def test_errors(): pytest.raises(ValueError, lambda: SparseMatrix(1.4, 2, lambda i, j: 0)) pytest.raises(ValueError, lambda: SparseMatrix(2, 2, 1)) pytest.raises(TypeError, lambda: SparseMatrix([1, 2, 3], [1, 2])) pytest.raises(ValueError, lambda: SparseMatrix([[1, 2], [3, 4]])[(1, 2, 3)]) pytest.raises(IndexError, lambda: SparseMatrix([[1, 2], [3, 4]])[5]) pytest.raises(ValueError, lambda: SparseMatrix([[1, 2], [3, 4]])[1, 2, 3]) pytest.raises( TypeError, lambda: SparseMatrix([[1, 2], [3, 4]]).copyin_list([0, 1], set())) pytest.raises(IndexError, lambda: SparseMatrix([[1, 2], [3, 4]])[1, 2]) pytest.raises(TypeError, lambda: SparseMatrix([1, 2, 3]).cross(1)) pytest.raises(IndexError, lambda: SparseMatrix(1, 2, [1, 2])[3]) pytest.raises( ShapeError, lambda: SparseMatrix(1, 2, [1, 2]) + SparseMatrix(2, 1, [2, 1])) pytest.raises(IndexError, lambda: SparseMatrix([1, 2, 3])[3, 0]) pytest.raises(TypeError, lambda: SparseMatrix([1, 2, 3]).applyfunc(1)) pytest.raises(ValueError, lambda: SparseMatrix([1, 2, 3]).reshape(2, 2)) pytest.raises(ValueError, lambda: SparseMatrix([[2, 3], [4, 1]]).cholesky()) pytest.raises(ValueError, lambda: SparseMatrix([[2, 3], [4, 1]]).LDLdecomposition()) pytest.raises(ValueError, lambda: SparseMatrix([[2, 3], [4, 1]]).add(1)) pytest.raises( ShapeError, lambda: SparseMatrix([[1, 2], [3, 4]]).row_join(Matrix([[1, 2]]))) pytest.raises( ShapeError, lambda: SparseMatrix([[1, 2], [3, 4]]).col_join(Matrix([1, 2]))) pytest.raises( ShapeError, lambda: SparseMatrix([[1, 2], [3, 4]]).copyin_matrix([1, 0], Matrix([1, 2])))
def prde_linear_constraints(a, b, G, DE): """ Parametric Risch Differential Equation - Generate linear constraints on the constants. Given a derivation D on k[t], a, b, in k[t] with gcd(a, b) == 1, and G = [g1, ..., gm] in k(t)^m, return Q = [q1, ..., qm] in k[t]^m and a matrix M with entries in k(t) such that for any solution c1, ..., cm in Const(k) and p in k[t] of a*Dp + b*p == Sum(ci*gi, (i, 1, m)), (c1, ..., cm) is a solution of Mx == 0, and p and the ci satisfy a*Dp + b*p == Sum(ci*qi, (i, 1, m)). Because M has entries in k(t), and because Matrix doesn't play well with Poly, M will be a Matrix of Basic expressions. """ m = len(G) Gns, Gds = list(zip(*G)) d = reduce(lambda i, j: i.lcm(j), Gds) d = Poly(d, field=True) Q = [(ga*(d).quo(gd)).div(d) for ga, gd in G] if not all([ri.is_zero for _, ri in Q]): N = max([ri.degree(DE.t) for _, ri in Q]) M = Matrix(N + 1, m, lambda i, j: Q[j][1].nth(i)) else: M = Matrix() # No constraints, return the empty matrix. qs, _ = list(zip(*Q)) return qs, M
def test_transpose(): Sq = MatrixSymbol('Sq', n, n) assert transpose(A) == Transpose(A) assert Transpose(A).shape == (m, n) assert Transpose(A*B).shape == (l, n) assert transpose(Transpose(A)) == A assert isinstance(Transpose(Transpose(A)), Transpose) assert adjoint(Transpose(A)) == Adjoint(Transpose(A)) assert conjugate(Transpose(A)) == Adjoint(A) assert Transpose(eye(3)).doit() == eye(3) assert Transpose(eye(3)).doit(deep=False) == eye(3) assert Transpose(Integer(5)).doit() == Integer(5) assert Transpose(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3], [2, 4]]) assert transpose(trace(Sq)) == trace(Sq) assert trace(Transpose(Sq)) == trace(Sq) assert Transpose(Sq)[0, 1] == Sq[1, 0] assert Transpose(A*B).doit() == Transpose(B) * Transpose(A)
def test_Matrix_printing(): # Test returning a Matrix mat = Matrix([x*y, Piecewise((2 + x, y > 0), (y, True)), sin(z)]) A = MatrixSymbol('A', 3, 1) assert fcode(mat, A) == ( " A(1, 1) = x*y\n" " if (y > 0) then\n" " A(2, 1) = x + 2\n" " else\n" " A(2, 1) = y\n" " end if\n" " A(3, 1) = sin(z)") # Test using MatrixElements in expressions expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0] assert fcode(expr, standard=95) == ( " merge(2*A(3, 1), A(3, 1), x > 0) + sin(A(2, 1)) + A(1, 1)") # Test using MatrixElements in a Matrix q = MatrixSymbol('q', 5, 1) M = MatrixSymbol('M', 3, 3) m = Matrix([[sin(q[1, 0]), 0, cos(q[2, 0])], [q[1, 0] + q[2, 0], q[3, 0], 5], [2*q[4, 0]/q[1, 0], sqrt(q[0, 0]) + 4, 0]]) assert fcode(m, M) == ( " M(1, 1) = sin(q(2, 1))\n" " M(2, 1) = q(2, 1) + q(3, 1)\n" " M(3, 1) = 2*q(5, 1)*1.0/q(2, 1)\n" " M(1, 2) = 0\n" " M(2, 2) = q(4, 1)\n" " M(3, 2) = 4 + sqrt(q(1, 1))\n" " M(1, 3) = cos(q(3, 1))\n" " M(2, 3) = 5\n" " M(3, 3) = 0")
def test_functional_diffgeom_ch2(): x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', extended_real=True) x, y = symbols('x, y', extended_real=True) f = Function('f') assert (R2_p.point_to_coords(R2_r.point([x0, y0])) == Matrix( [sqrt(x0**2 + y0**2), atan2(y0, x0)])) assert (R2_r.point_to_coords(R2_p.point([r0, theta0])) == Matrix( [r0 * cos(theta0), r0 * sin(theta0)])) assert R2_p.jacobian(R2_r, [r0, theta0]) == Matrix( [[cos(theta0), -r0 * sin(theta0)], [sin(theta0), r0 * cos(theta0)]]) field = f(R2.x, R2.y) p1_in_rect = R2_r.point([x0, y0]) p1_in_polar = R2_p.point([sqrt(x0**2 + y0**2), atan2(y0, x0)]) assert field.rcall(p1_in_rect) == f(x0, y0) assert field.rcall(p1_in_polar) == f(x0, y0) p_r = R2_r.point([x0, y0]) p_p = R2_p.point([r0, theta0]) assert R2.x(p_r) == x0 assert R2.x(p_p) == r0 * cos(theta0) assert R2.r(p_p) == r0 assert R2.r(p_r) == sqrt(x0**2 + y0**2) assert R2.theta(p_r) == atan2(y0, x0) h = R2.x * R2.r**2 + R2.y**3 assert h.rcall(p_r) == x0 * (x0**2 + y0**2) + y0**3 assert h.rcall(p_p) == r0**3 * sin(theta0)**3 + r0**3 * cos(theta0)
def test_Matrix(): assert mcode(Matrix()) == '{}' m = Matrix([[1, 2], [3, 4444]]) assert mcode(m) == mcode(m.as_immutable()) == '{{1, 2}, {3, 4444}}' m = SparseMatrix(m) assert mcode(m) == mcode(m.as_immutable()) == '{{1, 2}, {3, 4444}}'
def _eval_transpose(self): # Flip all the individual matrices matrices = [transpose(matrix) for matrix in self.blocks] # Make a copy M = Matrix(self.blockshape[0], self.blockshape[1], matrices) # Transpose the block structure M = M.transpose() return BlockMatrix(M)
def test_slices(): md = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4)) assert md[:] == md._array assert md[:, :, 0].tomatrix() == Matrix([[10, 14, 18], [22, 26, 30]]) assert md[0, 1:2, :].tomatrix() == Matrix([[14, 15, 16, 17]]) assert md[0, 1:3, :].tomatrix() == Matrix([[14, 15, 16, 17], [18, 19, 20, 21]]) assert md[:, :, :] == md
def test_adjoint(): assert adjoint(A*B) == Adjoint(B)*Adjoint(A) assert adjoint(2*A*B) == 2*Adjoint(B)*Adjoint(A) assert adjoint(2*I*C) == -2*I*Adjoint(C) M = Matrix(2, 2, [1, 2 + I, 3, 4]).as_immutable() MA = Matrix(2, 2, [1, 3, 2 - I, 4]) assert adjoint(M) == MA assert adjoint(2*M) == 2*MA assert adjoint(MatMul(2, M)) == MatMul(2, MA).doit()
def is_scalar_multiple(self, other): """Returns whether `self` and `other` are scalar multiples of each other. """ # if the vectors self and other are linearly dependent, then they must # be scalar multiples of each other m = Matrix([self.args, other.args]) # XXX: issue sympy/sympy#9480 we need `simplify=True` otherwise the # rank may be computed incorrectly return m.rank(simplify=True) < 2
def test_transpose(): assert transpose(A*B) == Transpose(B)*Transpose(A) assert transpose(2*A*B) == 2*Transpose(B)*Transpose(A) assert transpose(2*I*C) == 2*I*Transpose(C) M = Matrix(2, 2, [1, 2 + I, 3, 4]).as_immutable() MT = Matrix(2, 2, [1, 3, 2 + I, 4]) assert transpose(M) == MT assert transpose(2*M) == 2*MT assert transpose(MatMul(2, M)) == MatMul(2, MT).doit()
def test_sparse_solve(): A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11))) assert A.cholesky() == Matrix([ [ 5, 0, 0], [ 3, 3, 0], [-1, 1, 3]]) assert A.cholesky() * A.cholesky().T == Matrix([ [25, 15, -5], [15, 18, 0], [-5, 0, 11]]) A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11))) L, D = A.LDLdecomposition() assert 15*L == Matrix([ [15, 0, 0], [ 9, 15, 0], [-3, 5, 15]]) assert D == Matrix([ [25, 0, 0], [ 0, 9, 0], [ 0, 0, 9]]) assert L * D * L.T == A A = SparseMatrix(((3, 0, 2), (0, 0, 1), (1, 2, 0))) assert A.inv() * A == SparseMatrix(eye(3)) A = SparseMatrix([ [ 2, -1, 0], [-1, 2, -1], [ 0, 0, 2]]) ans = SparseMatrix([ [Rational(2, 3), Rational(1, 3), Rational(1, 6)], [Rational(1, 3), Rational(2, 3), Rational(1, 3)], [ 0, 0, Rational(1, 2)]]) assert A.inv(method='CH') == ans assert A.inv(method='LDL') == ans assert A * ans == SparseMatrix(eye(3)) s = A.solve(A[:, 0], 'LDL') assert A*s == A[:, 0] s = A.solve(A[:, 0], 'CH') assert A*s == A[:, 0] A = A.col_join(A) s = A.solve_least_squares(A[:, 0], 'CH') assert A*s == A[:, 0] s = A.solve_least_squares(A[:, 0], 'LDL') assert A*s == A[:, 0] pytest.raises(ValueError, lambda: SparseMatrix([[1, 0, 1], [0, 0, 1]]).solve([1, 1])) pytest.raises(ValueError, lambda: SparseMatrix([[1, 0], [0, 0], [2, 1]]).solve([1, 1, 1]))
def test_converting_functions(): arr_list = [1, 2, 3, 4] arr_matrix = Matrix(((1, 2), (3, 4))) # list arr_ndim_array = ImmutableDenseNDimArray(arr_list, (2, 2)) assert isinstance(arr_ndim_array, ImmutableDenseNDimArray) assert arr_matrix.tolist() == arr_ndim_array.tolist() # Matrix arr_ndim_array = ImmutableDenseNDimArray(arr_matrix) assert isinstance(arr_ndim_array, ImmutableDenseNDimArray) assert arr_matrix.tolist() == arr_ndim_array.tolist() assert arr_matrix.shape == arr_ndim_array.shape
def test_converting_functions(): arr_list = [1, 2, 3, 4] arr_matrix = Matrix(((1, 2), (3, 4))) # list arr_ndim_array = MutableDenseNDimArray(arr_list, (2, 2)) assert (isinstance(arr_ndim_array, MutableDenseNDimArray)) assert arr_matrix.tolist() == arr_ndim_array.tolist() # Matrix arr_ndim_array = MutableDenseNDimArray(arr_matrix) assert (isinstance(arr_ndim_array, MutableDenseNDimArray)) assert arr_matrix.tolist() == arr_ndim_array.tolist() assert arr_matrix.shape == arr_ndim_array.shape
def distance(self, o): """Distance between the plane and another geometric entity. Parameters ========== Point3D, LinearEntity3D, Plane. Returns ======= distance Notes ===== This method accepts only 3D entities as it's parameter, but if you want to calculate the distance between a 2D entity and a plane you should first convert to a 3D entity by projecting onto a desired plane and then proceed to calculate the distance. Examples ======== >>> from diofant import Point, Point3D, Line, Line3D, Plane >>> a = Plane(Point3D(1, 1, 1), normal_vector=(1, 1, 1)) >>> b = Point3D(1, 2, 3) >>> a.distance(b) sqrt(3) >>> c = Line3D(Point3D(2, 3, 1), Point3D(1, 2, 2)) >>> a.distance(c) 0 """ from diofant.geometry.line3d import LinearEntity3D x, y, z = map(Dummy, 'xyz') if self.intersection(o) != []: return S.Zero if isinstance(o, Point3D): x, y, z = map(Dummy, 'xyz') k = self.equation(x, y, z) a, b, c = [k.coeff(i) for i in (x, y, z)] d = k.xreplace({x: o.args[0], y: o.args[1], z: o.args[2]}) t = abs(d / sqrt(a**2 + b**2 + c**2)) return t if isinstance(o, LinearEntity3D): a, b = o.p1, self.p1 c = Matrix(a.direction_ratio(b)) d = Matrix(self.normal_vector) e = c.dot(d) f = sqrt(sum(i**2 for i in self.normal_vector)) return abs(e / f) if isinstance(o, Plane): a, b = o.p1, self.p1 c = Matrix(a.direction_ratio(b)) d = Matrix(self.normal_vector) e = c.dot(d) f = sqrt(sum(i**2 for i in self.normal_vector)) return abs(e / f)
def test_Matrices(): assert mcode(Matrix(1, 1, [10])) == "10" A = Matrix([[1, sin(x / 2), abs(x)], [0, 1, pi], [0, exp(1), ceiling(x)]]) expected = ("[1 sin(x/2) abs(x);\n" "0 1 pi;\n" "0 exp(1) ceil(x)]") assert mcode(A) == expected # row and columns assert mcode(A[:, 0]) == "[1; 0; 0]" assert mcode(A[0, :]) == "[1 sin(x/2) abs(x)]" # empty matrices assert mcode(Matrix(0, 0, [])) == '[]' assert mcode(Matrix(0, 3, [])) == 'zeros(0, 3)' # annoying to read but correct assert mcode(Matrix([[x, x - y, -y]])) == "[x x - y -y]"
def test_inverse_laplace_transform(): from diofant import sinh, cosh, besselj, besseli, simplify, factor_terms ILT = inverse_laplace_transform a, b, c, = symbols('a b c', positive=True, finite=True) t = symbols('t') def simp_hyp(expr): return factor_terms(expand_mul(expr)).rewrite(sin) # just test inverses of all of the above assert ILT(1 / s, s, t) == Heaviside(t) assert ILT(1 / s**2, s, t) == t * Heaviside(t) assert ILT(1 / s**5, s, t) == t**4 * Heaviside(t) / 24 assert ILT(exp(-a * s) / s, s, t) == Heaviside(t - a) assert ILT(exp(-a * s) / (s + b), s, t) == exp(b * (a - t)) * Heaviside(-a + t) assert ILT(a / (s**2 + a**2), s, t) == sin(a * t) * Heaviside(t) assert ILT(s / (s**2 + a**2), s, t) == cos(a * t) * Heaviside(t) # TODO is there a way around simp_hyp? assert simp_hyp(ILT(a / (s**2 - a**2), s, t)) == sinh(a * t) * Heaviside(t) assert simp_hyp(ILT(s / (s**2 - a**2), s, t)) == cosh(a * t) * Heaviside(t) assert ILT(a / ((s + b)**2 + a**2), s, t) == exp(-b * t) * sin(a * t) * Heaviside(t) assert ILT((s + b) / ((s + b)**2 + a**2), s, t) == exp(-b * t) * cos(a * t) * Heaviside(t) # TODO sinh/cosh shifted come out a mess. also delayed trig is a mess # TODO should this simplify further? assert ILT(exp(-a*s)/s**b, s, t) == \ (t - a)**(b - 1)*Heaviside(t - a)/gamma(b) assert ILT(exp(-a*s)/sqrt(1 + s**2), s, t) == \ Heaviside(t - a)*besselj(0, a - t) # note: besselj(0, x) is even # XXX ILT turns these branch factor into trig functions ... assert simplify(ILT(a**b*(s + sqrt(s**2 - a**2))**(-b)/sqrt(s**2 - a**2), s, t).rewrite(exp)) == \ Heaviside(t)*besseli(b, a*t) assert ILT(a**b*(s + sqrt(s**2 + a**2))**(-b)/sqrt(s**2 + a**2), s, t).rewrite(exp) == \ Heaviside(t)*besselj(b, a*t) assert ILT(1 / (s * sqrt(s + 1)), s, t) == Heaviside(t) * erf(sqrt(t)) # TODO can we make erf(t) work? assert ILT(1 / (s**2 * (s**2 + 1)), s, t) == (t - sin(t)) * Heaviside(t) assert ILT( (s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t) ==\ Matrix([[exp(t)*Heaviside(t), 0], [0, exp(2*t)*Heaviside(t)]])
def test_BlockMatrix(): A = MatrixSymbol('A', n, m) B = MatrixSymbol('B', n, k) C = MatrixSymbol('C', l, m) D = MatrixSymbol('D', l, k) M = MatrixSymbol('M', m + k, p) N = MatrixSymbol('N', l + n, k + m) X = BlockMatrix(Matrix([[A, B], [C, D]])) assert X.__class__(*X.args) == X # block_collapse does nothing on normal inputs E = MatrixSymbol('E', n, m) assert block_collapse(A + 2 * E) == A + 2 * E F = MatrixSymbol('F', m, m) assert block_collapse(E.T * A * F) == E.T * A * F assert X.shape == (l + n, k + m) assert X.blockshape == (2, 2) assert transpose(X) == BlockMatrix(Matrix([[A.T, C.T], [B.T, D.T]])) assert transpose(X).shape == X.shape[::-1] # Test that BlockMatrices and MatrixSymbols can still mix assert (X * M).is_MatMul assert X._blockmul(M).is_MatMul assert (X * M).shape == (n + l, p) assert (X + N).is_MatAdd assert X._blockadd(N).is_MatAdd assert (X + N).shape == X.shape E = MatrixSymbol('E', m, 1) F = MatrixSymbol('F', k, 1) Y = BlockMatrix(Matrix([[E], [F]])) assert (X * Y).shape == (l + n, 1) assert block_collapse(X * Y).blocks[0, 0] == A * E + B * F assert block_collapse(X * Y).blocks[1, 0] == C * E + D * F # block_collapse passes down into container objects, transposes, and inverse assert block_collapse(transpose(X * Y)) == transpose(block_collapse(X * Y)) assert block_collapse(Tuple(X * Y, 2 * X)) == (block_collapse(X * Y), block_collapse(2 * X)) # Make sure that MatrixSymbols will enter 1x1 BlockMatrix if it simplifies Ab = BlockMatrix([[A]]) Z = MatrixSymbol('Z', *A.shape) assert block_collapse(Ab + Z) == A + Z
def test_functional_diffgeom_ch6(): u0, u1, u2, v0, v1, v2, w0, w1, w2 = symbols('u0:3, v0:3, w0:3', extended_real=True) u = u0 * R2.e_x + u1 * R2.e_y v = v0 * R2.e_x + v1 * R2.e_y wp = WedgeProduct(R2.dx, R2.dy) assert wp(u, v) == u0 * v1 - u1 * v0 u = u0 * R3_r.e_x + u1 * R3_r.e_y + u2 * R3_r.e_z v = v0 * R3_r.e_x + v1 * R3_r.e_y + v2 * R3_r.e_z w = w0 * R3_r.e_x + w1 * R3_r.e_y + w2 * R3_r.e_z wp = WedgeProduct(R3_r.dx, R3_r.dy, R3_r.dz) assert wp(u, v, w) == Matrix(3, 3, [u0, u1, u2, v0, v1, v2, w0, w1, w2]).det() a, b, c = symbols('a, b, c', cls=Function) a_f = a(R3_r.x, R3_r.y, R3_r.z) b_f = b(R3_r.x, R3_r.y, R3_r.z) c_f = c(R3_r.x, R3_r.y, R3_r.z) theta = a_f * R3_r.dx + b_f * R3_r.dy + c_f * R3_r.dz dtheta = Differential(theta) da = Differential(a_f) db = Differential(b_f) dc = Differential(c_f) expr = dtheta - WedgeProduct(da, R3_r.dx) - WedgeProduct( db, R3_r.dy) - WedgeProduct(dc, R3_r.dz) assert expr.rcall(R3_r.e_x, R3_r.e_y) == 0
def rotate(self, angle=0, pt=None): """Rotate ``angle`` radians counterclockwise about Point ``pt``. The default pt is the origin, Point(0, 0). Examples ======== >>> from diofant.geometry.curve import Curve >>> from diofant.abc import x >>> from diofant import pi >>> Curve((x, x), (x, 0, 1)).rotate(pi/2) Curve((-x, x), (x, 0, 1)) """ from diofant.matrices import Matrix, rot_axis3 pt = -Point(pt or (0, 0)) rv = self.translate(*pt.args) f = list(rv.functions) f.append(0) f = Matrix(1, 3, f) f *= rot_axis3(angle) rv = self.func(f[0, :2].tolist()[0], self.limits) if pt is not None: pt = -pt return rv.translate(*pt.args) return rv
def prde_no_cancel_b_large(b, Q, n, DE): """ Parametric Poly Risch Differential Equation - No cancellation: deg(b) large enough. Given a derivation D on k[t], n in ZZ, and b, q1, ..., qm in k[t] with b != 0 and either D == d/dt or deg(b) > max(0, deg(D) - 1), returns h1, ..., hr in k[r] and a matrix A with coefficients in Const(k) such that if c1, ..., cm in Const(k) and q in k[t] satisfy deg(q) <= n and Dq + b*Q == Sum(ci*qi, (i, 1, m)), then q = Sum(dj*hj, (j, 1, r)), where d1, ..., dr in Const(k) and A*Matrix([[c1, ..., cm, d1, ..., dr]]).T == 0. """ db = b.degree(DE.t) m = len(Q) H = [Poly(0, DE.t)]*m for N in range(n, -1, -1): # [n, ..., 0] for i in range(m): si = Q[i].nth(N + db)/b.LC() sitn = Poly(si*DE.t**N, DE.t) H[i] = H[i] + sitn Q[i] = Q[i] - derivation(sitn, DE) - b*sitn if all(qi.is_zero for qi in Q): dc = -1 M = zeros(0, 2) else: dc = max([qi.degree(t) for qi in Q]) M = Matrix(dc + 1, m, lambda i, j: Q[j].nth(i)) A, u = constant_system(M, zeros(dc + 1, 1), DE) c = eye(m) A = A.row_join(zeros(A.rows, m)).col_join(c.row_join(-c)) return H, A
def test_Trace(): assert isinstance(Trace(A), Trace) assert not isinstance(Trace(A), MatrixExpr) pytest.raises(ShapeError, lambda: Trace(C)) assert trace(eye(3)) == 3 assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15 assert adjoint(Trace(A)) == trace(Adjoint(A)) assert conjugate(Trace(A)) == trace(Adjoint(A)) assert transpose(Trace(A)) == Trace(A) A / Trace(A) # Make sure this is possible # Some easy simplifications assert trace(Identity(5)) == 5 assert trace(ZeroMatrix(5, 5)) == 0 assert trace(2 * A * B) == 2 * Trace(A * B) assert trace(A.T) == trace(A) i, j = symbols('i j') F = FunctionMatrix(3, 3, Lambda((i, j), i + j)) assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2) pytest.raises(TypeError, lambda: Trace(S.One)) assert Trace(A).arg is A assert str(trace(A)) == str(Trace(A).doit())
def test_Assignment(): x, y = symbols("x, y") A = MatrixSymbol('A', 3, 1) mat = Matrix([1, 2, 3]) B = IndexedBase('B') n = symbols("n", integer=True) i = Idx("i", n) # Here we just do things to show they don't error Assignment(x, y) Assignment(x, 0) Assignment(A, mat) Assignment(A[1, 0], 0) Assignment(A[1, 0], x) Assignment(B[i], x) Assignment(B[i], 0) # Here we test things to show that they error # Matrix to scalar pytest.raises(ValueError, lambda: Assignment(B[i], A)) pytest.raises(ValueError, lambda: Assignment(B[i], mat)) pytest.raises(ValueError, lambda: Assignment(x, mat)) pytest.raises(ValueError, lambda: Assignment(x, A)) pytest.raises(ValueError, lambda: Assignment(A[1, 0], mat)) # Scalar to matrix pytest.raises(ValueError, lambda: Assignment(A, x)) pytest.raises(ValueError, lambda: Assignment(A, 0)) # Non-atomic lhs pytest.raises(TypeError, lambda: Assignment(mat, A)) pytest.raises(TypeError, lambda: Assignment(0, x)) pytest.raises(TypeError, lambda: Assignment(x * x, 1)) pytest.raises(TypeError, lambda: Assignment(A + A, mat)) pytest.raises(TypeError, lambda: Assignment(B, 0))
def limited_integrate(fa, fd, G, DE): """ Solves the limited integration problem: f = Dv + Sum(ci*wi, (i, 1, n)) """ fa, fd = fa*Poly(1/fd.LC(), DE.t), fd.monic() A, B, h, N, g, V = limited_integrate_reduce(fa, fd, G, DE) V = [g] + V g = A.gcd(B) A, B, V = A.quo(g), B.quo(g), [via.cancel(vid*g, include=True) for via, vid in V] Q, M = prde_linear_constraints(A, B, V, DE) M, _ = constant_system(M, zeros(M.rows, 1), DE) l = M.nullspace() if M == Matrix() or len(l) > 1: # Continue with param_rischDE() raise NotImplementedError("param_rischDE() is required to solve this " "integral.") elif len(l) == 0: raise NonElementaryIntegralException elif len(l) == 1: # The c1 == 1. In this case, we can assume a normal Risch DE if l[0][0].is_zero: raise NonElementaryIntegralException else: l[0] *= 1/l[0][0] C = sum(Poly(i, DE.t)*q for (i, q) in zip(l[0], Q)) # Custom version of rischDE() that uses the already computed # denominator and degree bound from above. B, C, m, alpha, beta = spde(A, B, C, N, DE) y = solve_poly_rde(B, C, m, DE) return (alpha*y + beta, h), list(l[0][1:]) else: raise NotImplementedError
def test_m_matrix_output_autoname_2(): e1 = (x + y) e2 = Matrix([[2 * x, 2 * y, 2 * z]]) e3 = Matrix([[x], [y], [z]]) e4 = Matrix([[x, y], [z, 16]]) name_expr = ("test", (e1, e2, e3, e4)) result, = codegen(name_expr, "Octave", header=False, empty=False) source = result[1] expected = ("function [out1, out2, out3, out4] = test(x, y, z)\n" " out1 = x + y;\n" " out2 = [2*x 2*y 2*z];\n" " out3 = [x; y; z];\n" " out4 = [x y;\n" " z 16];\n" "end\n") assert source == expected
def test_MatrixElement_with_values(): M = Matrix([[x, y], [z, w]]) Mij = M[i, j] assert isinstance(Mij, MatrixElement) Ms = SparseMatrix([[2, 3], [4, 5]]) msij = Ms[i, j] assert isinstance(msij, MatrixElement) for oi, oj in [(0, 0), (0, 1), (1, 0), (1, 1)]: assert Mij.subs({i: oi, j: oj}) == M[oi, oj] assert msij.subs({i: oi, j: oj}) == Ms[oi, oj] A = MatrixSymbol("A", 2, 2) assert A[0, 0].subs({A: M}) == x assert A[i, j].subs({A: M}) == M[i, j] assert M[i, j].subs([(M, A)]) == A[i, j] assert isinstance(M[3 * i - 2, j], MatrixElement) assert M[3 * i - 2, j].subs({i: 1, j: 0}) == M[1, 0] assert isinstance(M[i, 0], MatrixElement) assert M[i, 0].subs({i: 0}) == M[0, 0] assert M[0, i].subs({i: 1}) == M[0, 1] pytest.raises(ValueError, lambda: M[i, 2]) pytest.raises(ValueError, lambda: M[i, -1]) pytest.raises(ValueError, lambda: M[2, i]) pytest.raises(ValueError, lambda: M[-1, i]) pytest.raises(ValueError, lambda: Ms[i, 2]) pytest.raises(ValueError, lambda: Ms[i, -1]) pytest.raises(ValueError, lambda: Ms[2, i]) pytest.raises(ValueError, lambda: Ms[-1, i])
def test_coords(): r, theta = symbols('r, theta') m = Manifold('M', 2) patch = Patch('P', m) rect = CoordSystem('rect', patch) polar = CoordSystem('polar', patch) polar.connect_to(rect, [r, theta], [r * cos(theta), r * sin(theta)]) polar.coord_tuple_transform_to(rect, [0, 2]) == Matrix([[0], [0]])
def test_Trace_doit_deep_False(): X = Matrix([[1, 2], [3, 4]]) q = MatPow(X, 2) assert Trace(q).doit(deep=False).arg == q q = MatAdd(X, 2 * X) assert Trace(q).doit(deep=False).arg == q q = MatMul(X, 2 * X) assert Trace(q).doit(deep=False).arg == q
def test_octave_matrix_elements(): A = Matrix([[x, 2, x * y]]) assert mcode(A[0, 0]**2 + A[0, 1] + A[0, 2]) == "x.^2 + x.*y + 2" A = MatrixSymbol('AA', 1, 3) assert mcode(A) == "AA" assert mcode(A[0, 0]**2 + sin(A[0, 1]) + A[0, 2]) == \ "sin(AA(1, 2)) + AA(1, 1).^2 + AA(1, 3)" assert mcode(sum(A)) == "AA(1, 1) + AA(1, 2) + AA(1, 3)"
def test_octave_matrix_1x1(): A = Matrix([[3]]) B = MatrixSymbol('B', 1, 1) C = MatrixSymbol('C', 1, 2) assert mcode(A, assign_to=B) == "B = 3;" # FIXME? # assert mcode(A, assign_to=x) == "x = 3;" pytest.raises(ValueError, lambda: mcode(A, assign_to=C))
def test_octave_matrix_assign_to_more(): # assigning to Symbol or MatrixSymbol requires lhs/rhs match A = Matrix([[1, 2, 3]]) B = MatrixSymbol('B', 1, 3) C = MatrixSymbol('C', 2, 3) assert mcode(A, assign_to=B) == "B = [1 2 3];" pytest.raises(ValueError, lambda: mcode(A, assign_to=x)) pytest.raises(ValueError, lambda: mcode(A, assign_to=C))
def test_Matrices_entries_not_hadamard(): # For Matrix with col >= 2, row >= 2, they need to be scalars # FIXME: is it worth worrying about this? Its not wrong, just # leave it user's responsibility to put scalar data for x. A = Matrix([[1, sin(2 / x), 3 * pi / x / 5], [1, 2, x * y]]) expected = ("[1 sin(2/x) 3*pi/(5*x);\n" "1 2 x*y]") # <- we give x.*y assert mcode(A) == expected
def test_sparse_matrix(): def sparse_eye(n): return SparseMatrix.eye(n) def sparse_zeros(n): return SparseMatrix.zeros(n) # creation args pytest.raises(TypeError, lambda: SparseMatrix(1, 2)) pytest.raises(ValueError, lambda: SparseMatrix(2, 2, (1, 3, 4, 5, 6))) a = SparseMatrix(( (1, 0), (0, 1) )) assert SparseMatrix(a) == a a = MutableSparseMatrix([]) b = MutableDenseMatrix([1, 2]) assert a.row_join(b) == b assert a.col_join(b) == b assert type(a.row_join(b)) == type(a) assert type(a.col_join(b)) == type(a) # test element assignment a = SparseMatrix(( (1, 0), (0, 1) )) a[3] = 4 assert a[1, 1] == 4 a[3] = 1 a[0, 0] = 2 assert a == SparseMatrix(( (2, 0), (0, 1) )) a[1, 0] = 5 assert a == SparseMatrix(( (2, 0), (5, 1) )) a[1, 1] = 0 assert a == SparseMatrix(( (2, 0), (5, 0) )) assert a._smat == {(0, 0): 2, (1, 0): 5} # test_multiplication a = SparseMatrix(( (1, 2), (3, 1), (0, 6), )) b = SparseMatrix(( (1, 2), (3, 0), )) c = a*b assert c[0, 0] == 7 assert c[0, 1] == 2 assert c[1, 0] == 6 assert c[1, 1] == 6 assert c[2, 0] == 18 assert c[2, 1] == 0 c = b * x assert isinstance(c, SparseMatrix) assert c[0, 0] == x assert c[0, 1] == 2*x assert c[1, 0] == 3*x assert c[1, 1] == 0 c = 5 * b assert isinstance(c, SparseMatrix) assert c[0, 0] == 5 assert c[0, 1] == 2*5 assert c[1, 0] == 3*5 assert c[1, 1] == 0 # test_power A = SparseMatrix([[2, 3], [4, 5]]) assert (A**5)[:] == [6140, 8097, 10796, 14237] A = SparseMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]]) assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433] # test_creation a = SparseMatrix([[x, 0], [0, 0]]) m = a assert m.cols == m.rows assert m.cols == 2 assert m[:] == [x, 0, 0, 0] b = SparseMatrix(2, 2, [x, 0, 0, 0]) m = b assert m.cols == m.rows assert m.cols == 2 assert m[:] == [x, 0, 0, 0] assert a == b S = sparse_eye(3) del S[1, :] assert S == SparseMatrix([ [1, 0, 0], [0, 0, 1]]) S = sparse_eye(3) del S[:, 1] assert S == SparseMatrix([ [1, 0], [0, 0], [0, 1]]) S = SparseMatrix.eye(3) S[2, 1] = 2 S.col_swap(1, 0) assert S == SparseMatrix([[0, 1, 0], [1, 0, 0], [2, 0, 1]]) S.row_swap(0, 1) assert S == SparseMatrix([[1, 0, 0], [0, 1, 0], [2, 0, 1]]) S.col_swap(0, 1) assert S == SparseMatrix([[0, 1, 0], [1, 0, 0], [0, 2, 1]]) S.row_swap(0, 2) assert S == SparseMatrix([[0, 2, 1], [1, 0, 0], [0, 1, 0]]) S.col_swap(0, 2) assert S == SparseMatrix([[1, 2, 0], [0, 0, 1], [0, 1, 0]]) a = SparseMatrix(1, 2, [1, 2]) b = a.copy() c = a.copy() assert a[0] == 1 del a[0, :] assert a == SparseMatrix(0, 2, []) del b[:, 1] assert b == SparseMatrix(1, 1, [1]) # test_determinant assert SparseMatrix(1, 1, [0]).det() == 0 assert SparseMatrix([[1]]).det() == 1 assert SparseMatrix(((-3, 2), (8, -5))).det() == -1 assert SparseMatrix(((x, 1), (y, 2*y))).det() == 2*x*y - y assert SparseMatrix(( (1, 1, 1), (1, 2, 3), (1, 3, 6) )).det() == 1 assert SparseMatrix(( ( 3, -2, 0, 5), (-2, 1, -2, 2), ( 0, -2, 5, 0), ( 5, 0, 3, 4) )).det() == -289 assert SparseMatrix(( ( 1, 2, 3, 4), ( 5, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16) )).det() == 0 assert SparseMatrix(( (3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0), (0, 0, 0, 3, 2), (2, 0, 0, 0, 3) )).det() == 275 assert SparseMatrix(( (1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3), (3, 2, -1, 1, 8), (1, 1, 1, 0, 6) )).det() == -55 assert SparseMatrix(( (-5, 2, 3, 4, 5), ( 1, -4, 3, 4, 5), ( 1, 2, -3, 4, 5), ( 1, 2, 3, -2, 5), ( 1, 2, 3, 4, -1) )).det() == 11664 assert SparseMatrix(( ( 2, 7, -1, 3, 2), ( 0, 0, 1, 0, 1), (-2, 0, 7, 0, 2), (-3, -2, 4, 5, 3), ( 1, 0, 0, 0, 1) )).det() == 123 # test_slicing m0 = sparse_eye(4) assert m0[:3, :3] == sparse_eye(3) assert m0[2:4, 0:2] == sparse_zeros(2) m1 = SparseMatrix(3, 3, lambda i, j: i + j) assert m1[0, :] == SparseMatrix(1, 3, (0, 1, 2)) assert m1[1:3, 1] == SparseMatrix(2, 1, (2, 3)) m2 = SparseMatrix( [[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15]]) assert m2[:, -1] == SparseMatrix(4, 1, [3, 7, 11, 15]) assert m2[-2:, :] == SparseMatrix([[8, 9, 10, 11], [12, 13, 14, 15]]) assert SparseMatrix([[1, 2], [3, 4]])[[1], [1]] == Matrix([[4]]) # test_submatrix_assignment m = sparse_zeros(4) m[2:4, 2:4] = sparse_eye(2) assert m == SparseMatrix([(0, 0, 0, 0), (0, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]) assert len(m._smat) == 2 m[:2, :2] = sparse_eye(2) assert m == sparse_eye(4) m[:, 0] = SparseMatrix(4, 1, (1, 2, 3, 4)) assert m == SparseMatrix([(1, 0, 0, 0), (2, 1, 0, 0), (3, 0, 1, 0), (4, 0, 0, 1)]) m[:, :] = sparse_zeros(4) assert m == sparse_zeros(4) m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)) assert m == SparseMatrix((( 1, 2, 3, 4), ( 5, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16))) m[:2, 0] = [0, 0] assert m == SparseMatrix((( 0, 2, 3, 4), ( 0, 6, 7, 8), ( 9, 10, 11, 12), (13, 14, 15, 16))) # test_reshape m0 = sparse_eye(3) assert m0.reshape(1, 9) == SparseMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1)) m1 = SparseMatrix(3, 4, lambda i, j: i + j) assert m1.reshape(4, 3) == \ SparseMatrix([(0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)]) assert m1.reshape(2, 6) == \ SparseMatrix([(0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)]) # test_applyfunc m0 = sparse_eye(3) assert m0.applyfunc(lambda x: 2*x) == sparse_eye(3)*2 assert m0.applyfunc(lambda x: 0 ) == sparse_zeros(3) # test_LUdecomp testmat = SparseMatrix([[ 0, 2, 5, 3], [ 3, 3, 7, 4], [ 8, 4, 0, 2], [-2, 6, 3, 4]]) L, U, p = testmat.LUdecomposition() assert L.is_lower assert U.is_upper assert (L*U).permuteBkwd(p) - testmat == sparse_zeros(4) testmat = SparseMatrix([[ 6, -2, 7, 4], [ 0, 3, 6, 7], [ 1, -2, 7, 4], [-9, 2, 6, 3]]) L, U, p = testmat.LUdecomposition() assert L.is_lower assert U.is_upper assert (L*U).permuteBkwd(p) - testmat == sparse_zeros(4) M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z))) L, U, p = M.LUdecomposition() assert L.is_lower assert U.is_upper assert (L*U).permuteBkwd(p) - M == sparse_zeros(3) # test_LUsolve A = SparseMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]]) B = SparseMatrix(3, 1, [3, 7, 5]) b = A*B soln = A.LUsolve(b) assert soln == B A = SparseMatrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]]) B = SparseMatrix(3, 1, [-1, 2, 5]) b = A*B soln = A.LUsolve(b) assert soln == B # test_inverse A = sparse_eye(4) assert A.inv() == sparse_eye(4) assert A.inv(method="CH") == sparse_eye(4) assert A.inv(method="LDL") == sparse_eye(4) A = SparseMatrix([[2, 3, 5], [3, 6, 2], [7, 2, 6]]) Ainv = SparseMatrix(Matrix(A).inv()) assert A*Ainv == sparse_eye(3) assert A.inv(method="CH") == Ainv assert A.inv(method="LDL") == Ainv A = SparseMatrix([[2, 3, 5], [3, 6, 2], [5, 2, 6]]) Ainv = SparseMatrix(Matrix(A).inv()) assert A*Ainv == sparse_eye(3) assert A.inv(method="CH") == Ainv assert A.inv(method="LDL") == Ainv # test_cross v1 = Matrix(1, 3, [1, 2, 3]) v2 = Matrix(1, 3, [3, 4, 5]) assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2]) assert v1.norm(2)**2 == 14 # conjugate a = SparseMatrix(((1, 2 + I), (3, 4))) assert a.C == SparseMatrix([ [1, 2 - I], [3, 4] ]) # mul assert a*Matrix(2, 2, [1, 0, 0, 1]) == a assert a + Matrix(2, 2, [1, 1, 1, 1]) == SparseMatrix([ [2, 3 + I], [4, 5] ]) assert a*0 == Matrix([[0, 0], [0, 0]]) # col join assert a.col_join(sparse_eye(2)) == SparseMatrix([ [1, 2 + I], [3, 4], [1, 0], [0, 1] ]) A = SparseMatrix(ones(3)) B = eye(3) assert A.col_join(B) == Matrix([[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 0, 0], [0, 1, 0], [0, 0, 1]]) # row join A = SparseMatrix(((1, 0, 1), (0, 1, 0), (1, 1, 0))) B = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 1))) assert A.row_join(B) == Matrix([[1, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0], [1, 1, 0, 0, 0, 1]]) # symmetric assert not a.is_symmetric(simplify=False) assert sparse_eye(3).is_symmetric(simplify=False) # test_cofactor assert sparse_eye(3) == sparse_eye(3).cofactorMatrix() test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]]) assert test.cofactorMatrix() == \ SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]]) test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) assert test.cofactorMatrix() == \ SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]]) # test_jacobian L = SparseMatrix(1, 2, [x**2*y, 2*y**2 + x*y]) syms = [x, y] assert L.jacobian(syms) == Matrix([[2*x*y, x**2], [y, 4*y + x]]) L = SparseMatrix(1, 2, [x, x**2*y**3]) assert L.jacobian(syms) == SparseMatrix([[1, 0], [2*x*y**3, x**2*3*y**2]]) # test_QR A = Matrix([[1, 2], [2, 3]]) Q, S = A.QRdecomposition() R = Rational assert Q == Matrix([ [ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)], [2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]]) assert S == Matrix([ [5**R(1, 2), 8*5**R(-1, 2)], [ 0, (R(1)/5)**R(1, 2)]]) assert Q*S == A assert Q.T * Q == sparse_eye(2) R = Rational # test nullspace # first test reduced row-ech form M = SparseMatrix([[5, 7, 2, 1], [1, 6, 2, -1]]) out, tmp = M.rref() assert out == Matrix([[1, 0, -R(2)/23, R(13)/23], [0, 1, R(8)/23, R(-6)/23]]) M = SparseMatrix([[ 1, 3, 0, 2, 6, 3, 1], [-2, -6, 0, -2, -8, 3, 1], [ 3, 9, 0, 0, 6, 6, 2], [-1, -3, 0, 1, 0, 9, 3]]) out, tmp = M.rref() assert out == Matrix([[1, 3, 0, 0, 2, 0, 0], [0, 0, 0, 1, 2, 0, 0], [0, 0, 0, 0, 0, 1, R(1)/3], [0, 0, 0, 0, 0, 0, 0]]) # now check the vectors basis = M.nullspace() assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0]) assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0]) assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0]) assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1]) # test eigen sparse_eye3 = sparse_eye(3) assert sparse_eye3.charpoly(x) == PurePoly(((x - 1)**3)) assert sparse_eye3.charpoly(y) == PurePoly(((y - 1)**3)) # test values M = Matrix([( 0, 1, -1), ( 1, 1, 0), (-1, 0, 1)]) vals = M.eigenvals() assert sorted(vals) == [-1, 1, 2] R = Rational M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) assert M.eigenvects() == [(1, 3, [ Matrix([1, 0, 0]), Matrix([0, 1, 0]), Matrix([0, 0, 1])])] M = Matrix([[5, 0, 2], [3, 2, 0], [0, 0, 1]]) assert M.eigenvects() == [(1, 1, [Matrix([R(-1)/2, R(3)/2, 1])]), (2, 1, [Matrix([0, 1, 0])]), (5, 1, [Matrix([1, 1, 0])])] assert M.zeros(3, 5) == SparseMatrix(3, 5, {}) A = SparseMatrix(10, 10, {(0, 0): 18, (0, 9): 12, (1, 4): 18, (2, 7): 16, (3, 9): 12, (4, 2): 19, (5, 7): 16, (6, 2): 12, (9, 7): 18}) assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16), (3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12), (9, 7, 18)] assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18), (2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12), (3, 9, 12)] assert SparseMatrix.eye(2).nnz() == 2 M = SparseMatrix.eye(3)*2 M[1, 0] = -1 M.col_op(1, lambda v, i: v + 2*M[i, 0]) assert M == Matrix([[ 2, 4, 0], [-1, 0, 0], [ 0, 0, 2]]) M = SparseMatrix.zeros(3) M.fill(1) assert M == ones(3) assert SparseMatrix(ones(0, 3)).tolist() == []