators for sampled data cumtrapz : cumulative integration for sampled data ode : ODE integrator odeint : ODE integrator """ x, w = _cached_p_roots(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=0), 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 xrange(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 simps: integrator for sampled data cumtrapz: cumulative integration for sampled data ode: ODE integrator odeint: ODE integrator """ 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 xrange(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) def cumtrapz(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 : int, 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, uses this value as the first value in 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 matplotlib.pyplot as plt >>> x = np.linspace(-2, 2, num=20) >>> y = x >>> y_int = integrate.cumtrapz(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.ones(shape, dtype=res.dtype) * initial, res], axis=axis) return res def _basic_simps(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(dx/3.0 * (y[slice0]+4*y[slice1]+y[slice2]), axis=axis) 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] h1 = h[sl1] hsum = h0 + h1 hprod = h0 * h1 h0divh1 = h0 / h1 tmp = hsum/6.0 * (y[slice0]*(2-1.0/h0divh1) + y[slice1]*hsum*hsum/hprod + y[slice2]*(2-h0divh1)) result = np.sum(tmp, axis=axis) return result def simps(y, x=None, dx=1, axis=-1, even='avg'): """ 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 : int, optional Spacing of integration points along axis of `y`. Only used when `x` is None. Default is 1. axis : int, optional Axis along which to integrate. Default is the last axis. even : {'avg', 'first', 'str'}, 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. 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 cumtrapz: 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. """ 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.") if N % 2 == 0: val = 0.0 result = 0.0 slice1 = (slice(None),)*nd slice2 = (slice(None),)*nd if even not in ['avg', 'last', 'first']: raise ValueError("Parameter 'even' must be " "'avg', 'last', or 'first'.") # Compute using Simpson's rule on first intervals if even in ['avg', 'first']: 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_simps(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_simps(y, 1, N-2, x, dx, axis) if even == 'avg': val /= 2.0 result /= 2.0 result = result + val else: result = _basic_simps(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 simps : integrators for sampled data cumtrapz : cumulative integration for sampled data ode : ODE integrators odeint : ODE integrators """ 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 xrange(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 xrange(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="") for i in xrange(k+1): for j in xrange(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 # last revision: 14 Nov 98 # # Cosmetic changes by Konrad Hinsen # last revision: 1999-7-21 # # Adapted to scipy by Travis Oliphant # 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 xrange(len(resmat)): print('%6d %9f' % (2**i, (interval[1]-interval[0])/(2.**i)), end=' ') for j in xrange(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. simps : Integrators for sampled data. cumtrapz : Cumulative integration for sampled data. ode : ODE integrator. odeint : ODE integrator. References ---------- .. [1] 'Romberg's method' http://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 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 xrange(1, divmax+1): n *= 2 ordsum += _difftrap(vfunc, interval, n) row = [intrange * ordsum / n] for k in xrange(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 Netwon-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): """ 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. """ try: N = len(rn)-1 if equal: rn = np.arange(N+1) elif np.all(np.diff(rn) == 1): equal = 1 except: 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