from __future__ import absolute_import, division, print_function import warnings import numpy as np from numpy import pi, sqrt, ones, zeros from scipy import integrate as intg import scipy.special.orthogonal as ort from scipy import special as sp from scipy.integrate import simps, trapz from wafo.plotbackend import plotbackend as plt from wafo.demos import humps from wafo.misc import dea3 from wafo.dctpack import dct # from pychebfun import Chebfun _EPS = np.finfo(float).eps _NODES_AND_WEIGHTS = {} __all__ = ['dea3', 'clencurt', 'romberg', 'h_roots', 'j_roots', 'la_roots', 'p_roots', 'qrule', 'gaussq', 'richardson', 'quadgr', 'qdemo'] def _assert(cond, msg): if not cond: raise ValueError(msg) def _assert_warn(cond, msg): if not cond: warnings.warn(msg) def clencurt(fun, a, b, n=5, trace=False): """ Numerical evaluation of an integral, Clenshaw-Curtis method. Parameters ---------- fun : callable a, b : array-like Lower and upper integration limit, respectively. n : integer defines number of evaluation points (default 5) Returns ------- q_val = evaluated integral tol = Estimate of the approximation error Notes ----- CLENCURT approximates the integral of f(x) from a to b using an 2*n+1 points Clenshaw-Curtis formula. The error estimate is usually a conservative estimate of the approximation error. The integral is exact for polynomials of degree 2*n or less. Example ------- >>> import numpy as np >>> val, err = clencurt(np.exp, 0, 2) >>> np.allclose(val, np.expm1(2)), err[0] < 1e-10 (True, True) See also -------- simpson, gaussq References ---------- [1] Goodwin, E.T. (1961), "Modern Computing Methods", 2nd edition, New yourk: Philosophical Library, pp. 78--79 [2] Clenshaw, C.W. and Curtis, A.R. (1960), Numerische Matematik, Vol. 2, pp. 197--205 """ # make sure n_2 is even n_2 = 2 * int(n) a, b = np.atleast_1d(a, b) a_shape = a.shape a = a.ravel() b = b.ravel() a_size = np.prod(a_shape) s = np.c_[0:n_2 + 1:1] s_2 = np.c_[0:n_2 + 1:2] x = np.cos(np.pi * s / n_2) * (b - a) / 2. + (b + a) / 2 if hasattr(fun, '__call__'): f = fun(x) else: x_0 = np.flipud(fun[:, 0]) n_2 = len(x_0) - 1 _assert(abs(x - x_0) <= 1e-8, 'Input vector x must equal cos(pi*s/n_2)*(b-a)/2+(b+a)/2') f = np.flipud(fun[:, 1::]) if trace: plt.plot(x, f, '+') # using a Gauss-Lobatto variant, i.e., first and last # term f(a) and f(b) is multiplied with 0.5 f[0, :] = f[0, :] / 2 f[n_2, :] = f[n_2, :] / 2 # x = cos(pi*0:n_2/n_2) # f = f(x) # # N+1 # c(k) = (2/N) sum f''(n)*cos(pi*(2*k-2)*(n-1)/N), 1 <= k <= N/2+1. # n=1 n = n_2 // 2 fft = np.fft.fft tmp = np.real(fft(f[:n_2, :], axis=0)) c = 2 / n_2 * (tmp[0:n + 1, :] + np.cos(np.pi * s_2) * f[n_2, :]) c[0, :] = c[0, :] / 2 c[n, :] = c[n, :] / 2 c = c[0:n + 1, :] / ((s_2 - 1) * (s_2 + 1)) q_val = (a - b) * np.sum(c, axis=0) abserr = (b - a) * np.abs(c[n, :]) if a_size > 1: abserr = np.reshape(abserr, a_shape) q_val = np.reshape(q_val, a_shape) return q_val, abserr def romberg(fun, a, b, releps=1e-3, abseps=1e-3): """ Numerical integration with the Romberg method Parameters ---------- fun : callable function to integrate a, b : real scalars lower and upper integration limits, respectively. releps, abseps : scalar, optional requested relative and absolute error, respectively. Returns ------- Q : scalar value of integral abserr : scalar estimated absolute error of integral ROMBERG approximates the integral of F(X) from A to B using Romberg's method of integration. The function F must return a vector of output values if a vector of input values is given. Example ------- >>> import numpy as np >>> [q,err] = romberg(np.sqrt,0,10,0,1e-4) >>> np.allclose(q, 21.08185107) True >>> err[0] < 1e-4 True """ h = b - a h_min = 1.0e-9 # Max size of extrapolation table table_limit = max(min(np.round(np.log2(h / h_min)), 30), 3) rom = zeros((2, table_limit)) rom[0, 0] = h * (fun(a) + fun(b)) / 2 ipower = 1 f_p = ones(table_limit) * 4 # q_val1 = 0 q_val2 = 0. q_val4 = rom[0, 0] abserr = q_val4 # epstab = zeros(1,decdigs+7) # newflg = 1 # [res,abserr,epstab,newflg] = dea(newflg,q_val4,abserr,epstab) two = 1 one = 0 converged = False for i in range(1, table_limit): h *= 0.5 u_n5 = np.sum(fun(a + np.arange(1, 2 * ipower, 2) * h)) * h # trapezoidal approximations # T2n = 0.5 * (Tn + Un) = 0.5*Tn + u_n5 rom[two, 0] = 0.5 * rom[one, 0] + u_n5 f_p[i] = 4 * f_p[i - 1] # Richardson extrapolation for k in range(i): rom[two, k + 1] = (rom[two, k] + (rom[two, k] - rom[one, k]) / (f_p[k] - 1)) q_val1 = q_val2 q_val2 = q_val4 q_val4 = rom[two, i] if 2 <= i: res, abserr = dea3(q_val1, q_val2, q_val4) # q_val4 = res converged = abserr <= max(abseps, releps * abs(res)) if converged: break # rom(1,1:i) = rom(2,1:i) two = one one = (one + 1) % 2 ipower *= 2 _assert(converged, "Integral did not converge to the required accuracy!") return res, abserr def _h_roots_newton(n, releps=3e-14, max_iter=10): # pim4=0.7511255444649425 pim4 = np.pi ** (-1. / 4) # The roots are symmetric about the origin, so we have to # find only half of them. m = int(np.fix((n + 1) / 2)) # Initial approximations to the roots go into z. anu = 2.0 * n + 1 rhs = np.arange(3, 4 * m, 4) * np.pi / anu theta = _get_theta(rhs) z = sqrt(anu) * np.cos(theta) p = zeros((3, len(z))) k_0 = 0 k_p1 = 1 for _i in range(max_iter): # Newtons method carried out simultaneously on the roots. p[k_0, :] = 0 p[k_p1, :] = pim4 for j in range(1, n + 1): # Loop up the recurrence relation to get the Hermite # polynomials evaluated at z. k_m1 = k_0 k_0 = k_p1 k_p1 = np.mod(k_p1 + 1, 3) p[k_p1, :] = (z * sqrt(2 / j) * p[k_0, :] - sqrt((j - 1) / j) * p[k_m1, :]) # p now contains the desired Hermite polynomials. # We next compute p_deriv, the derivatives, # by the relation (4.5.21) using p2, the polynomials # of one lower order. p_deriv = sqrt(2 * n) * p[k_0, :] d_z = p[k_p1, :] / p_deriv z = z - d_z # Newtons formula. converged = not np.any(abs(d_z) > releps) if converged: break _assert_warn(converged, 'Newton iteration did not converge!') weights = 2. / p_deriv ** 2 return _expand_roots(z, weights, n, m) def h_roots(n, method='newton'): """ Returns the roots (x) of the nth order Hermite polynomial, H_n(x), and weights (w) to use in Gaussian Quadrature over [-inf,inf] with weighting function exp(-x**2). Parameters ---------- n : integer number of roots method : 'newton' or 'eigenvalue' uses Newton Raphson to find zeros of the Hermite polynomial (Fast) or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and weights, respectively. Returns ------- x : ndarray roots w : ndarray weights Example ------- >>> import numpy as np >>> x, w = h_roots(10) >>> np.allclose(np.sum(x*w), -5.2516042729766621e-19) True See also -------- qrule, gaussq References ---------- [1] Golub, G. H. and Welsch, J. H. (1969) 'Calculation of Gaussian Quadrature Rules' Mathematics of Computation, vol 23,page 221-230, [2]. Stroud and Secrest (1966), 'gaussian quadrature formulas', prentice-hall, Englewood cliffs, n.j. """ if not method.startswith('n'): return ort.h_roots(n) return _h_roots_newton(n) def _j_roots_newton(n, alpha, beta, releps=3e-14, max_iter=10): # Initial approximations to the roots go into z. alfbet = alpha + beta z = np.cos(np.pi * (np.arange(1, n + 1) - 0.25 + 0.5 * alpha) / (n + 0.5 * (alfbet + 1))) p = zeros((3, len(z))) k_0 = 0 k_p1 = 1 for _i in range(max_iter): # Newton's method carried out simultaneously on the roots. tmp = 2 + alfbet p[k_0, :] = 1 p[k_p1, :] = (alpha - beta + tmp * z) / 2 for j in range(2, n + 1): # Loop up the recurrence relation to get the Jacobi # polynomials evaluated at z. k_m1 = k_0 k_0 = k_p1 k_p1 = np.mod(k_p1 + 1, 3) a = 2. * j * (j + alfbet) * tmp tmp = tmp + 2 c = 2 * (j - 1 + alpha) * (j - 1 + beta) * tmp b = (tmp - 1) * (alpha ** 2 - beta ** 2 + tmp * (tmp - 2) * z) p[k_p1, :] = (b * p[k_0, :] - c * p[k_m1, :]) / a # p now contains the desired Jacobi polynomials. # We next compute p_deriv, the derivatives with a standard # relation involving the polynomials of one lower order. p_deriv = ((n * (alpha - beta - tmp * z) * p[k_p1, :] + 2 * (n + alpha) * (n + beta) * p[k_0, :]) / (tmp * (1 - z ** 2))) d_z = p[k_p1, :] / p_deriv z = z - d_z # Newton's formula. converged = not any(abs(d_z) > releps * abs(z)) if converged: break _assert_warn(converged, 'too many iterations in jrule') x = z # Store the root and the weight. f = (sp.gammaln(alpha + n) + sp.gammaln(beta + n) - sp.gammaln(n + 1) - sp.gammaln(alpha + beta + n + 1)) weights = (np.exp(f) * tmp * 2 ** alfbet / (p_deriv * p[k_0, :])) return x, weights def j_roots(n, alpha, beta, method='newton'): """ Returns the roots of the nth order Jacobi polynomial, P^(alpha,beta)_n(x) and weights (w) to use in Gaussian Quadrature over [-1,1] with weighting function (1-x)**alpha (1+x)**beta with alpha,beta > -1. Parameters ---------- n : integer number of roots alpha,beta : scalars defining shape of Jacobi polynomial method : 'newton' or 'eigenvalue' uses Newton Raphson to find zeros of the Hermite polynomial (Fast) or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and weights, respectively. Returns ------- x : ndarray roots w : ndarray weights Example -------- >>> [x,w]= j_roots(10,0,0) >>> sum(x*w) 2.7755575615628914e-16 See also -------- qrule, gaussq Reference --------- [1] Golub, G. H. and Welsch, J. H. (1969) 'Calculation of Gaussian Quadrature Rules' Mathematics of Computation, vol 23,page 221-230, [2]. Stroud and Secrest (1966), 'gaussian quadrature formulas', prentice-hall, Englewood cliffs, n.j. """ _assert((-1 < alpha) & (-1 < beta), 'alpha and beta must be greater than -1') if not method.startswith('n'): return ort.j_roots(n, alpha, beta) return _j_roots_newton(n, alpha, beta) def la_roots(n, alpha=0, method='newton'): """ Returns the roots (x) of the nth order generalized (associated) Laguerre polynomial, L^(alpha)_n(x), and weights (w) to use in Gaussian quadrature over [0,inf] with weighting function exp(-x) x**alpha with alpha > -1. Parameters ---------- n : integer number of roots method : 'newton' or 'eigenvalue' uses Newton Raphson to find zeros of the Laguerre polynomial (Fast) or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and weights, respectively. Returns ------- x : ndarray roots w : ndarray weights Example ------- >>> import numpy as np >>> [x,w] = h_roots(10) >>> np.sum(x*w) 1.3352627380516791e-17 See also -------- qrule, gaussq References ---------- [1] Golub, G. H. and Welsch, J. H. (1969) 'Calculation of Gaussian Quadrature Rules' Mathematics of Computation, vol 23,page 221-230, [2]. Stroud and Secrest (1966), 'gaussian quadrature formulas', prentice-hall, Englewood cliffs, n.j. """ _assert(-1 < alpha, 'alpha must be greater than -1') if not method.startswith('n'): return ort.la_roots(n, alpha) return _la_roots_newton(n, alpha) def _get_theta(rhs): r_3 = rhs ** (1. / 3) r_2 = r_3 ** 2 c = [9.084064e-01, 5.214976e-02, 2.579930e-03, 3.986126e-03] theta = r_3 * (c[0] + r_2 * (c[1] + r_2 * (c[2] + r_2 * c[3]))) return theta def _la_roots_newton(n, alpha, releps=3e-14, max_iter=10): # Initial approximations to the roots go into z. anu = 4.0 * n + 2.0 * alpha + 2.0 rhs = np.arange(4 * n - 1, 2, -4) * np.pi / anu theta = _get_theta(rhs) z = anu * np.cos(theta) ** 2 d_z = zeros(len(z)) p = zeros((3, len(z))) p_previous = zeros((1, len(z))) p_deriv = zeros((1, len(z))) k_0 = 0 k_p1 = 1 k = slice(len(z)) for _i in range(max_iter): # Newton's method carried out simultaneously on the roots. p[k_0, k] = 0. p[k_p1, k] = 1. for j in range(1, n + 1): # Loop up the recurrence relation to get the Laguerre # polynomials evaluated at z. km1 = k_0 k_0 = k_p1 k_p1 = np.mod(k_p1 + 1, 3) p[k_p1, k] = ((2 * j - 1 + alpha - z[k]) * p[k_0, k] - (j - 1 + alpha) * p[km1, k]) / j # end # p now contains the desired Laguerre polynomials. # We next compute p_deriv, the derivatives with a standard # relation involving the polynomials of one lower order. p_previous[k] = p[k_0, k] p_deriv[k] = (n * p[k_p1, k] - (n + alpha) * p_previous[k]) / z[k] d_z[k] = p[k_p1, k] / p_deriv[k] z[k] = z[k] - d_z[k] # Newton?s formula. # k = find((abs(d_z) > releps.*z)) converged = not np.any(abs(d_z) > releps) if converged: break _assert_warn(converged, 'too many iterations!') nodes = z weights = -np.exp(sp.gammaln(alpha + n) - sp.gammaln(n)) / (p_deriv * n * p_previous) return nodes, weights def _p_roots_newton_start(n): m = int(np.fix((n + 1) / 2)) t = (np.pi / (4 * n + 2)) * np.arange(3, 4 * m, 4) a = 1 - (1 - 1 / n) / (8 * n * n) x = a * np.cos(t) return m, x def _p_roots_newton(n): """ Algorithm given by Davis and Rabinowitz in 'Methods of Numerical Integration', page 365, Academic Press, 1975. """ m, x = _p_roots_newton_start(n) e_1 = n * (n + 1) for _j in range(2): p_km1 = 1 p_k = x for k in range(2, n + 1): t_1 = x * p_k p_kp1 = t_1 - p_km1 - (t_1 - p_km1) / k + t_1 p_km1 = p_k p_k = p_kp1 den = 1. - x * x d_1 = n * (p_km1 - x * p_k) d_pn = d_1 / den d_2pn = (2. * x * d_pn - e_1 * p_k) / den d_3pn = (4. * x * d_2pn + (2 - e_1) * d_pn) / den d_4pn = (6. * x * d_3pn + (6 - e_1) * d_2pn) / den u = p_k / d_pn v = d_2pn / d_pn h = -u * (1 + (.5 * u) * (v + u * (v * v - u * d_3pn / (3 * d_pn)))) p = p_k + h * (d_pn + (.5 * h) * (d_2pn + (h / 3) * (d_3pn + .25 * h * d_4pn))) d_p = d_pn + h * (d_2pn + (.5 * h) * (d_3pn + h * d_4pn / 3)) h = h - p / d_p x = x + h nodes = -x - h f_x = d_1 - h * e_1 * (p_k + (h / 2) * (d_pn + (h / 3) * (d_2pn + (h / 4) * (d_3pn + (.2 * h) * d_4pn)))) weights = 2 * (1 - nodes ** 2) / (f_x ** 2) return _expand_roots(nodes, weights, n, m) def _p_roots_newton1(n, releps=1e-15, max_iter=100): m, x = _p_roots_newton_start(n) # Compute the zeros of the N+1 Legendre Polynomial # using the recursion relation and the Newton-Raphson method # Legendre-Gauss Polynomials p = zeros((3, m)) # Derivative of LGP p_deriv = zeros((m,)) d_x = zeros((m,)) # Compute the zeros of the N+1 Legendre Polynomial # using the recursion relation and the Newton-Raphson method # Iterate until new points are uniformly within epsilon of old # points k = slice(m) k_0 = 0 k_p1 = 1 for _ix in range(max_iter): p[k_0, k] = 1 p[k_p1, k] = x[k] for j in range(2, n + 1): k_m1 = k_0 k_0 = k_p1 k_p1 = np.mod(k_0 + 1, 3) p[k_p1, k] = ((2 * j - 1) * x[k] * p[k_0, k] - (j - 1) * p[k_m1, k]) / j p_deriv[k] = n * (p[k_0, k] - x[k] * p[k_p1, k]) / (1 - x[k] ** 2) d_x[k] = p[k_p1, k] / p_deriv[k] x[k] = x[k] - d_x[k] k, = np.nonzero((abs(d_x) > releps * np.abs(x))) converged = len(k) == 0 if converged: break _assert(converged, 'Too many iterations!') nodes = -x weights = 2. / ((1 - nodes ** 2) * (p_deriv ** 2)) return _expand_roots(nodes, weights, n, m) def _expand_roots(x, w, n, m): if (m + m) > n: x[m - 1] = 0.0 if not (m + m) == n: m = m - 1 x = np.hstack((x, -x[m - 1::-1])) w = np.hstack((w, w[m - 1::-1])) return x, w def p_roots(n, method='newton', a=-1, b=1): """ Returns the roots (x) of the nth order Legendre polynomial, P_n(x), and weights to use in Gaussian Quadrature over [-1,1] with weighting function 1. Parameters ---------- n : integer number of roots method : 'newton' or 'eigenvalue' uses Newton Raphson to find zeros of the Hermite polynomial (Fast) or eigenvalue of the jacobi matrix (Slow) to obtain the nodes and weights, respectively. Returns ------- nodes : ndarray roots weights : ndarray weights Example ------- Integral of exp(x) from a = 0 to b = 3 is: exp(3)-exp(0)= >>> import numpy as np >>> nodes, weights = p_roots(11, a=0, b=3) >>> np.allclose(np.sum(np.exp(nodes) * weights), 19.085536923187668) True >>> nodes, weights = p_roots(11, method='newton1', a=0, b=3) >>> np.allclose(np.sum(np.exp(nodes) * weights), 19.085536923187668) True >>> nodes, weights = p_roots(11, method='eigenvalue', a=0, b=3) >>> np.allclose(np.sum(np.exp(nodes) * weights), 19.085536923187668) True See also -------- quadg. References ---------- [1] Davis and Rabinowitz (1975) 'Methods of Numerical Integration', page 365, Academic Press. [2] Golub, G. H. and Welsch, J. H. (1969) 'Calculation of Gaussian Quadrature Rules' Mathematics of Computation, vol 23,page 221-230, [3] Stroud and Secrest (1966), 'gaussian quadrature formulas', prentice-hall, Englewood cliffs, n.j. """ if not method.startswith('n'): nodes, weights = ort.p_roots(n) else: if method.endswith('1'): nodes, weights = _p_roots_newton1(n) else: nodes, weights = _p_roots_newton(n) if (a != -1) | (b != 1): # Linear map from[-1,1] to [a,b] d_h = (b - a) / 2 nodes = d_h * (nodes + 1) + a weights = weights * d_h return nodes, weights def q5_roots(n): """ 5 : p(x) = 1/sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 1'st kind """ j = np.arange(1, n + 1) weights = ones(n) * np.pi / n nodes = np.cos((2 * j - 1) * np.pi / (2 * n)) return nodes, weights def q6_roots(n): """ 6 : p(x) = sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 2'nd kind """ j = np.arange(1, n + 1) x_j = j * np.pi / (n + 1) weights = np.pi / (n + 1) * np.sin(x_j) ** 2 nodes = np.cos(x_j) return nodes, weights def q7_roots(n): """ 7 : p(x) = sqrt((x-a)/(b-x)), a = 0, b = 1 """ j = np.arange(1, n + 1) x_j = (j - 0.5) * pi / (2 * n + 1) nodes = np.cos(x_j) ** 2 weights = 2 * np.pi * nodes / (2 * n + 1) return nodes, weights def q8_roots(n): """ 8 : p(x) = 1/sqrt(b-x), a = 0, b = 1 """ nodes_1, weights_1 = p_roots(2 * n) k, = np.where(0 <= nodes_1) weights = 2 * weights_1[k] nodes = 1 - nodes_1[k] ** 2 return nodes, weights def q9_roots(n): """ 9 : p(x) = sqrt(b-x), a = 0, b = 1 """ nodes_1, weights_1 = p_roots(2 * n + 1) k, = np.where(0 < nodes_1) weights = 2 * nodes_1[k] ** 2 * weights_1[k] nodes = 1 - nodes_1[k] ** 2 return nodes, weights def qrule(n, wfun=1, alpha=0, beta=0): """ Return nodes and weights for Gaussian quadratures. Parameters ---------- n : integer number of base points wfun : integer defining the weight function, p(x). (default wfun = 1) 1 : p(x) = 1 a =-1, b = 1 Gauss-Legendre 2 : p(x) = exp(-x^2) a =-inf, b = inf Hermite 3 : p(x) = x^alpha*exp(-x) a = 0, b = inf Laguerre 4 : p(x) = (x-a)^alpha*(b-x)^beta a =-1, b = 1 Jacobi 5 : p(x) = 1/sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 1'st kind 6 : p(x) = sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 2'nd kind 7 : p(x) = sqrt((x-a)/(b-x)), a = 0, b = 1 8 : p(x) = 1/sqrt(b-x), a = 0, b = 1 9 : p(x) = sqrt(b-x), a = 0, b = 1 Returns ------- bp = base points (abscissas) wf = weight factors The Gaussian Quadrature integrates a (2n-1)th order polynomial exactly and the integral is of the form b n Int ( p(x)* F(x) ) dx = Sum ( wf_j* F( bp_j ) ) a j=1 where p(x) is the weight function. For Jacobi and Laguerre: alpha, beta >-1 (default alpha=beta=0) Examples: --------- >>> import numpy as np # integral of x^2 from a = -1 to b = 1 >>> bp, wf = qrule(10) >>> np.allclose(sum(bp**2*wf), 0.66666666666666641) True # integral of exp(-x**2)*x**2 from a = -inf to b = inf >>> bp, wf = qrule(10,2) >>> np.allclose(sum(bp ** 2 * wf), 0.88622692545275772) True # integral of (x+1)*(1-x)**2 from a = -1 to b = 1 >>> bp, wf = qrule(10,4,1,2) >>> np.allclose((bp * wf).sum(), 0.26666666666666755) True See also -------- gaussq Reference --------- Abromowitz and Stegun (1954) (for method 5 to 9) """ if wfun == 3: # Generalized Laguerre return la_roots(n, alpha) if wfun == 4: # Gauss-Jacobi return j_roots(n, alpha, beta) _assert(0 < wfun < 10, 'unknown weight function') root_fun = [None, p_roots, h_roots, la_roots, j_roots, q5_roots, q6_roots, q7_roots, q8_roots, q9_roots][wfun] return root_fun(n) class _Gaussq(object): """ Numerically evaluate integral, Gauss quadrature. Parameters ---------- fun : callable a,b : array-like lower and upper integration limits, respectively. releps, abseps : real scalars, optional relative and absolute tolerance, respectively. (default releps=abseps=1e-3). wfun : scalar integer, optional defining the weight function, p(x). (default wfun = 1) 1 : p(x) = 1 a =-1, b = 1 Gauss-Legendre 2 : p(x) = exp(-x^2) a =-inf, b = inf Hermite 3 : p(x) = x^alpha*exp(-x) a = 0, b = inf Laguerre 4 : p(x) = (x-a)^alpha*(b-x)^beta a =-1, b = 1 Jacobi 5 : p(x) = 1/sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 1'st kind 6 : p(x) = sqrt((x-a)*(b-x)), a =-1, b = 1 Chebyshev 2'nd kind 7 : p(x) = sqrt((x-a)/(b-x)), a = 0, b = 1 8 : p(x) = 1/sqrt(b-x), a = 0, b = 1 9 : p(x) = sqrt(b-x), a = 0, b = 1 trace : bool, optional If non-zero a point plot of the integrand (default False). gn : scalar integer number of base points to start the integration with (default 2). alpha, beta : real scalars, optional Shape parameters of Laguerre or Jacobi weight function (alpha,beta>-1) (default alpha=beta=0) Returns ------- val : ndarray evaluated integral err : ndarray error estimate, absolute tolerance abs(int-intold) Notes ----- GAUSSQ numerically evaluate integral using a Gauss quadrature. The Quadrature integrates a (2m-1)th order polynomial exactly and the integral is of the form b Int (p(x)* Fun(x)) dx a GAUSSQ is vectorized to accept integration limits A, B and coefficients P1,P2,...Pn, as matrices or scalars and the result is the common size of A, B and P1,P2,...,Pn. Examples --------- integration of x**2 from 0 to 2 and from 1 to 4 >>> import numpy as np >>> A = [0, 1] >>> B = [2, 4] >>> fun = lambda x: x**2 >>> val1, err1 = gaussq(fun,A,B) >>> np.allclose(val1, [ 2.6666667, 21. ]) True >>> np.allclose(err1, [ 1.7763568e-15, 1.0658141e-14]) True Integration of x^2*exp(-x) from zero to infinity: >>> fun2 = lambda x : np.ones(np.shape(x)) >>> val2, err2 = gaussq(fun2, 0, np.inf, wfun=3, alpha=2) >>> val3, err3 = gaussq(lambda x: x**2,0, np.inf, wfun=3, alpha=0) >>> np.allclose(val2, 2), err2[0] < 1e-14 (True, True) >>> np.allclose(val3, 2), err3[0] < 1e-14 (True, True) Integrate humps from 0 to 2 and from 1 to 4 >>> val4, err4 = gaussq(humps,A,B) See also -------- qrule gaussq2d """ @staticmethod def _get_dx(wfun, jacob, alpha, beta): def fun1(x): return x if wfun == 4: d_x = jacob ** (alpha + beta + 1) else: d_x = [None, fun1, fun1, fun1, None, lambda x: ones(np.shape(x)), lambda x: x ** 2, fun1, sqrt, lambda x: sqrt(x) ** 3][wfun](jacob) return d_x.ravel() @staticmethod def _nodes_and_weights(num_nodes, wfun, alpha, beta): global _NODES_AND_WEIGHTS name = 'wfun{:d}_{:d}_{:g}_{:g}'.format(wfun, num_nodes, alpha, beta) nodes_and_weights = _NODES_AND_WEIGHTS.setdefault(name, []) if len(nodes_and_weights) == 0: nodes_and_weights.extend(qrule(num_nodes, wfun, alpha, beta)) nodes, weights = nodes_and_weights return nodes, weights def _initialize_trace(self, max_iter): if self.trace: self.x_trace = [0] * max_iter self.y_trace = [0] * max_iter def _plot_trace(self, x, y): if self.trace: self.x_trace.append(x.ravel()) self.y_trace.append(y.ravel()) hfig = plt.plot(x, y, 'r.') plt.setp(hfig, 'color', 'b') def _plot_final_trace(self): if self.trace > 0: plt.clf() plt.plot(np.hstack(self.x_trace), np.hstack(self.y_trace), '+') @staticmethod def _get_jacob(wfun, a, b): if wfun in [2, 3]: jacob = ones((np.size(a), 1)) else: jacob = (b - a) * 0.5 if wfun in [7, 8, 9]: jacob *= 2 return jacob @staticmethod def _warn(k, a_shape): n = len(k) if n > 1: if n == np.prod(a_shape): tmptxt = 'All integrals did not converge' else: tmptxt = '%d integrals did not converge' % (n, ) tmptxt = tmptxt + '--singularities likely!' else: tmptxt = 'Integral did not converge--singularity likely!' warnings.warn(tmptxt) @staticmethod def _initialize(wfun, a, b, args): args = np.broadcast_arrays(*np.atleast_1d(a, b, *args)) a_shape = args[0].shape args = [np.reshape(x, (-1, 1)) for x in args] a_out, b_out = args[:2] args = args[2:] if wfun in [2, 3]: a_out = zeros((a_out.size, 1)) return a_out, b_out, args, a_shape def __call__(self, fun, a, b, releps=1e-3, abseps=1e-3, alpha=0, beta=0, wfun=1, trace=False, args=(), max_iter=11): self.trace = trace num_nodes = 2 a_0, b_0, args, a_shape = self._initialize(wfun, a, b, args) jacob = self._get_jacob(wfun, a_0, b_0) shift = int(wfun in [1, 4, 5, 6]) d_x = self._get_dx(wfun, jacob, alpha, beta) self._initialize_trace(max_iter) # Break out of the iteration loop for three reasons: # 1) the last update is very small (compared to int and to releps) # 2) There are more than 11 iterations. This should NEVER happen. dtype = np.result_type(fun((a_0+b_0)*0.5, *args)) n_k = np.prod(a_shape) # # of integrals we have to compute k = np.arange(n_k) opts = (n_k, dtype) val, val_old, abserr = zeros(*opts), ones(*opts), zeros(*opts) nodes_and_weights = self._nodes_and_weights for i in range(max_iter): x_n, weights = nodes_and_weights(num_nodes, wfun, alpha, beta) x = (x_n + shift) * jacob[k, :] + a_0[k, :] params = [xi[k, :] for xi in args] y = fun(x, *params) self._plot_trace(x, y) val[k] = np.sum(weights * y, axis=1) * d_x[k] # do the integration if any(np.isnan(val)): val[np.isnan(val)] = val_old[np.isnan(val)] if 1 < i: abserr[k] = abs(val_old[k] - val[k]) # absolute tolerance k, = np.where(abserr > np.maximum(abs(releps * val), abseps)) n_k = len(k) # of integrals we have to compute again if n_k == 0: break val_old[k] = val[k] num_nodes *= 2 # double the # of basepoints and weights else: self._warn(k, a_shape) # make sure int is the same size as the integration limits val.shape = a_shape abserr.shape = a_shape self._plot_final_trace() return val, abserr gaussq = _Gaussq() def richardson(q_val, k): # license BSD # Richardson extrapolation with parameter estimation c = np.real((q_val[k - 1] - q_val[k - 2]) / (q_val[k] - q_val[k - 1])) - 1. # The lower bound 0.07 admits the singularity x.^-0.9 c = max(c, 0.07) return q_val[k] + (q_val[k] - q_val[k - 1]) / c class _Quadgr(object): """ Gauss-Legendre quadrature with Richardson extrapolation. [q_val,ERR] = QUADGR(FUN,A,B,TOL) approximates the integral of a function FUN from A to B with an absolute error tolerance TOL. FUN is a function handle and must accept vector arguments. TOL is 1e-6 by default. q_val is the integral approximation and ERR is an estimate of the absolute error. QUADGR uses a 12-point Gauss-Legendre quadrature. The error estimate is based on successive interval bisection. Richardson extrapolation accelerates the convergence for some integrals, especially integrals with endpoint singularities. Examples -------- >>> import numpy as np >>> q_val, err = quadgr(np.log,0,1) >>> q, err = quadgr(np.exp,0,9999*1j*np.pi) >>> np.allclose(q, -2.0000000000122662), err < 1.0e-08 (True, True) >>> q, err = quadgr(lambda x: np.sqrt(4-x**2), 0, 2, abseps=1e-12) >>> np.allclose(q, 3.1415926535897811), err < 1.0e-12 (True, True) >>> q, err = quadgr(lambda x: x**-0.75, 0, 1) >>> np.allclose(q, 4), err < 1.e-13 (True, True) >>> q, err = quadgr(lambda x: 1./np.sqrt(1-x**2), -1, 1) >>> np.allclose(q, 3.141596056985029), err < 1.0e-05 (True, True) >>> q, err = quadgr(lambda x: np.exp(-x**2), -np.inf, np.inf, 1e-9) >>> np.allclose(q, np.sqrt(np.pi)), err < 1e-9 (True, True) >>> q, err = quadgr(lambda x: np.cos(x)*np.exp(-x), 0, np.inf, 1e-9) >>> np.allclose(q, 0.5), err < 1e-9 (True, True) >>> q, err = quadgr(lambda x: np.cos(x)*np.exp(-x), np.inf, 0, 1e-9) >>> np.allclose(q, -0.5), err < 1e-9 (True, True) >>> q, err = quadgr(lambda x: np.cos(x)*np.exp(x), -np.inf, 0, 1e-9) >>> np.allclose(q, 0.5), err < 1e-9 (True, True) See also -------- QUAD, QUADGK """ # Author: jonas.lundgren@saabgroup.com, 2009. license BSD # Order limits (required if infinite limits) def _change_variable_and_integrate(self, fun, a, b, abseps, max_iter): isreal = np.isreal(a) & np.isreal(b) & ~np.isnan(a) & ~np.isnan(b) _assert(isreal, 'Infinite intervals must be real.') integrate = self._integrate # Change of variable if np.isfinite(a) & np.isinf(b): # a to inf val, err = integrate(lambda t: fun(a + t / (1 - t)) / (1 - t) ** 2, 0, 1, abseps, max_iter) elif np.isinf(a) & np.isfinite(b): # -inf to b val, err = integrate(lambda t: fun(b + t / (1 + t)) / (1 + t) ** 2, -1, 0, abseps, max_iter) else: # -inf to inf val1, err1 = integrate(lambda t: fun(t / (1 - t)) / (1 - t) ** 2, 0, 1, abseps / 2, max_iter) val2, err2 = integrate(lambda t: fun(t / (1 + t)) / (1 + t) ** 2, -1, 0, abseps / 2, max_iter) val = val1 + val2 err = err1 + err2 return val, err @staticmethod def _nodes_and_weights(): # Gauss-Legendre quadrature (12-point) x = np.asarray( [0.12523340851146894, 0.36783149899818018, 0.58731795428661748, 0.76990267419430469, 0.9041172563704748, 0.98156063424671924]) w = np.asarray( [0.24914704581340288, 0.23349253653835478, 0.20316742672306584, 0.16007832854334636, 0.10693932599531818, 0.047175336386511842]) nodes = np.hstack((x, -x)) weights = np.hstack((w, w)) return nodes, weights @staticmethod def _get_best_estimate(vals0, vals1, vals2, k): if k >= 6: q_v = np.hstack((vals0[k], vals1[k], vals2[k])) q_w = np.hstack((vals0[k - 1], vals1[k - 1], vals2[k - 1])) elif k >= 4: q_v = np.hstack((vals0[k], vals1[k])) q_w = np.hstack((vals0[k - 1], vals1[k - 1])) else: q_v = np.atleast_1d(vals0[k]) q_w = vals0[k - 1] # Estimate absolute error errors = np.atleast_1d(abs(q_v - q_w)) j = errors.argmin() err = errors[j] q_val = q_v[j] # if k >= 2: # and not iscomplex: # _val, err1 = dea3(vals0[k - 2], vals0[k - 1], vals0[k]) return q_val, err def _integrate(self, fun, a, b, abseps, max_iter): dtype = np.result_type(fun((a+b)/2), fun((a+b)/4)) # Initiate vectors val0 = zeros(max_iter, dtype=dtype) # Quadrature val1 = zeros(max_iter, dtype=dtype) # First Richardson extrapolation val2 = zeros(max_iter, dtype=dtype) # Second Richardson extrapolation x_n, weights = self._nodes_and_weights() n = len(x_n) # One interval d_x = (b - a) / 2 # Half interval length x = (a + b) / 2 + d_x * x_n # Nodes # Quadrature val0[0] = d_x * np.sum(weights * fun(x), axis=0) # Successive bisection of intervals for k in range(1, max_iter): # Interval bisection d_x = d_x / 2 x = np.hstack([x + a, x + b]) / 2 # Quadrature val0[k] = np.sum(np.sum(np.reshape(fun(x), (-1, n)), axis=0) * weights, axis=0) * d_x # Richardson extrapolation if k >= 5: val1[k] = richardson(val0, k) val2[k] = richardson(val1, k) elif k >= 3: val1[k] = richardson(val0, k) q_val, err = self._get_best_estimate(val0, val1, val2, k) converged = (err < abseps) | ~np.isfinite(q_val) if converged: break _assert_warn(converged, 'Max number of iterations reached without ' 'convergence.') _assert_warn(np.isfinite(q_val), 'Integral approximation is Infinite or NaN.') # The error estimate should not be zero err = err + 2 * np.finfo(q_val).eps return q_val, err @staticmethod def _order_limits(a, b): if np.real(a) > np.real(b): return b, a, True return a, b, False def __call__(self, fun, a, b, abseps=1e-5, max_iter=17): a = np.asarray(a) b = np.asarray(b) if a == b: q_val = b - a err = b - a return q_val, err a, b, reverse = self._order_limits(a, b) improper_integral = np.isinf(a) | np.isinf(b) if improper_integral: # Infinite limits q_val, err = self._change_variable_and_integrate(fun, a, b, abseps, max_iter) else: q_val, err = self._integrate(fun, a, b, abseps, max_iter) # Reverse direction if reverse: q_val = -q_val return q_val, err quadgr = _Quadgr() def boole(y, x): a, b = x[0], x[-1] n = len(x) h = (b - a) / (n - 1) return (2 * h / 45) * (7 * (y[0] + y[-1]) + 12 * np.sum(y[2:n - 1:4]) + 32 * np.sum(y[1:n - 1:2]) + 14 * np.sum(y[4:n - 3:4])) def _display(neval, vals_dic, err_dic, plot_error): # display results kmax = len(neval) names = sorted(vals_dic.keys()) num_cols = 2 formats = ['%4.0f, '] + ['%10.10f, '] * num_cols * 2 formats[-1] = formats[-1].split(',')[0] formats_h = ['%4s, '] + ['%20s, '] * num_cols formats_h[-1] = formats_h[-1].split(',')[0] headers = ['evals'] + ['%12s %12s' % ('approx', 'error')] * num_cols while len(names) > 0: print(''.join(fi % t for (fi, t) in zip(formats_h, ['ftn'] + names[:num_cols]))) print(' '.join(headers)) data = [neval] for name in names[:num_cols]: data.append(vals_dic[name]) data.append(err_dic[name]) data = np.vstack(tuple(data)).T for k in range(kmax): tmp = data[k].tolist() print(''.join(fi % t for (fi, t) in zip(formats, tmp))) if plot_error: plt.figure(0) for name in names[:num_cols]: plt.loglog(neval, err_dic[name], label=name) names = names[num_cols:] if plot_error: plt.xlabel('number of function evaluations') plt.ylabel('error') plt.legend() def qdemo(f, a, b, kmax=9, plot_error=False): """ Compares different quadrature rules. Parameters ---------- f : callable function a,b : scalars lower and upper integration limits Details ------- qdemo(f,a,b) computes and compares various approximations to the integral of f from a to b. Three approximations are used, the composite trapezoid, Simpson's, and Boole's rules, all with equal length subintervals. In a case like qdemo(exp,0,3) one can see the expected convergence rates for each of the three methods. In a case like qdemo(sqrt,0,3), the convergence rate is limited not by the method, but by the singularity of the integrand. Example ------- >>> import numpy as np >>> qdemo(np.exp,0,3, plot_error=True) true value = 19.08553692 ftn, Boole, Chebychev evals approx error approx error 3, 19.4008539142, 0.3153169910, 19.5061466023, 0.4206096791 5, 19.0910191534, 0.0054822302, 19.0910191534, 0.0054822302 9, 19.0856414320, 0.0001045088, 19.0855374134, 0.0000004902 17, 19.0855386464, 0.0000017232, 19.0855369232, 0.0000000000 33, 19.0855369505, 0.0000000273, 19.0855369232, 0.0000000000 65, 19.0855369236, 0.0000000004, 19.0855369232, 0.0000000000 129, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 257, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 513, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 ftn, Clenshaw-Curtis, Gauss-Legendre evals approx error approx error 3, 19.5061466023, 0.4206096791, 19.0803304585, 0.0052064647 5, 19.0834145766, 0.0021223465, 19.0855365951, 0.0000003281 9, 19.0855369150, 0.0000000082, 19.0855369232, 0.0000000000 17, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 33, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 65, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 129, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 257, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 513, 19.0855369232, 0.0000000000, 19.0855369232, 0.0000000000 ftn, Simps, Trapz evals approx error approx error 3, 19.5061466023, 0.4206096791, 22.5366862979, 3.4511493747 5, 19.1169646189, 0.0314276957, 19.9718950387, 0.8863581155 9, 19.0875991312, 0.0020622080, 19.3086731081, 0.2231361849 17, 19.0856674267, 0.0001305035, 19.1414188470, 0.0558819239 33, 19.0855451052, 0.0000081821, 19.0995135407, 0.0139766175 65, 19.0855374350, 0.0000005118, 19.0890314614, 0.0034945382 129, 19.0855369552, 0.0000000320, 19.0864105817, 0.0008736585 257, 19.0855369252, 0.0000000020, 19.0857553393, 0.0002184161 513, 19.0855369233, 0.0000000001, 19.0855915273, 0.0000546041 """ true_val, _tol = intg.quad(f, a, b) print('true value = %12.8f' % (true_val,)) neval = zeros(kmax, dtype=int) vals_dic = {} err_dic = {} # try various approximations methods = [trapz, simps, boole, ] for k in range(kmax): n = 2 ** (k + 1) + 1 neval[k] = n x = np.linspace(a, b, n) y = f(x) for method in methods: name = method.__name__.title() q = method(y, x) vals_dic.setdefault(name, []).append(q) err_dic.setdefault(name, []).append(abs(q - true_val)) name = 'Clenshaw-Curtis' q = clencurt(f, a, b, (n - 1) // 2)[0] vals_dic.setdefault(name, []).append(q[0]) err_dic.setdefault(name, []).append(abs(q[0] - true_val)) name = 'Chebychev' c_k = np.polynomial.chebyshev.chebfit(x, y, deg=min(n-1, 36)) c_ki = np.polynomial.chebyshev.chebint(c_k) q = np.polynomial.chebyshev.chebval(x[-1], c_ki) vals_dic.setdefault(name, []).append(q) err_dic.setdefault(name, []).append(abs(q - true_val)) # c_k = chebfit(f,n,a,b) # q = chebval(b,chebint(c_k,a,b),a,b) # qc2[k] = q; ec2[k] = abs(q - true) name = 'Gauss-Legendre' # quadrature q = intg.fixed_quad(f, a, b, n=n)[0] # [x, w]=qrule(n,1) # x = (b-a)/2*x + (a+b)/2 % Transform base points X. # w = (b-a)/2*w % Adjust weigths. # q = sum(feval(f,x)*w) vals_dic.setdefault(name, []).append(q) err_dic.setdefault(name, []).append(abs(q - true_val)) _display(neval, vals_dic, err_dic, plot_error) def main(): # val, err = clencurt(np.exp, 0, 2) # valt = np.exp(2) - np.exp(0) # [Q, err] = quadgr(lambda x: x ** 2, 1, 4, 1e-9) # [Q, err] = quadgr(humps, 1, 4, 1e-9) # # [x, w] = h_roots(11, 'newton') # sum(w) # [x2, w2] = la_roots(11, 1, 't') # # from scitools import numpyutils as npu #@UnresolvedImport # fun = npu.wrap2callable('x**2') # p0 = fun(0) # A = [0, 1, 1]; B = [2, 4, 3] # area, err = gaussq(fun, A, B) # # fun = npu.wrap2callable('x**2') # [val1, err1] = gaussq(fun, A, B) # # # Integration of x^2*exp(-x) from zero to infinity: # fun2 = npu.wrap2callable('1') # [val2, err2] = gaussq(fun2, 0, np.inf, wfun=3, alpha=2) # [val2, err2] = gaussq(lambda x: x ** 2, 0, np.inf, wfun=3, alpha=0) # # Integrate humps from 0 to 2 and from 1 to 4 # [val3, err3] = gaussq(humps, A, B) # # [x, w] = p_roots(11, 'newton', 1, 3) # y = np.sum(x ** 2 * w) x = np.linspace(0, np.pi / 2) _q0 = np.trapz(humps(x), x) [q, err] = romberg(humps, 0, np.pi / 2, 1e-4) print(q, err) if __name__ == '__main__': from wafo.testing import test_docstrings test_docstrings(__file__) # qdemo(np.exp, 0, 3, plot_error=True) # plt.show('hold') # main()