from __future__ import division, print_function, absolute_import import sys import math import numpy as np from numpy import sqrt, cos, sin, arctan, exp, log, pi, Inf from numpy.testing import (assert_, TestCase, run_module_suite, dec, assert_allclose, assert_array_less, assert_almost_equal) from scipy.integrate import quad, dblquad, tplquad, nquad from scipy.lib.six import xrange try: import ctypes import ctypes.util _ctypes_missing = False except ImportError: _ctypes_missing = True def assert_quad(value_and_err, tabled_value, errTol=1.5e-8): value, err = value_and_err assert_allclose(value, tabled_value, atol=err, rtol=0) if errTol is not None: assert_array_less(err, errTol) class TestCtypesQuad(TestCase): @dec.skipif(_ctypes_missing, msg="Ctypes library could not be found") def setUp(self): if sys.platform == 'win32': file = ctypes.util.find_msvcrt() elif sys.platform == 'darwin': file = 'libm.dylib' else: file = 'libm.so' self.lib = ctypes.CDLL(file) restype = ctypes.c_double argtypes = (ctypes.c_double,) for name in ['sin', 'cos', 'tan']: func = getattr(self.lib, name) func.restype = restype func.argtypes = argtypes @dec.skipif(_ctypes_missing, msg="Ctypes library could not be found") def test_typical(self): assert_quad(quad(self.lib.sin,0,5),quad(math.sin,0,5)[0]) assert_quad(quad(self.lib.cos,0,5),quad(math.cos,0,5)[0]) assert_quad(quad(self.lib.tan,0,1),quad(math.tan,0,1)[0]) #@dec.skipif(_ctypes_missing, msg="Ctypes library could not be found") # This doesn't seem to always work. Need a better way to figure out # whether the fast path is called. @dec.knownfailureif(True, msg="Unreliable test, see ticket 1684.") def test_improvement(self): import time start = time.time() for i in xrange(100): quad(self.lib.sin, 0, 100) fast = time.time() - start start = time.time() for i in xrange(100): quad(math.sin, 0, 100) slow = time.time() - start assert_(fast < 0.5*slow, (fast, slow)) class TestQuad(TestCase): def test_typical(self): # 1) Typical function with two extra arguments: def myfunc(x,n,z): # Bessel function integrand return cos(n*x-z*sin(x))/pi assert_quad(quad(myfunc,0,pi,(2,1.8)), 0.30614353532540296487) def test_indefinite(self): # 2) Infinite integration limits --- Euler's constant def myfunc(x): # Euler's constant integrand return -exp(-x)*log(x) assert_quad(quad(myfunc,0,Inf), 0.577215664901532860606512) def test_singular(self): # 3) Singular points in region of integration. def myfunc(x): if x > 0 and x < 2.5: return sin(x) elif x >= 2.5 and x <= 5.0: return exp(-x) else: return 0.0 assert_quad(quad(myfunc,0,10,points=[2.5,5.0]), 1 - cos(2.5) + exp(-2.5) - exp(-5.0)) def test_sine_weighted_finite(self): # 4) Sine weighted integral (finite limits) def myfunc(x,a): return exp(a*(x-1)) ome = 2.0**3.4 assert_quad(quad(myfunc,0,1,args=20,weight='sin',wvar=ome), (20*sin(ome)-ome*cos(ome)+ome*exp(-20))/(20**2 + ome**2)) def test_sine_weighted_infinite(self): # 5) Sine weighted integral (infinite limits) def myfunc(x,a): return exp(-x*a) a = 4.0 ome = 3.0 assert_quad(quad(myfunc,0,Inf,args=a,weight='sin',wvar=ome), ome/(a**2 + ome**2)) def test_cosine_weighted_infinite(self): # 6) Cosine weighted integral (negative infinite limits) def myfunc(x,a): return exp(x*a) a = 2.5 ome = 2.3 assert_quad(quad(myfunc,-Inf,0,args=a,weight='cos',wvar=ome), a/(a**2 + ome**2)) def test_algebraic_log_weight(self): # 6) Algebraic-logarithmic weight. def myfunc(x,a): return 1/(1+x+2**(-a)) a = 1.5 assert_quad(quad(myfunc,-1,1,args=a,weight='alg',wvar=(-0.5,-0.5)), pi/sqrt((1+2**(-a))**2 - 1)) def test_cauchypv_weight(self): # 7) Cauchy prinicpal value weighting w(x) = 1/(x-c) def myfunc(x,a): return 2.0**(-a)/((x-1)**2+4.0**(-a)) a = 0.4 tabledValue = (2.0**(-0.4)*log(1.5)-2.0**(-1.4)*log((4.0**(-a)+16)/(4.0**(-a)+1)) - arctan(2.0**(a+2)) - arctan(2.0**a))/(4.0**(-a) + 1) assert_quad(quad(myfunc,0,5,args=0.4,weight='cauchy',wvar=2.0), tabledValue, errTol=1.9e-8) def test_double_integral(self): # 8) Double Integral test def simpfunc(y,x): # Note order of arguments. return x+y a, b = 1.0, 2.0 assert_quad(dblquad(simpfunc,a,b,lambda x: x, lambda x: 2*x), 5/6.0 * (b**3.0-a**3.0)) def test_triple_integral(self): # 9) Triple Integral test def simpfunc(z,y,x): # Note order of arguments. return x+y+z a, b = 1.0, 2.0 assert_quad(tplquad(simpfunc,a,b, lambda x: x, lambda x: 2*x, lambda x,y: x-y, lambda x,y: x+y), 8/3.0 * (b**4.0 - a**4.0)) class TestNQuad(TestCase): def test_fixed_limits(self): def func1(x0, x1, x2, x3): return x0**2 + x1*x2 - x3**3 + np.sin(x0) + (1 if (x0 - 0.2*x3 - 0.5 - 0.25*x1 > 0) else 0) def opts_basic(*args): return {'points': [0.2*args[2] + 0.5 + 0.25*args[0]]} res = nquad(func1, [[0,1], [-1,1], [.13,.8], [-.15,1]], opts=[opts_basic, {}, {}, {}]) assert_quad(res, 1.5267454070738635) def test_variable_limits(self): scale = .1 def func2(x0, x1, x2, x3, t0, t1): return x0*x1*x3**2 + np.sin(x2) + 1 + (1 if x0 + t1*x1 - t0 > 0 else 0) def lim0(x1, x2, x3, t0, t1): return [scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) - 1, scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) + 1] def lim1(x2, x3, t0, t1): return [scale * (t0*x2 + t1*x3) - 1, scale * (t0*x2 + t1*x3) + 1] def lim2(x3, t0, t1): return [scale * (x3 + t0**2*t1**3) - 1, scale * (x3 + t0**2*t1**3) + 1] def lim3(t0, t1): return [scale * (t0 + t1) - 1, scale * (t0 + t1) + 1] def opts0(x1, x2, x3, t0, t1): return {'points':[t0 - t1*x1]} def opts1(x2, x3, t0, t1): return {} def opts2(x3, t0, t1): return {} def opts3(t0, t1): return {} res = nquad(func2, [lim0, lim1, lim2, lim3], args=(0,0), opts=[opts0, opts1, opts2, opts3]) assert_quad(res, 25.066666666666663) def test_square_separate_ranges_and_opts(self): def f(y, x): return 1.0 assert_quad(nquad(f, [[-1, 1], [-1, 1]], opts=[{}, {}]), 4.0) def test_square_aliased_ranges_and_opts(self): def f(y, x): return 1.0 r = [-1, 1] opt = {} assert_quad(nquad(f, [r, r], opts=[opt, opt]), 4.0) def test_square_separate_fn_ranges_and_opts(self): def f(y, x): return 1.0 def fn_range0(*args): return (-1, 1) def fn_range1(*args): return (-1, 1) def fn_opt0(*args): return {} def fn_opt1(*args): return {} ranges = [fn_range0, fn_range1] opts = [fn_opt0, fn_opt1] assert_quad(nquad(f, ranges, opts=opts), 4.0) def test_square_aliased_fn_ranges_and_opts(self): def f(y, x): return 1.0 def fn_range(*args): return (-1, 1) def fn_opt(*args): return {} ranges = [fn_range, fn_range] opts = [fn_opt, fn_opt] assert_quad(nquad(f, ranges, opts=opts), 4.0) def test_matching_quad(self): def func(x): return x**2 + 1 res, reserr = quad(func, 0, 4) res2, reserr2 = nquad(func, ranges=[[0, 4]]) assert_almost_equal(res, res2) assert_almost_equal(reserr, reserr2) def test_matching_dblquad(self): def func2d(x0, x1): return x0**2 + x1**3 - x0 * x1 + 1 res, reserr = dblquad(func2d, -2, 2, lambda x:-3, lambda x:3) res2, reserr2 = nquad(func2d, [[-3, 3], (-2, 2)]) assert_almost_equal(res, res2) assert_almost_equal(reserr, reserr2) def test_matching_tplquad(self): def func3d(x0, x1, x2, c0, c1): return x0**2 + c0 * x1**3 - x0 * x1 + 1 + c1 * np.sin(x2) res = tplquad(func3d, -1, 2, lambda x: -2, lambda x: 2, lambda x, y: -np.pi, lambda x, y: np.pi, args=(2, 3)) res2 = nquad(func3d, [[-np.pi, np.pi], [-2, 2], (-1, 2)], args=(2, 3)) assert_almost_equal(res, res2) if __name__ == "__main__": run_module_suite()