import random from sympy.core.basic import Basic from sympy.core.singleton import S from sympy.core.symbol import Symbol from sympy.core.sympify import sympify from sympy.functions.elementary.trigonometric import cos, sin from sympy.utilities.decorator import doctest_depends_on from sympy.utilities.exceptions import sympy_deprecation_warning from sympy.utilities.iterables import is_sequence from .common import ShapeError from .decompositions import _cholesky, _LDLdecomposition from .matrices import MatrixBase from .repmatrix import MutableRepMatrix, RepMatrix from .solvers import _lower_triangular_solve, _upper_triangular_solve def _iszero(x): """Returns True if x is zero.""" return x.is_zero class DenseMatrix(RepMatrix): """Matrix implementation based on DomainMatrix as the internal representation""" # # DenseMatrix is a superclass for both MutableDenseMatrix and # ImmutableDenseMatrix. Methods shared by both classes but not for the # Sparse classes should be implemented here. # is_MatrixExpr = False # type: bool _op_priority = 10.01 _class_priority = 4 @property def _mat(self): sympy_deprecation_warning( """ The private _mat attribute of Matrix is deprecated. Use the .flat() method instead. """, deprecated_since_version="1.9", active_deprecations_target="deprecated-private-matrix-attributes" ) return self.flat() def _eval_inverse(self, **kwargs): return self.inv(method=kwargs.get('method', 'GE'), iszerofunc=kwargs.get('iszerofunc', _iszero), try_block_diag=kwargs.get('try_block_diag', False)) def as_immutable(self): """Returns an Immutable version of this Matrix """ from .immutable import ImmutableDenseMatrix as cls return cls._fromrep(self._rep.copy()) def as_mutable(self): """Returns a mutable version of this matrix Examples ======== >>> from sympy import ImmutableMatrix >>> X = ImmutableMatrix([[1, 2], [3, 4]]) >>> Y = X.as_mutable() >>> Y[1, 1] = 5 # Can set values in Y >>> Y Matrix([ [1, 2], [3, 5]]) """ return Matrix(self) def cholesky(self, hermitian=True): return _cholesky(self, hermitian=hermitian) def LDLdecomposition(self, hermitian=True): return _LDLdecomposition(self, hermitian=hermitian) def lower_triangular_solve(self, rhs): return _lower_triangular_solve(self, rhs) def upper_triangular_solve(self, rhs): return _upper_triangular_solve(self, rhs) cholesky.__doc__ = _cholesky.__doc__ LDLdecomposition.__doc__ = _LDLdecomposition.__doc__ lower_triangular_solve.__doc__ = _lower_triangular_solve.__doc__ upper_triangular_solve.__doc__ = _upper_triangular_solve.__doc__ def _force_mutable(x): """Return a matrix as a Matrix, otherwise return x.""" if getattr(x, 'is_Matrix', False): return x.as_mutable() elif isinstance(x, Basic): return x elif hasattr(x, '__array__'): a = x.__array__() if len(a.shape) == 0: return sympify(a) return Matrix(x) return x class MutableDenseMatrix(DenseMatrix, MutableRepMatrix): def simplify(self, **kwargs): """Applies simplify to the elements of a matrix in place. This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure)) See Also ======== sympy.simplify.simplify.simplify """ from sympy.simplify.simplify import simplify as _simplify for (i, j), element in self.todok().items(): self[i, j] = _simplify(element, **kwargs) MutableMatrix = Matrix = MutableDenseMatrix ########### # Numpy Utility Functions: # list2numpy, matrix2numpy, symmarray ########### def list2numpy(l, dtype=object): # pragma: no cover """Converts Python list of SymPy expressions to a NumPy array. See Also ======== matrix2numpy """ from numpy import empty a = empty(len(l), dtype) for i, s in enumerate(l): a[i] = s return a def matrix2numpy(m, dtype=object): # pragma: no cover """Converts SymPy's matrix to a NumPy array. See Also ======== list2numpy """ from numpy import empty a = empty(m.shape, dtype) for i in range(m.rows): for j in range(m.cols): a[i, j] = m[i, j] return a ########### # Rotation matrices: # rot_givens, rot_axis[123], rot_ccw_axis[123] ########### def rot_givens(i, j, theta, dim=3): r"""Returns a a Givens rotation matrix, a a rotation in the plane spanned by two coordinates axes. Explanation =========== The Givens rotation corresponds to a generalization of rotation matrices to any number of dimensions, given by: .. math:: G(i, j, \theta) = \begin{bmatrix} 1 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & & \vdots & & \vdots \\ 0 & \cdots & c & \cdots & -s & \cdots & 0 \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ 0 & \cdots & s & \cdots & c & \cdots & 0 \\ \vdots & & \vdots & & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \cdots & 0 & \cdots & 1 \end{bmatrix} Where $c = \cos(\theta)$ and $s = \sin(\theta)$ appear at the intersections ``i``\th and ``j``\th rows and columns. For fixed ``i > j``\, the non-zero elements of a Givens matrix are given by: - $g_{kk} = 1$ for $k \ne i,\,j$ - $g_{kk} = c$ for $k = i,\,j$ - $g_{ji} = -g_{ij} = -s$ Parameters ========== i : int between ``0`` and ``dim - 1`` Represents first axis j : int between ``0`` and ``dim - 1`` Represents second axis dim : int bigger than 1 Number of dimentions. Defaults to 3. Examples ======== >>> from sympy import pi, rot_givens A counterclockwise rotation of pi/3 (60 degrees) around the third axis (z-axis): >>> rot_givens(1, 0, pi/3) Matrix([ [ 1/2, -sqrt(3)/2, 0], [sqrt(3)/2, 1/2, 0], [ 0, 0, 1]]) If we rotate by pi/2 (90 degrees): >>> rot_givens(1, 0, pi/2) Matrix([ [0, -1, 0], [1, 0, 0], [0, 0, 1]]) This can be generalized to any number of dimensions: >>> rot_givens(1, 0, pi/2, dim=4) Matrix([ [0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]) References ========== .. [1] https://en.wikipedia.org/wiki/Givens_rotation See Also ======== rot_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (clockwise around the x axis) rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (clockwise around the y axis) rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (clockwise around the z axis) rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (counterclockwise around the x axis) rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (counterclockwise around the y axis) rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (counterclockwise around the z axis) """ if not isinstance(dim, int) or dim < 2: raise ValueError('dim must be an integer biggen than one, ' 'got {}.'.format(dim)) if i == j: raise ValueError('i and j must be different, ' 'got ({}, {})'.format(i, j)) for ij in [i, j]: if not isinstance(ij, int) or ij < 0 or ij > dim - 1: raise ValueError('i and j must be integers between 0 and ' '{}, got i={} and j={}.'.format(dim-1, i, j)) theta = sympify(theta) c = cos(theta) s = sin(theta) M = eye(dim) M[i, i] = c M[j, j] = c M[i, j] = s M[j, i] = -s return M def rot_axis3(theta): r"""Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis. Explanation =========== For a right-handed coordinate system, this corresponds to a clockwise rotation around the `z`-axis, given by: .. math:: R = \begin{bmatrix} \cos(\theta) & \sin(\theta) & 0 \\ -\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix} Examples ======== >>> from sympy import pi, rot_axis3 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_axis3(theta) Matrix([ [ 1/2, sqrt(3)/2, 0], [-sqrt(3)/2, 1/2, 0], [ 0, 0, 1]]) If we rotate by pi/2 (90 degrees): >>> rot_axis3(pi/2) Matrix([ [ 0, 1, 0], [-1, 0, 0], [ 0, 0, 1]]) See Also ======== rot_givens: Returns a Givens rotation matrix (generalized rotation for any number of dimensions) rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (counterclockwise around the z axis) rot_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (clockwise around the x axis) rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (clockwise around the y axis) """ return rot_givens(0, 1, theta, dim=3) def rot_axis2(theta): r"""Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis. Explanation =========== For a right-handed coordinate system, this corresponds to a clockwise rotation around the `y`-axis, given by: .. math:: R = \begin{bmatrix} \cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & \cos(\theta) \end{bmatrix} Examples ======== >>> from sympy import pi, rot_axis2 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_axis2(theta) Matrix([ [ 1/2, 0, -sqrt(3)/2], [ 0, 1, 0], [sqrt(3)/2, 0, 1/2]]) If we rotate by pi/2 (90 degrees): >>> rot_axis2(pi/2) Matrix([ [0, 0, -1], [0, 1, 0], [1, 0, 0]]) See Also ======== rot_givens: Returns a Givens rotation matrix (generalized rotation for any number of dimensions) rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (clockwise around the y axis) rot_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (counterclockwise around the x axis) rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (counterclockwise around the z axis) """ return rot_givens(2, 0, theta, dim=3) def rot_axis1(theta): r"""Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis. Explanation =========== For a right-handed coordinate system, this corresponds to a clockwise rotation around the `x`-axis, given by: .. math:: R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & \sin(\theta) \\ 0 & -\sin(\theta) & \cos(\theta) \end{bmatrix} Examples ======== >>> from sympy import pi, rot_axis1 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_axis1(theta) Matrix([ [1, 0, 0], [0, 1/2, sqrt(3)/2], [0, -sqrt(3)/2, 1/2]]) If we rotate by pi/2 (90 degrees): >>> rot_axis1(pi/2) Matrix([ [1, 0, 0], [0, 0, 1], [0, -1, 0]]) See Also ======== rot_givens: Returns a Givens rotation matrix (generalized rotation for any number of dimensions) rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (counterclockwise around the x axis) rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (clockwise around the y axis) rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (clockwise around the z axis) """ return rot_givens(1, 2, theta, dim=3) def rot_ccw_axis3(theta): r"""Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis. Explanation =========== For a right-handed coordinate system, this corresponds to a counterclockwise rotation around the `z`-axis, given by: .. math:: R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix} Examples ======== >>> from sympy import pi, rot_ccw_axis3 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_ccw_axis3(theta) Matrix([ [ 1/2, -sqrt(3)/2, 0], [sqrt(3)/2, 1/2, 0], [ 0, 0, 1]]) If we rotate by pi/2 (90 degrees): >>> rot_ccw_axis3(pi/2) Matrix([ [0, -1, 0], [1, 0, 0], [0, 0, 1]]) See Also ======== rot_givens: Returns a Givens rotation matrix (generalized rotation for any number of dimensions) rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (clockwise around the z axis) rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (counterclockwise around the x axis) rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (counterclockwise around the y axis) """ return rot_givens(1, 0, theta, dim=3) def rot_ccw_axis2(theta): r"""Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis. Explanation =========== For a right-handed coordinate system, this corresponds to a counterclockwise rotation around the `y`-axis, given by: .. math:: R = \begin{bmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{bmatrix} Examples ======== >>> from sympy import pi, rot_ccw_axis2 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_ccw_axis2(theta) Matrix([ [ 1/2, 0, sqrt(3)/2], [ 0, 1, 0], [-sqrt(3)/2, 0, 1/2]]) If we rotate by pi/2 (90 degrees): >>> rot_ccw_axis2(pi/2) Matrix([ [ 0, 0, 1], [ 0, 1, 0], [-1, 0, 0]]) See Also ======== rot_givens: Returns a Givens rotation matrix (generalized rotation for any number of dimensions) rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (clockwise around the y axis) rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (counterclockwise around the x axis) rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (counterclockwise around the z axis) """ return rot_givens(0, 2, theta, dim=3) def rot_ccw_axis1(theta): r"""Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis. Explanation =========== For a right-handed coordinate system, this corresponds to a counterclockwise rotation around the `x`-axis, given by: .. math:: R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{bmatrix} Examples ======== >>> from sympy import pi, rot_ccw_axis1 A rotation of pi/3 (60 degrees): >>> theta = pi/3 >>> rot_ccw_axis1(theta) Matrix([ [1, 0, 0], [0, 1/2, -sqrt(3)/2], [0, sqrt(3)/2, 1/2]]) If we rotate by pi/2 (90 degrees): >>> rot_ccw_axis1(pi/2) Matrix([ [1, 0, 0], [0, 0, -1], [0, 1, 0]]) See Also ======== rot_givens: Returns a Givens rotation matrix (generalized rotation for any number of dimensions) rot_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis (clockwise around the x axis) rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis (counterclockwise around the y axis) rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis (counterclockwise around the z axis) """ return rot_givens(2, 1, theta, dim=3) @doctest_depends_on(modules=('numpy',)) def symarray(prefix, shape, **kwargs): # pragma: no cover r"""Create a numpy ndarray of symbols (as an object array). The created symbols are named ``prefix_i1_i2_``... You should thus provide a non-empty prefix if you want your symbols to be unique for different output arrays, as SymPy symbols with identical names are the same object. Parameters ---------- prefix : string A prefix prepended to the name of every symbol. shape : int or tuple Shape of the created array. If an int, the array is one-dimensional; for more than one dimension the shape must be a tuple. \*\*kwargs : dict keyword arguments passed on to Symbol Examples ======== These doctests require numpy. >>> from sympy import symarray >>> symarray('', 3) [_0 _1 _2] If you want multiple symarrays to contain distinct symbols, you *must* provide unique prefixes: >>> a = symarray('', 3) >>> b = symarray('', 3) >>> a[0] == b[0] True >>> a = symarray('a', 3) >>> b = symarray('b', 3) >>> a[0] == b[0] False Creating symarrays with a prefix: >>> symarray('a', 3) [a_0 a_1 a_2] For more than one dimension, the shape must be given as a tuple: >>> symarray('a', (2, 3)) [[a_0_0 a_0_1 a_0_2] [a_1_0 a_1_1 a_1_2]] >>> symarray('a', (2, 3, 2)) [[[a_0_0_0 a_0_0_1] [a_0_1_0 a_0_1_1] [a_0_2_0 a_0_2_1]] [[a_1_0_0 a_1_0_1] [a_1_1_0 a_1_1_1] [a_1_2_0 a_1_2_1]]] For setting assumptions of the underlying Symbols: >>> [s.is_real for s in symarray('a', 2, real=True)] [True, True] """ from numpy import empty, ndindex arr = empty(shape, dtype=object) for index in ndindex(shape): arr[index] = Symbol('%s_%s' % (prefix, '_'.join(map(str, index))), **kwargs) return arr ############### # Functions ############### def casoratian(seqs, n, zero=True): """Given linear difference operator L of order 'k' and homogeneous equation Ly = 0 we want to compute kernel of L, which is a set of 'k' sequences: a(n), b(n), ... z(n). Solutions of L are linearly independent iff their Casoratian, denoted as C(a, b, ..., z), do not vanish for n = 0. Casoratian is defined by k x k determinant:: + a(n) b(n) . . . z(n) + | a(n+1) b(n+1) . . . z(n+1) | | . . . . | | . . . . | | . . . . | + a(n+k-1) b(n+k-1) . . . z(n+k-1) + It proves very useful in rsolve_hyper() where it is applied to a generating set of a recurrence to factor out linearly dependent solutions and return a basis: >>> from sympy import Symbol, casoratian, factorial >>> n = Symbol('n', integer=True) Exponential and factorial are linearly independent: >>> casoratian([2**n, factorial(n)], n) != 0 True """ seqs = list(map(sympify, seqs)) if not zero: f = lambda i, j: seqs[j].subs(n, n + i) else: f = lambda i, j: seqs[j].subs(n, i) k = len(seqs) return Matrix(k, k, f).det() def eye(*args, **kwargs): """Create square identity matrix n x n See Also ======== diag zeros ones """ return Matrix.eye(*args, **kwargs) def diag(*values, strict=True, unpack=False, **kwargs): """Returns a matrix with the provided values placed on the diagonal. If non-square matrices are included, they will produce a block-diagonal matrix. Examples ======== This version of diag is a thin wrapper to Matrix.diag that differs in that it treats all lists like matrices -- even when a single list is given. If this is not desired, either put a `*` before the list or set `unpack=True`. >>> from sympy import diag >>> diag([1, 2, 3], unpack=True) # = diag(1,2,3) or diag(*[1,2,3]) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> diag([1, 2, 3]) # a column vector Matrix([ [1], [2], [3]]) See Also ======== .common.MatrixCommon.eye .common.MatrixCommon.diagonal .common.MatrixCommon.diag .expressions.blockmatrix.BlockMatrix """ return Matrix.diag(*values, strict=strict, unpack=unpack, **kwargs) def GramSchmidt(vlist, orthonormal=False): """Apply the Gram-Schmidt process to a set of vectors. Parameters ========== vlist : List of Matrix Vectors to be orthogonalized for. orthonormal : Bool, optional If true, return an orthonormal basis. Returns ======= vlist : List of Matrix Orthogonalized vectors Notes ===== This routine is mostly duplicate from ``Matrix.orthogonalize``, except for some difference that this always raises error when linearly dependent vectors are found, and the keyword ``normalize`` has been named as ``orthonormal`` in this function. See Also ======== .matrices.MatrixSubspaces.orthogonalize References ========== .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process """ return MutableDenseMatrix.orthogonalize( *vlist, normalize=orthonormal, rankcheck=True ) def hessian(f, varlist, constraints=()): """Compute Hessian matrix for a function f wrt parameters in varlist which may be given as a sequence or a row/column vector. A list of constraints may optionally be given. Examples ======== >>> from sympy import Function, hessian, pprint >>> from sympy.abc import x, y >>> f = Function('f')(x, y) >>> g1 = Function('g')(x, y) >>> g2 = x**2 + 3*y >>> pprint(hessian(f, (x, y), [g1, g2])) [ d d ] [ 0 0 --(g(x, y)) --(g(x, y)) ] [ dx dy ] [ ] [ 0 0 2*x 3 ] [ ] [ 2 2 ] [d d d ] [--(g(x, y)) 2*x ---(f(x, y)) -----(f(x, y))] [dx 2 dy dx ] [ dx ] [ ] [ 2 2 ] [d d d ] [--(g(x, y)) 3 -----(f(x, y)) ---(f(x, y)) ] [dy dy dx 2 ] [ dy ] References ========== .. [1] https://en.wikipedia.org/wiki/Hessian_matrix See Also ======== sympy.matrices.matrices.MatrixCalculus.jacobian wronskian """ # f is the expression representing a function f, return regular matrix if isinstance(varlist, MatrixBase): if 1 not in varlist.shape: raise ShapeError("`varlist` must be a column or row vector.") if varlist.cols == 1: varlist = varlist.T varlist = varlist.tolist()[0] if is_sequence(varlist): n = len(varlist) if not n: raise ShapeError("`len(varlist)` must not be zero.") else: raise ValueError("Improper variable list in hessian function") if not getattr(f, 'diff'): # check differentiability raise ValueError("Function `f` (%s) is not differentiable" % f) m = len(constraints) N = m + n out = zeros(N) for k, g in enumerate(constraints): if not getattr(g, 'diff'): # check differentiability raise ValueError("Function `f` (%s) is not differentiable" % f) for i in range(n): out[k, i + m] = g.diff(varlist[i]) for i in range(n): for j in range(i, n): out[i + m, j + m] = f.diff(varlist[i]).diff(varlist[j]) for i in range(N): for j in range(i + 1, N): out[j, i] = out[i, j] return out def jordan_cell(eigenval, n): """ Create a Jordan block: Examples ======== >>> from sympy import jordan_cell >>> from sympy.abc import x >>> jordan_cell(x, 4) Matrix([ [x, 1, 0, 0], [0, x, 1, 0], [0, 0, x, 1], [0, 0, 0, x]]) """ return Matrix.jordan_block(size=n, eigenvalue=eigenval) def matrix_multiply_elementwise(A, B): """Return the Hadamard product (elementwise product) of A and B >>> from sympy import Matrix, matrix_multiply_elementwise >>> A = Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = Matrix([[1, 10, 100], [100, 10, 1]]) >>> matrix_multiply_elementwise(A, B) Matrix([ [ 0, 10, 200], [300, 40, 5]]) See Also ======== sympy.matrices.common.MatrixCommon.__mul__ """ return A.multiply_elementwise(B) def ones(*args, **kwargs): """Returns a matrix of ones with ``rows`` rows and ``cols`` columns; if ``cols`` is omitted a square matrix will be returned. See Also ======== zeros eye diag """ if 'c' in kwargs: kwargs['cols'] = kwargs.pop('c') return Matrix.ones(*args, **kwargs) def randMatrix(r, c=None, min=0, max=99, seed=None, symmetric=False, percent=100, prng=None): """Create random matrix with dimensions ``r`` x ``c``. If ``c`` is omitted the matrix will be square. If ``symmetric`` is True the matrix must be square. If ``percent`` is less than 100 then only approximately the given percentage of elements will be non-zero. The pseudo-random number generator used to generate matrix is chosen in the following way. * If ``prng`` is supplied, it will be used as random number generator. It should be an instance of ``random.Random``, or at least have ``randint`` and ``shuffle`` methods with same signatures. * if ``prng`` is not supplied but ``seed`` is supplied, then new ``random.Random`` with given ``seed`` will be created; * otherwise, a new ``random.Random`` with default seed will be used. Examples ======== >>> from sympy import randMatrix >>> randMatrix(3) # doctest:+SKIP [25, 45, 27] [44, 54, 9] [23, 96, 46] >>> randMatrix(3, 2) # doctest:+SKIP [87, 29] [23, 37] [90, 26] >>> randMatrix(3, 3, 0, 2) # doctest:+SKIP [0, 2, 0] [2, 0, 1] [0, 0, 1] >>> randMatrix(3, symmetric=True) # doctest:+SKIP [85, 26, 29] [26, 71, 43] [29, 43, 57] >>> A = randMatrix(3, seed=1) >>> B = randMatrix(3, seed=2) >>> A == B False >>> A == randMatrix(3, seed=1) True >>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP [77, 70, 0], [70, 0, 0], [ 0, 0, 88] """ # Note that ``Random()`` is equivalent to ``Random(None)`` prng = prng or random.Random(seed) if c is None: c = r if symmetric and r != c: raise ValueError('For symmetric matrices, r must equal c, but %i != %i' % (r, c)) ij = range(r * c) if percent != 100: ij = prng.sample(ij, int(len(ij)*percent // 100)) m = zeros(r, c) if not symmetric: for ijk in ij: i, j = divmod(ijk, c) m[i, j] = prng.randint(min, max) else: for ijk in ij: i, j = divmod(ijk, c) if i <= j: m[i, j] = m[j, i] = prng.randint(min, max) return m def wronskian(functions, var, method='bareiss'): """ Compute Wronskian for [] of functions :: | f1 f2 ... fn | | f1' f2' ... fn' | | . . . . | W(f1, ..., fn) = | . . . . | | . . . . | | (n) (n) (n) | | D (f1) D (f2) ... D (fn) | see: https://en.wikipedia.org/wiki/Wronskian See Also ======== sympy.matrices.matrices.MatrixCalculus.jacobian hessian """ functions = [sympify(f) for f in functions] n = len(functions) if n == 0: return S.One W = Matrix(n, n, lambda i, j: functions[i].diff(var, j)) return W.det(method) def zeros(*args, **kwargs): """Returns a matrix of zeros with ``rows`` rows and ``cols`` columns; if ``cols`` is omitted a square matrix will be returned. See Also ======== ones eye diag """ if 'c' in kwargs: kwargs['cols'] = kwargs.pop('c') return Matrix.zeros(*args, **kwargs)