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| """Tests for polynomial module. | |
| """ | |
| from functools import reduce | |
| import numpy as np | |
| import numpy.polynomial.polynomial as poly | |
| from numpy.testing import ( | |
| assert_almost_equal, assert_raises, assert_equal, assert_, | |
| assert_warns, assert_array_equal, assert_raises_regex) | |
| def trim(x): | |
| return poly.polytrim(x, tol=1e-6) | |
| T0 = [1] | |
| T1 = [0, 1] | |
| T2 = [-1, 0, 2] | |
| T3 = [0, -3, 0, 4] | |
| T4 = [1, 0, -8, 0, 8] | |
| T5 = [0, 5, 0, -20, 0, 16] | |
| T6 = [-1, 0, 18, 0, -48, 0, 32] | |
| T7 = [0, -7, 0, 56, 0, -112, 0, 64] | |
| T8 = [1, 0, -32, 0, 160, 0, -256, 0, 128] | |
| T9 = [0, 9, 0, -120, 0, 432, 0, -576, 0, 256] | |
| Tlist = [T0, T1, T2, T3, T4, T5, T6, T7, T8, T9] | |
| class TestConstants: | |
| def test_polydomain(self): | |
| assert_equal(poly.polydomain, [-1, 1]) | |
| def test_polyzero(self): | |
| assert_equal(poly.polyzero, [0]) | |
| def test_polyone(self): | |
| assert_equal(poly.polyone, [1]) | |
| def test_polyx(self): | |
| assert_equal(poly.polyx, [0, 1]) | |
| class TestArithmetic: | |
| def test_polyadd(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 = poly.polyadd([0]*i + [1], [0]*j + [1]) | |
| assert_equal(trim(res), trim(tgt), err_msg=msg) | |
| def test_polysub(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 = poly.polysub([0]*i + [1], [0]*j + [1]) | |
| assert_equal(trim(res), trim(tgt), err_msg=msg) | |
| def test_polymulx(self): | |
| assert_equal(poly.polymulx([0]), [0]) | |
| assert_equal(poly.polymulx([1]), [0, 1]) | |
| for i in range(1, 5): | |
| ser = [0]*i + [1] | |
| tgt = [0]*(i + 1) + [1] | |
| assert_equal(poly.polymulx(ser), tgt) | |
| def test_polymul(self): | |
| for i in range(5): | |
| for j in range(5): | |
| msg = f"At i={i}, j={j}" | |
| tgt = np.zeros(i + j + 1) | |
| tgt[i + j] += 1 | |
| res = poly.polymul([0]*i + [1], [0]*j + [1]) | |
| assert_equal(trim(res), trim(tgt), err_msg=msg) | |
| def test_polydiv(self): | |
| # check zero division | |
| assert_raises(ZeroDivisionError, poly.polydiv, [1], [0]) | |
| # check scalar division | |
| quo, rem = poly.polydiv([2], [2]) | |
| assert_equal((quo, rem), (1, 0)) | |
| quo, rem = poly.polydiv([2, 2], [2]) | |
| assert_equal((quo, rem), ((1, 1), 0)) | |
| # check rest. | |
| for i in range(5): | |
| for j in range(5): | |
| msg = f"At i={i}, j={j}" | |
| ci = [0]*i + [1, 2] | |
| cj = [0]*j + [1, 2] | |
| tgt = poly.polyadd(ci, cj) | |
| quo, rem = poly.polydiv(tgt, ci) | |
| res = poly.polyadd(poly.polymul(quo, ci), rem) | |
| assert_equal(res, tgt, err_msg=msg) | |
| def test_polypow(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(poly.polymul, [c]*j, np.array([1])) | |
| res = poly.polypow(c, j) | |
| assert_equal(trim(res), trim(tgt), err_msg=msg) | |
| class TestEvaluation: | |
| # coefficients of 1 + 2*x + 3*x**2 | |
| c1d = np.array([1., 2., 3.]) | |
| 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 = poly.polyval(x, [1., 2., 3.]) | |
| def test_polyval(self): | |
| #check empty input | |
| assert_equal(poly.polyval([], [1]).size, 0) | |
| #check normal input) | |
| x = np.linspace(-1, 1) | |
| y = [x**i for i in range(5)] | |
| for i in range(5): | |
| tgt = y[i] | |
| res = poly.polyval(x, [0]*i + [1]) | |
| assert_almost_equal(res, tgt) | |
| tgt = x*(x**2 - 1) | |
| res = poly.polyval(x, [0, -1, 0, 1]) | |
| assert_almost_equal(res, tgt) | |
| #check that shape is preserved | |
| for i in range(3): | |
| dims = [2]*i | |
| x = np.zeros(dims) | |
| assert_equal(poly.polyval(x, [1]).shape, dims) | |
| assert_equal(poly.polyval(x, [1, 0]).shape, dims) | |
| assert_equal(poly.polyval(x, [1, 0, 0]).shape, dims) | |
| #check masked arrays are processed correctly | |
| mask = [False, True, False] | |
| mx = np.ma.array([1, 2, 3], mask=mask) | |
| res = np.polyval([7, 5, 3], mx) | |
| assert_array_equal(res.mask, mask) | |
| #check subtypes of ndarray are preserved | |
| class C(np.ndarray): | |
| pass | |
| cx = np.array([1, 2, 3]).view(C) | |
| assert_equal(type(np.polyval([2, 3, 4], cx)), C) | |
| def test_polyvalfromroots(self): | |
| # check exception for broadcasting x values over root array with | |
| # too few dimensions | |
| assert_raises(ValueError, poly.polyvalfromroots, | |
| [1], [1], tensor=False) | |
| # check empty input | |
| assert_equal(poly.polyvalfromroots([], [1]).size, 0) | |
| assert_(poly.polyvalfromroots([], [1]).shape == (0,)) | |
| # check empty input + multidimensional roots | |
| assert_equal(poly.polyvalfromroots([], [[1] * 5]).size, 0) | |
| assert_(poly.polyvalfromroots([], [[1] * 5]).shape == (5, 0)) | |
| # check scalar input | |
| assert_equal(poly.polyvalfromroots(1, 1), 0) | |
| assert_(poly.polyvalfromroots(1, np.ones((3, 3))).shape == (3,)) | |
| # check normal input) | |
| x = np.linspace(-1, 1) | |
| y = [x**i for i in range(5)] | |
| for i in range(1, 5): | |
| tgt = y[i] | |
| res = poly.polyvalfromroots(x, [0]*i) | |
| assert_almost_equal(res, tgt) | |
| tgt = x*(x - 1)*(x + 1) | |
| res = poly.polyvalfromroots(x, [-1, 0, 1]) | |
| assert_almost_equal(res, tgt) | |
| # check that shape is preserved | |
| for i in range(3): | |
| dims = [2]*i | |
| x = np.zeros(dims) | |
| assert_equal(poly.polyvalfromroots(x, [1]).shape, dims) | |
| assert_equal(poly.polyvalfromroots(x, [1, 0]).shape, dims) | |
| assert_equal(poly.polyvalfromroots(x, [1, 0, 0]).shape, dims) | |
| # check compatibility with factorization | |
| ptest = [15, 2, -16, -2, 1] | |
| r = poly.polyroots(ptest) | |
| x = np.linspace(-1, 1) | |
| assert_almost_equal(poly.polyval(x, ptest), | |
| poly.polyvalfromroots(x, r)) | |
| # check multidimensional arrays of roots and values | |
| # check tensor=False | |
| rshape = (3, 5) | |
| x = np.arange(-3, 2) | |
| r = np.random.randint(-5, 5, size=rshape) | |
| res = poly.polyvalfromroots(x, r, tensor=False) | |
| tgt = np.empty(r.shape[1:]) | |
| for ii in range(tgt.size): | |
| tgt[ii] = poly.polyvalfromroots(x[ii], r[:, ii]) | |
| assert_equal(res, tgt) | |
| # check tensor=True | |
| x = np.vstack([x, 2*x]) | |
| res = poly.polyvalfromroots(x, r, tensor=True) | |
| tgt = np.empty(r.shape[1:] + x.shape) | |
| for ii in range(r.shape[1]): | |
| for jj in range(x.shape[0]): | |
| tgt[ii, jj, :] = poly.polyvalfromroots(x[jj], r[:, ii]) | |
| assert_equal(res, tgt) | |
| def test_polyval2d(self): | |
| x1, x2, x3 = self.x | |
| y1, y2, y3 = self.y | |
| #test exceptions | |
| assert_raises_regex(ValueError, 'incompatible', | |
| poly.polyval2d, x1, x2[:2], self.c2d) | |
| #test values | |
| tgt = y1*y2 | |
| res = poly.polyval2d(x1, x2, self.c2d) | |
| assert_almost_equal(res, tgt) | |
| #test shape | |
| z = np.ones((2, 3)) | |
| res = poly.polyval2d(z, z, self.c2d) | |
| assert_(res.shape == (2, 3)) | |
| def test_polyval3d(self): | |
| x1, x2, x3 = self.x | |
| y1, y2, y3 = self.y | |
| #test exceptions | |
| assert_raises_regex(ValueError, 'incompatible', | |
| poly.polyval3d, x1, x2, x3[:2], self.c3d) | |
| #test values | |
| tgt = y1*y2*y3 | |
| res = poly.polyval3d(x1, x2, x3, self.c3d) | |
| assert_almost_equal(res, tgt) | |
| #test shape | |
| z = np.ones((2, 3)) | |
| res = poly.polyval3d(z, z, z, self.c3d) | |
| assert_(res.shape == (2, 3)) | |
| def test_polygrid2d(self): | |
| x1, x2, x3 = self.x | |
| y1, y2, y3 = self.y | |
| #test values | |
| tgt = np.einsum('i,j->ij', y1, y2) | |
| res = poly.polygrid2d(x1, x2, self.c2d) | |
| assert_almost_equal(res, tgt) | |
| #test shape | |
| z = np.ones((2, 3)) | |
| res = poly.polygrid2d(z, z, self.c2d) | |
| assert_(res.shape == (2, 3)*2) | |
| def test_polygrid3d(self): | |
| x1, x2, x3 = self.x | |
| y1, y2, y3 = self.y | |
| #test values | |
| tgt = np.einsum('i,j,k->ijk', y1, y2, y3) | |
| res = poly.polygrid3d(x1, x2, x3, self.c3d) | |
| assert_almost_equal(res, tgt) | |
| #test shape | |
| z = np.ones((2, 3)) | |
| res = poly.polygrid3d(z, z, z, self.c3d) | |
| assert_(res.shape == (2, 3)*3) | |
| class TestIntegral: | |
| def test_polyint(self): | |
| # check exceptions | |
| assert_raises(TypeError, poly.polyint, [0], .5) | |
| assert_raises(ValueError, poly.polyint, [0], -1) | |
| assert_raises(ValueError, poly.polyint, [0], 1, [0, 0]) | |
| assert_raises(ValueError, poly.polyint, [0], lbnd=[0]) | |
| assert_raises(ValueError, poly.polyint, [0], scl=[0]) | |
| assert_raises(TypeError, poly.polyint, [0], axis=.5) | |
| with assert_warns(DeprecationWarning): | |
| poly.polyint([1, 1], 1.) | |
| # test integration of zero polynomial | |
| for i in range(2, 5): | |
| k = [0]*(i - 2) + [1] | |
| res = poly.polyint([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] | |
| res = poly.polyint(pol, m=1, k=[i]) | |
| 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] | |
| res = poly.polyint(pol, m=1, k=[i], lbnd=-1) | |
| assert_almost_equal(poly.polyval(-1, res), 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] | |
| res = poly.polyint(pol, m=1, k=[i], scl=2) | |
| 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 = poly.polyint(tgt, m=1) | |
| res = poly.polyint(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 = poly.polyint(tgt, m=1, k=[k]) | |
| res = poly.polyint(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 = poly.polyint(tgt, m=1, k=[k], lbnd=-1) | |
| res = poly.polyint(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 = poly.polyint(tgt, m=1, k=[k], scl=2) | |
| res = poly.polyint(pol, m=j, k=list(range(j)), scl=2) | |
| assert_almost_equal(trim(res), trim(tgt)) | |
| def test_polyint_axis(self): | |
| # check that axis keyword works | |
| c2d = np.random.random((3, 4)) | |
| tgt = np.vstack([poly.polyint(c) for c in c2d.T]).T | |
| res = poly.polyint(c2d, axis=0) | |
| assert_almost_equal(res, tgt) | |
| tgt = np.vstack([poly.polyint(c) for c in c2d]) | |
| res = poly.polyint(c2d, axis=1) | |
| assert_almost_equal(res, tgt) | |
| tgt = np.vstack([poly.polyint(c, k=3) for c in c2d]) | |
| res = poly.polyint(c2d, k=3, axis=1) | |
| assert_almost_equal(res, tgt) | |
| class TestDerivative: | |
| def test_polyder(self): | |
| # check exceptions | |
| assert_raises(TypeError, poly.polyder, [0], .5) | |
| assert_raises(ValueError, poly.polyder, [0], -1) | |
| # check that zeroth derivative does nothing | |
| for i in range(5): | |
| tgt = [0]*i + [1] | |
| res = poly.polyder(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 = poly.polyder(poly.polyint(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 = poly.polyder(poly.polyint(tgt, m=j, scl=2), m=j, scl=.5) | |
| assert_almost_equal(trim(res), trim(tgt)) | |
| def test_polyder_axis(self): | |
| # check that axis keyword works | |
| c2d = np.random.random((3, 4)) | |
| tgt = np.vstack([poly.polyder(c) for c in c2d.T]).T | |
| res = poly.polyder(c2d, axis=0) | |
| assert_almost_equal(res, tgt) | |
| tgt = np.vstack([poly.polyder(c) for c in c2d]) | |
| res = poly.polyder(c2d, axis=1) | |
| assert_almost_equal(res, tgt) | |
| class TestVander: | |
| # some random values in [-1, 1) | |
| x = np.random.random((3, 5))*2 - 1 | |
| def test_polyvander(self): | |
| # check for 1d x | |
| x = np.arange(3) | |
| v = poly.polyvander(x, 3) | |
| assert_(v.shape == (3, 4)) | |
| for i in range(4): | |
| coef = [0]*i + [1] | |
| assert_almost_equal(v[..., i], poly.polyval(x, coef)) | |
| # check for 2d x | |
| x = np.array([[1, 2], [3, 4], [5, 6]]) | |
| v = poly.polyvander(x, 3) | |
| assert_(v.shape == (3, 2, 4)) | |
| for i in range(4): | |
| coef = [0]*i + [1] | |
| assert_almost_equal(v[..., i], poly.polyval(x, coef)) | |
| def test_polyvander2d(self): | |
| # also tests polyval2d for non-square coefficient array | |
| x1, x2, x3 = self.x | |
| c = np.random.random((2, 3)) | |
| van = poly.polyvander2d(x1, x2, [1, 2]) | |
| tgt = poly.polyval2d(x1, x2, c) | |
| res = np.dot(van, c.flat) | |
| assert_almost_equal(res, tgt) | |
| # check shape | |
| van = poly.polyvander2d([x1], [x2], [1, 2]) | |
| assert_(van.shape == (1, 5, 6)) | |
| def test_polyvander3d(self): | |
| # also tests polyval3d for non-square coefficient array | |
| x1, x2, x3 = self.x | |
| c = np.random.random((2, 3, 4)) | |
| van = poly.polyvander3d(x1, x2, x3, [1, 2, 3]) | |
| tgt = poly.polyval3d(x1, x2, x3, c) | |
| res = np.dot(van, c.flat) | |
| assert_almost_equal(res, tgt) | |
| # check shape | |
| van = poly.polyvander3d([x1], [x2], [x3], [1, 2, 3]) | |
| assert_(van.shape == (1, 5, 24)) | |
| def test_polyvandernegdeg(self): | |
| x = np.arange(3) | |
| assert_raises(ValueError, poly.polyvander, x, -1) | |
| class TestCompanion: | |
| def test_raises(self): | |
| assert_raises(ValueError, poly.polycompanion, []) | |
| assert_raises(ValueError, poly.polycompanion, [1]) | |
| def test_dimensions(self): | |
| for i in range(1, 5): | |
| coef = [0]*i + [1] | |
| assert_(poly.polycompanion(coef).shape == (i, i)) | |
| def test_linear_root(self): | |
| assert_(poly.polycompanion([1, 2])[0, 0] == -.5) | |
| class TestMisc: | |
| def test_polyfromroots(self): | |
| res = poly.polyfromroots([]) | |
| 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]) | |
| tgt = Tlist[i] | |
| res = poly.polyfromroots(roots)*2**(i-1) | |
| assert_almost_equal(trim(res), trim(tgt)) | |
| def test_polyroots(self): | |
| assert_almost_equal(poly.polyroots([1]), []) | |
| assert_almost_equal(poly.polyroots([1, 2]), [-.5]) | |
| for i in range(2, 5): | |
| tgt = np.linspace(-1, 1, i) | |
| res = poly.polyroots(poly.polyfromroots(tgt)) | |
| assert_almost_equal(trim(res), trim(tgt)) | |
| def test_polyfit(self): | |
| def f(x): | |
| return x*(x - 1)*(x - 2) | |
| def f2(x): | |
| return x**4 + x**2 + 1 | |
| # Test exceptions | |
| assert_raises(ValueError, poly.polyfit, [1], [1], -1) | |
| assert_raises(TypeError, poly.polyfit, [[1]], [1], 0) | |
| assert_raises(TypeError, poly.polyfit, [], [1], 0) | |
| assert_raises(TypeError, poly.polyfit, [1], [[[1]]], 0) | |
| assert_raises(TypeError, poly.polyfit, [1, 2], [1], 0) | |
| assert_raises(TypeError, poly.polyfit, [1], [1, 2], 0) | |
| assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[[1]]) | |
| assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[1, 1]) | |
| assert_raises(ValueError, poly.polyfit, [1], [1], [-1,]) | |
| assert_raises(ValueError, poly.polyfit, [1], [1], [2, -1, 6]) | |
| assert_raises(TypeError, poly.polyfit, [1], [1], []) | |
| # Test fit | |
| x = np.linspace(0, 2) | |
| y = f(x) | |
| # | |
| coef3 = poly.polyfit(x, y, 3) | |
| assert_equal(len(coef3), 4) | |
| assert_almost_equal(poly.polyval(x, coef3), y) | |
| coef3 = poly.polyfit(x, y, [0, 1, 2, 3]) | |
| assert_equal(len(coef3), 4) | |
| assert_almost_equal(poly.polyval(x, coef3), y) | |
| # | |
| coef4 = poly.polyfit(x, y, 4) | |
| assert_equal(len(coef4), 5) | |
| assert_almost_equal(poly.polyval(x, coef4), y) | |
| coef4 = poly.polyfit(x, y, [0, 1, 2, 3, 4]) | |
| assert_equal(len(coef4), 5) | |
| assert_almost_equal(poly.polyval(x, coef4), y) | |
| # | |
| coef2d = poly.polyfit(x, np.array([y, y]).T, 3) | |
| assert_almost_equal(coef2d, np.array([coef3, coef3]).T) | |
| coef2d = poly.polyfit(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 | |
| yw[0::2] = 0 | |
| wcoef3 = poly.polyfit(x, yw, 3, w=w) | |
| assert_almost_equal(wcoef3, coef3) | |
| wcoef3 = poly.polyfit(x, yw, [0, 1, 2, 3], w=w) | |
| assert_almost_equal(wcoef3, coef3) | |
| # | |
| wcoef2d = poly.polyfit(x, np.array([yw, yw]).T, 3, w=w) | |
| assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) | |
| wcoef2d = poly.polyfit(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(poly.polyfit(x, x, 1), [0, 1]) | |
| assert_almost_equal(poly.polyfit(x, x, [0, 1]), [0, 1]) | |
| # test fitting only even Polyendre polynomials | |
| x = np.linspace(-1, 1) | |
| y = f2(x) | |
| coef1 = poly.polyfit(x, y, 4) | |
| assert_almost_equal(poly.polyval(x, coef1), y) | |
| coef2 = poly.polyfit(x, y, [0, 2, 4]) | |
| assert_almost_equal(poly.polyval(x, coef2), y) | |
| assert_almost_equal(coef1, coef2) | |
| def test_polytrim(self): | |
| coef = [2, -1, 1, 0] | |
| # Test exceptions | |
| assert_raises(ValueError, poly.polytrim, coef, -1) | |
| # Test results | |
| assert_equal(poly.polytrim(coef), coef[:-1]) | |
| assert_equal(poly.polytrim(coef, 1), coef[:-3]) | |
| assert_equal(poly.polytrim(coef, 2), [0]) | |
| def test_polyline(self): | |
| assert_equal(poly.polyline(3, 4), [3, 4]) | |
| def test_polyline_zero(self): | |
| assert_equal(poly.polyline(3, 0), [3]) | |