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diff --git a/.venv/lib/python3.12/site-packages/numpy/polynomial/tests/test_legendre.py b/.venv/lib/python3.12/site-packages/numpy/polynomial/tests/test_legendre.py
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+"""Tests for legendre module.
+
+"""
+from functools import reduce
+
+import numpy as np
+import numpy.polynomial.legendre as leg
+from numpy.polynomial.polynomial import polyval
+from numpy.testing import (
+ assert_almost_equal, assert_raises, assert_equal, assert_,
+ )
+
+L0 = np.array([1])
+L1 = np.array([0, 1])
+L2 = np.array([-1, 0, 3])/2
+L3 = np.array([0, -3, 0, 5])/2
+L4 = np.array([3, 0, -30, 0, 35])/8
+L5 = np.array([0, 15, 0, -70, 0, 63])/8
+L6 = np.array([-5, 0, 105, 0, -315, 0, 231])/16
+L7 = np.array([0, -35, 0, 315, 0, -693, 0, 429])/16
+L8 = np.array([35, 0, -1260, 0, 6930, 0, -12012, 0, 6435])/128
+L9 = np.array([0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155])/128
+
+Llist = [L0, L1, L2, L3, L4, L5, L6, L7, L8, L9]
+
+
+def trim(x):
+ return leg.legtrim(x, tol=1e-6)
+
+
+class TestConstants:
+
+ def test_legdomain(self):
+ assert_equal(leg.legdomain, [-1, 1])
+
+ def test_legzero(self):
+ assert_equal(leg.legzero, [0])
+
+ def test_legone(self):
+ assert_equal(leg.legone, [1])
+
+ def test_legx(self):
+ assert_equal(leg.legx, [0, 1])
+
+
+class TestArithmetic:
+ x = np.linspace(-1, 1, 100)
+
+ def test_legadd(self):
+ for i in range(5):
+ for j in range(5):
+ msg = f"At i={i}, j={j}"
+ tgt = np.zeros(max(i, j) + 1)
+ tgt[i] += 1
+ tgt[j] += 1
+ res = leg.legadd([0]*i + [1], [0]*j + [1])
+ assert_equal(trim(res), trim(tgt), err_msg=msg)
+
+ def test_legsub(self):
+ for i in range(5):
+ for j in range(5):
+ msg = f"At i={i}, j={j}"
+ tgt = np.zeros(max(i, j) + 1)
+ tgt[i] += 1
+ tgt[j] -= 1
+ res = leg.legsub([0]*i + [1], [0]*j + [1])
+ assert_equal(trim(res), trim(tgt), err_msg=msg)
+
+ def test_legmulx(self):
+ assert_equal(leg.legmulx([0]), [0])
+ assert_equal(leg.legmulx([1]), [0, 1])
+ for i in range(1, 5):
+ tmp = 2*i + 1
+ ser = [0]*i + [1]
+ tgt = [0]*(i - 1) + [i/tmp, 0, (i + 1)/tmp]
+ assert_equal(leg.legmulx(ser), tgt)
+
+ def test_legmul(self):
+ # check values of result
+ for i in range(5):
+ pol1 = [0]*i + [1]
+ val1 = leg.legval(self.x, pol1)
+ for j in range(5):
+ msg = f"At i={i}, j={j}"
+ pol2 = [0]*j + [1]
+ val2 = leg.legval(self.x, pol2)
+ pol3 = leg.legmul(pol1, pol2)
+ val3 = leg.legval(self.x, pol3)
+ assert_(len(pol3) == i + j + 1, msg)
+ assert_almost_equal(val3, val1*val2, err_msg=msg)
+
+ def test_legdiv(self):
+ for i in range(5):
+ for j in range(5):
+ msg = f"At i={i}, j={j}"
+ ci = [0]*i + [1]
+ cj = [0]*j + [1]
+ tgt = leg.legadd(ci, cj)
+ quo, rem = leg.legdiv(tgt, ci)
+ res = leg.legadd(leg.legmul(quo, ci), rem)
+ assert_equal(trim(res), trim(tgt), err_msg=msg)
+
+ def test_legpow(self):
+ for i in range(5):
+ for j in range(5):
+ msg = f"At i={i}, j={j}"
+ c = np.arange(i + 1)
+ tgt = reduce(leg.legmul, [c]*j, np.array([1]))
+ res = leg.legpow(c, j)
+ assert_equal(trim(res), trim(tgt), err_msg=msg)
+
+
+class TestEvaluation:
+ # coefficients of 1 + 2*x + 3*x**2
+ c1d = np.array([2., 2., 2.])
+ c2d = np.einsum('i,j->ij', c1d, c1d)
+ c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d)
+
+ # some random values in [-1, 1)
+ x = np.random.random((3, 5))*2 - 1
+ y = polyval(x, [1., 2., 3.])
+
+ def test_legval(self):
+ #check empty input
+ assert_equal(leg.legval([], [1]).size, 0)
+
+ #check normal input)
+ x = np.linspace(-1, 1)
+ y = [polyval(x, c) for c in Llist]
+ for i in range(10):
+ msg = f"At i={i}"
+ tgt = y[i]
+ res = leg.legval(x, [0]*i + [1])
+ assert_almost_equal(res, tgt, err_msg=msg)
+
+ #check that shape is preserved
+ for i in range(3):
+ dims = [2]*i
+ x = np.zeros(dims)
+ assert_equal(leg.legval(x, [1]).shape, dims)
+ assert_equal(leg.legval(x, [1, 0]).shape, dims)
+ assert_equal(leg.legval(x, [1, 0, 0]).shape, dims)
+
+ def test_legval2d(self):
+ x1, x2, x3 = self.x
+ y1, y2, y3 = self.y
+
+ #test exceptions
+ assert_raises(ValueError, leg.legval2d, x1, x2[:2], self.c2d)
+
+ #test values
+ tgt = y1*y2
+ res = leg.legval2d(x1, x2, self.c2d)
+ assert_almost_equal(res, tgt)
+
+ #test shape
+ z = np.ones((2, 3))
+ res = leg.legval2d(z, z, self.c2d)
+ assert_(res.shape == (2, 3))
+
+ def test_legval3d(self):
+ x1, x2, x3 = self.x
+ y1, y2, y3 = self.y
+
+ #test exceptions
+ assert_raises(ValueError, leg.legval3d, x1, x2, x3[:2], self.c3d)
+
+ #test values
+ tgt = y1*y2*y3
+ res = leg.legval3d(x1, x2, x3, self.c3d)
+ assert_almost_equal(res, tgt)
+
+ #test shape
+ z = np.ones((2, 3))
+ res = leg.legval3d(z, z, z, self.c3d)
+ assert_(res.shape == (2, 3))
+
+ def test_leggrid2d(self):
+ x1, x2, x3 = self.x
+ y1, y2, y3 = self.y
+
+ #test values
+ tgt = np.einsum('i,j->ij', y1, y2)
+ res = leg.leggrid2d(x1, x2, self.c2d)
+ assert_almost_equal(res, tgt)
+
+ #test shape
+ z = np.ones((2, 3))
+ res = leg.leggrid2d(z, z, self.c2d)
+ assert_(res.shape == (2, 3)*2)
+
+ def test_leggrid3d(self):
+ x1, x2, x3 = self.x
+ y1, y2, y3 = self.y
+
+ #test values
+ tgt = np.einsum('i,j,k->ijk', y1, y2, y3)
+ res = leg.leggrid3d(x1, x2, x3, self.c3d)
+ assert_almost_equal(res, tgt)
+
+ #test shape
+ z = np.ones((2, 3))
+ res = leg.leggrid3d(z, z, z, self.c3d)
+ assert_(res.shape == (2, 3)*3)
+
+
+class TestIntegral:
+
+ def test_legint(self):
+ # check exceptions
+ assert_raises(TypeError, leg.legint, [0], .5)
+ assert_raises(ValueError, leg.legint, [0], -1)
+ assert_raises(ValueError, leg.legint, [0], 1, [0, 0])
+ assert_raises(ValueError, leg.legint, [0], lbnd=[0])
+ assert_raises(ValueError, leg.legint, [0], scl=[0])
+ assert_raises(TypeError, leg.legint, [0], axis=.5)
+
+ # test integration of zero polynomial
+ for i in range(2, 5):
+ k = [0]*(i - 2) + [1]
+ res = leg.legint([0], m=i, k=k)
+ assert_almost_equal(res, [0, 1])
+
+ # check single integration with integration constant
+ for i in range(5):
+ scl = i + 1
+ pol = [0]*i + [1]
+ tgt = [i] + [0]*i + [1/scl]
+ legpol = leg.poly2leg(pol)
+ legint = leg.legint(legpol, m=1, k=[i])
+ res = leg.leg2poly(legint)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ # check single integration with integration constant and lbnd
+ for i in range(5):
+ scl = i + 1
+ pol = [0]*i + [1]
+ legpol = leg.poly2leg(pol)
+ legint = leg.legint(legpol, m=1, k=[i], lbnd=-1)
+ assert_almost_equal(leg.legval(-1, legint), i)
+
+ # check single integration with integration constant and scaling
+ for i in range(5):
+ scl = i + 1
+ pol = [0]*i + [1]
+ tgt = [i] + [0]*i + [2/scl]
+ legpol = leg.poly2leg(pol)
+ legint = leg.legint(legpol, m=1, k=[i], scl=2)
+ res = leg.leg2poly(legint)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ # check multiple integrations with default k
+ for i in range(5):
+ for j in range(2, 5):
+ pol = [0]*i + [1]
+ tgt = pol[:]
+ for k in range(j):
+ tgt = leg.legint(tgt, m=1)
+ res = leg.legint(pol, m=j)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ # check multiple integrations with defined k
+ for i in range(5):
+ for j in range(2, 5):
+ pol = [0]*i + [1]
+ tgt = pol[:]
+ for k in range(j):
+ tgt = leg.legint(tgt, m=1, k=[k])
+ res = leg.legint(pol, m=j, k=list(range(j)))
+ assert_almost_equal(trim(res), trim(tgt))
+
+ # check multiple integrations with lbnd
+ for i in range(5):
+ for j in range(2, 5):
+ pol = [0]*i + [1]
+ tgt = pol[:]
+ for k in range(j):
+ tgt = leg.legint(tgt, m=1, k=[k], lbnd=-1)
+ res = leg.legint(pol, m=j, k=list(range(j)), lbnd=-1)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ # check multiple integrations with scaling
+ for i in range(5):
+ for j in range(2, 5):
+ pol = [0]*i + [1]
+ tgt = pol[:]
+ for k in range(j):
+ tgt = leg.legint(tgt, m=1, k=[k], scl=2)
+ res = leg.legint(pol, m=j, k=list(range(j)), scl=2)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ def test_legint_axis(self):
+ # check that axis keyword works
+ c2d = np.random.random((3, 4))
+
+ tgt = np.vstack([leg.legint(c) for c in c2d.T]).T
+ res = leg.legint(c2d, axis=0)
+ assert_almost_equal(res, tgt)
+
+ tgt = np.vstack([leg.legint(c) for c in c2d])
+ res = leg.legint(c2d, axis=1)
+ assert_almost_equal(res, tgt)
+
+ tgt = np.vstack([leg.legint(c, k=3) for c in c2d])
+ res = leg.legint(c2d, k=3, axis=1)
+ assert_almost_equal(res, tgt)
+
+ def test_legint_zerointord(self):
+ assert_equal(leg.legint((1, 2, 3), 0), (1, 2, 3))
+
+
+class TestDerivative:
+
+ def test_legder(self):
+ # check exceptions
+ assert_raises(TypeError, leg.legder, [0], .5)
+ assert_raises(ValueError, leg.legder, [0], -1)
+
+ # check that zeroth derivative does nothing
+ for i in range(5):
+ tgt = [0]*i + [1]
+ res = leg.legder(tgt, m=0)
+ assert_equal(trim(res), trim(tgt))
+
+ # check that derivation is the inverse of integration
+ for i in range(5):
+ for j in range(2, 5):
+ tgt = [0]*i + [1]
+ res = leg.legder(leg.legint(tgt, m=j), m=j)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ # check derivation with scaling
+ for i in range(5):
+ for j in range(2, 5):
+ tgt = [0]*i + [1]
+ res = leg.legder(leg.legint(tgt, m=j, scl=2), m=j, scl=.5)
+ assert_almost_equal(trim(res), trim(tgt))
+
+ def test_legder_axis(self):
+ # check that axis keyword works
+ c2d = np.random.random((3, 4))
+
+ tgt = np.vstack([leg.legder(c) for c in c2d.T]).T
+ res = leg.legder(c2d, axis=0)
+ assert_almost_equal(res, tgt)
+
+ tgt = np.vstack([leg.legder(c) for c in c2d])
+ res = leg.legder(c2d, axis=1)
+ assert_almost_equal(res, tgt)
+
+ def test_legder_orderhigherthancoeff(self):
+ c = (1, 2, 3, 4)
+ assert_equal(leg.legder(c, 4), [0])
+
+class TestVander:
+ # some random values in [-1, 1)
+ x = np.random.random((3, 5))*2 - 1
+
+ def test_legvander(self):
+ # check for 1d x
+ x = np.arange(3)
+ v = leg.legvander(x, 3)
+ assert_(v.shape == (3, 4))
+ for i in range(4):
+ coef = [0]*i + [1]
+ assert_almost_equal(v[..., i], leg.legval(x, coef))
+
+ # check for 2d x
+ x = np.array([[1, 2], [3, 4], [5, 6]])
+ v = leg.legvander(x, 3)
+ assert_(v.shape == (3, 2, 4))
+ for i in range(4):
+ coef = [0]*i + [1]
+ assert_almost_equal(v[..., i], leg.legval(x, coef))
+
+ def test_legvander2d(self):
+ # also tests polyval2d for non-square coefficient array
+ x1, x2, x3 = self.x
+ c = np.random.random((2, 3))
+ van = leg.legvander2d(x1, x2, [1, 2])
+ tgt = leg.legval2d(x1, x2, c)
+ res = np.dot(van, c.flat)
+ assert_almost_equal(res, tgt)
+
+ # check shape
+ van = leg.legvander2d([x1], [x2], [1, 2])
+ assert_(van.shape == (1, 5, 6))
+
+ def test_legvander3d(self):
+ # also tests polyval3d for non-square coefficient array
+ x1, x2, x3 = self.x
+ c = np.random.random((2, 3, 4))
+ van = leg.legvander3d(x1, x2, x3, [1, 2, 3])
+ tgt = leg.legval3d(x1, x2, x3, c)
+ res = np.dot(van, c.flat)
+ assert_almost_equal(res, tgt)
+
+ # check shape
+ van = leg.legvander3d([x1], [x2], [x3], [1, 2, 3])
+ assert_(van.shape == (1, 5, 24))
+
+ def test_legvander_negdeg(self):
+ assert_raises(ValueError, leg.legvander, (1, 2, 3), -1)
+
+
+class TestFitting:
+
+ def test_legfit(self):
+ def f(x):
+ return x*(x - 1)*(x - 2)
+
+ def f2(x):
+ return x**4 + x**2 + 1
+
+ # Test exceptions
+ assert_raises(ValueError, leg.legfit, [1], [1], -1)
+ assert_raises(TypeError, leg.legfit, [[1]], [1], 0)
+ assert_raises(TypeError, leg.legfit, [], [1], 0)
+ assert_raises(TypeError, leg.legfit, [1], [[[1]]], 0)
+ assert_raises(TypeError, leg.legfit, [1, 2], [1], 0)
+ assert_raises(TypeError, leg.legfit, [1], [1, 2], 0)
+ assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, leg.legfit, [1], [1], [-1,])
+ assert_raises(ValueError, leg.legfit, [1], [1], [2, -1, 6])
+ assert_raises(TypeError, leg.legfit, [1], [1], [])
+
+ # Test fit
+ x = np.linspace(0, 2)
+ y = f(x)
+ #
+ coef3 = leg.legfit(x, y, 3)
+ assert_equal(len(coef3), 4)
+ assert_almost_equal(leg.legval(x, coef3), y)
+ coef3 = leg.legfit(x, y, [0, 1, 2, 3])
+ assert_equal(len(coef3), 4)
+ assert_almost_equal(leg.legval(x, coef3), y)
+ #
+ coef4 = leg.legfit(x, y, 4)
+ assert_equal(len(coef4), 5)
+ assert_almost_equal(leg.legval(x, coef4), y)
+ coef4 = leg.legfit(x, y, [0, 1, 2, 3, 4])
+ assert_equal(len(coef4), 5)
+ assert_almost_equal(leg.legval(x, coef4), y)
+ # check things still work if deg is not in strict increasing
+ coef4 = leg.legfit(x, y, [2, 3, 4, 1, 0])
+ assert_equal(len(coef4), 5)
+ assert_almost_equal(leg.legval(x, coef4), y)
+ #
+ coef2d = leg.legfit(x, np.array([y, y]).T, 3)
+ assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
+ coef2d = leg.legfit(x, np.array([y, y]).T, [0, 1, 2, 3])
+ assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
+ # test weighting
+ w = np.zeros_like(x)
+ yw = y.copy()
+ w[1::2] = 1
+ y[0::2] = 0
+ wcoef3 = leg.legfit(x, yw, 3, w=w)
+ assert_almost_equal(wcoef3, coef3)
+ wcoef3 = leg.legfit(x, yw, [0, 1, 2, 3], w=w)
+ assert_almost_equal(wcoef3, coef3)
+ #
+ wcoef2d = leg.legfit(x, np.array([yw, yw]).T, 3, w=w)
+ assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
+ wcoef2d = leg.legfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w)
+ assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
+ # test scaling with complex values x points whose square
+ # is zero when summed.
+ x = [1, 1j, -1, -1j]
+ assert_almost_equal(leg.legfit(x, x, 1), [0, 1])
+ assert_almost_equal(leg.legfit(x, x, [0, 1]), [0, 1])
+ # test fitting only even Legendre polynomials
+ x = np.linspace(-1, 1)
+ y = f2(x)
+ coef1 = leg.legfit(x, y, 4)
+ assert_almost_equal(leg.legval(x, coef1), y)
+ coef2 = leg.legfit(x, y, [0, 2, 4])
+ assert_almost_equal(leg.legval(x, coef2), y)
+ assert_almost_equal(coef1, coef2)
+
+
+class TestCompanion:
+
+ def test_raises(self):
+ assert_raises(ValueError, leg.legcompanion, [])
+ assert_raises(ValueError, leg.legcompanion, [1])
+
+ def test_dimensions(self):
+ for i in range(1, 5):
+ coef = [0]*i + [1]
+ assert_(leg.legcompanion(coef).shape == (i, i))
+
+ def test_linear_root(self):
+ assert_(leg.legcompanion([1, 2])[0, 0] == -.5)
+
+
+class TestGauss:
+
+ def test_100(self):
+ x, w = leg.leggauss(100)
+
+ # test orthogonality. Note that the results need to be normalized,
+ # otherwise the huge values that can arise from fast growing
+ # functions like Laguerre can be very confusing.
+ v = leg.legvander(x, 99)
+ vv = np.dot(v.T * w, v)
+ vd = 1/np.sqrt(vv.diagonal())
+ vv = vd[:, None] * vv * vd
+ assert_almost_equal(vv, np.eye(100))
+
+ # check that the integral of 1 is correct
+ tgt = 2.0
+ assert_almost_equal(w.sum(), tgt)
+
+
+class TestMisc:
+
+ def test_legfromroots(self):
+ res = leg.legfromroots([])
+ assert_almost_equal(trim(res), [1])
+ for i in range(1, 5):
+ roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2])
+ pol = leg.legfromroots(roots)
+ res = leg.legval(roots, pol)
+ tgt = 0
+ assert_(len(pol) == i + 1)
+ assert_almost_equal(leg.leg2poly(pol)[-1], 1)
+ assert_almost_equal(res, tgt)
+
+ def test_legroots(self):
+ assert_almost_equal(leg.legroots([1]), [])
+ assert_almost_equal(leg.legroots([1, 2]), [-.5])
+ for i in range(2, 5):
+ tgt = np.linspace(-1, 1, i)
+ res = leg.legroots(leg.legfromroots(tgt))
+ assert_almost_equal(trim(res), trim(tgt))
+
+ def test_legtrim(self):
+ coef = [2, -1, 1, 0]
+
+ # Test exceptions
+ assert_raises(ValueError, leg.legtrim, coef, -1)
+
+ # Test results
+ assert_equal(leg.legtrim(coef), coef[:-1])
+ assert_equal(leg.legtrim(coef, 1), coef[:-3])
+ assert_equal(leg.legtrim(coef, 2), [0])
+
+ def test_legline(self):
+ assert_equal(leg.legline(3, 4), [3, 4])
+
+ def test_legline_zeroscl(self):
+ assert_equal(leg.legline(3, 0), [3])
+
+ def test_leg2poly(self):
+ for i in range(10):
+ assert_almost_equal(leg.leg2poly([0]*i + [1]), Llist[i])
+
+ def test_poly2leg(self):
+ for i in range(10):
+ assert_almost_equal(leg.poly2leg(Llist[i]), [0]*i + [1])
+
+ def test_weight(self):
+ x = np.linspace(-1, 1, 11)
+ tgt = 1.
+ res = leg.legweight(x)
+ assert_almost_equal(res, tgt)
generated by cgit v1.2.3 (git 2.47.1) at 2025-07-27 05:49:45 +0000