o 8Va0O@sddlZddlmZddlmZddlmZddlmZddl m Z m Z ddl m ZddlmZdd lmZd d lmZd d lmZmZd d lmZd dlmZmZd dlmZmZddZ GdddeZ!ddZ"Gddde!eZ#e#Z$Z%e&fddZ'e&fddZ(ddZ)ddZ*d d!Z+ed"d#d$d%Z,dBd'd(Z-d)d*Z.d&d+d,d-d.Z/dCd/d0Z0gfd1d2Z1d3d4Z2d5d6Z3d7d8Z4 + dDd;d<Z5dEd>d?Z6d@dAZ7dS)FN)Basic) is_sequence)Symbol)sympify)cossin)simplify)doctest_depends_on)SymPyDeprecationWarning) ShapeError) _cholesky_LDLdecomposition) MatrixBase)MutableRepMatrix RepMatrix)_lower_triangular_solve_upper_triangular_solvecCs|jS)zReturns True if x is zero.)Zis_zero)xr6/usr/lib/python3/dist-packages/sympy/matrices/dense.py_iszerosrc@seZdZdZdZdZdZeddZddZ d d Z d d Z dddZ dddZ ddZddZeje _eje _eje_eje_dS) DenseMatrixzJMatrix implementation based on DomainMatrix as the internal representationFgQ$@cCstddddd|S)Nz$The private _mat attribute of Matrixzthe .flat() methodiTz1.9)ZfeatureZ useinsteadZissueZdeprecated_since_version)r warnZflatselfrrr_mat&szDenseMatrix._matcKs(|j|dd|dt|dddS)NmethodZGE iszerofunctry_block_diagF)rrr )invgetr)rkwargsrrr _eval_inverse1s  zDenseMatrix._eval_inversecCsddlm}||jS)z4Returns an Immutable version of this Matrix r )ImmutableDenseMatrix)Z immutabler%Z_fromrepZ_repcopy)rclsrrr as_immutable6s zDenseMatrix.as_immutablecCst|S)aBReturns 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]]) )Matrixrrrr as_mutable<szDenseMatrix.as_mutableTcC t||dSN) hermitian)r rr-rrrcholeskyM zDenseMatrix.choleskycCr+r,)rr.rrrLDLdecompositionPr0zDenseMatrix.LDLdecompositioncC t||SN)rrZrhsrrrlower_triangular_solveS z"DenseMatrix.lower_triangular_solvecCr2r3)rr4rrrupper_triangular_solveVr6z"DenseMatrix.upper_triangular_solveNT)__name__ __module__ __qualname____doc__Z is_MatrixExprZ _op_priorityZ_class_prioritypropertyrr$r(r*r/r1r5r7r rrrrrrrrs$     rcCsVt|ddr |St|tr|St|dr)|}t|jdkr%t|St |S|S)z0Return a matrix as a Matrix, otherwise return x.Z is_MatrixF __array__r) getattrr* isinstancerhasattrr>lenshaperr))rarrr_force_mutable_s   rEc@seZdZddZdS)MutableDenseMatrixcKs6|D]\\}}}t|fi||||f<qdS)zApplies 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 N)Ztodokitems _simplify)rr#ijelementrrrros zMutableDenseMatrix.simplifyN)r9r:r;rrrrrrFms rFcCs8ddlm}|t||}t|D]\}}|||<q|S)zmConverts python list of SymPy expressions to a NumPy array. See Also ======== matrix2numpy rempty)numpyrMrB enumerate)ldtyperMrDrIsrrr list2numpys  rScCsPddlm}||j|}t|jD]}t|jD] }|||f|||f<qq|S)zYConverts SymPy's matrix to a NumPy array. See Also ======== list2numpy rrL)rNrMrCrangeZrowscols)mrQrMrDrIrJrrr matrix2numpys  rWcCs0t|}t|}||df| |dfdf}t|S)aReturns a rotation matrix for a rotation of theta (in radians) about the 3-axis. Examples ======== >>> from sympy import pi >>> from sympy.matrices import 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_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis rot_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis r)rrr rrr)ZthetaZctstZlilrrr rot_axis3s# r[cCs0t|}t|}|d| fd|d|ff}t|S)aReturns a rotation matrix for a rotation of theta (in radians) about the 2-axis. Examples ======== >>> from sympy import pi >>> from sympy.matrices import 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_axis1: Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis r)rr rrXrYrrr rot_axis2s# r\cCs0t|}t|}dd||fd| |ff}t|S)aReturns a rotation matrix for a rotation of theta (in radians) about the 1-axis. Examples ======== >>> from sympy import pi >>> from sympy.matrices import 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_axis2: Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis rot_axis3: Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis )r rrrrXrYrrr rot_axis1s# r])rN)modulesc KsVddlm}m}||td}||D]}td|dtt|ffi|||<q|S)aICreate 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] r)rMndindex)rQz%s_%s_)rNrMr_objectrjoinmapstr)prefixrCr#rMr_Zarrindexrrrsymarray%sA   rgTcsHttt|sfdd}nfdd}t}t|||S)aYGiven 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 cs||Sr3ZsubsrIrJnseqsrrszcasoratian..cs||Sr3rhrirjrrrm)listrcrrBr)det)rlrkZzerofkrrjr casoratianrs rscOstj|i|S)z`Create square identity matrix n x n See Also ======== diag zeros ones )r)eyeargsr#rrrrts rtFstrictunpackcOstj|||d|S)a]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 - to extract a diagonal .common.MatrixCommon.diag .expressions.blockmatrix.BlockMatrix rw)r)diag)rxryvaluesr#rrrrzs"rzcCstj||ddS)a 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 T) normalizeZ rankcheck)rFZ orthogonalize)ZvlistZ orthonormalrrr GramSchmidts$r}c Cs\t|trd|jvrtd|jdkr|j}|d}t|r+t|}|s*tdnt dt |ds:t d|t|}||}t |}t |D]#\}}t |dsYt d|t |D]} ||| ||| |f<q]qJt |D]} t | |D]} ||| || || || |f<qyqrt |D]} t | d|D] } || | f|| | f<qq|S)aCompute 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 ========== https://en.wikipedia.org/wiki/Hessian_matrix See Also ======== sympy.matrices.matrices.MatrixCalculus.jacobian wronskian r z)`varlist` must be a column or row vector.rz `len(varlist)` must not be zero.z*Improper variable list in hessian functiondiffz'Function `f` (%s) is not differentiable)r@rrCr rUTtolistrrB ValueErrorr?zerosrOrTr~) rqZvarlistZ constraintsrkrVNoutrrgrIrJrrrhessians@ ,         * rcCstj||dS)z Create a Jordan block: Examples ======== >>> from sympy.matrices 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]]) )sizeZ eigenvalue)r)Z jordan_block)Zeigenvalrkrrr jordan_cellFsrcCs ||S)aReturn the Hadamard product (elementwise product) of A and B >>> from sympy.matrices import matrix_multiply_elementwise >>> from sympy.matrices import Matrix >>> 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__ )Zmultiply_elementwise)ABrrrmatrix_multiply_elementwiseZs rcO&d|vr |d|d<tj|i|S)zReturns 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 crU)popr)onesrurrrrn rcdc Cs|pt|}|dur |}|r||krtd||ft||}|dkr3||tt||d}t||} |sR|D]} t| |\} } | ||| | | f<q<| S|D]} t| |\} } | | krq| ||| | | f<| | | f<qT| S)aCreate 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.matrices 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] Nz4For symmetric matrices, r must equal c, but %i != %ir) randomZRandomrrTZsampleintrBrdivmodZrandint) rrminmaxZseedZ symmetricpercentZprngZijrVZijkrIrJrrr randMatrixs(1    rbareisscsXtdtD] }t||<qt}|dkrdSt||fdd}||S)ax 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 rr cs||Sr3)r~ri functionsvarrrrmrnzwronskian..)rTrBrr)rp)rrrrfrkWrrr wronskians rcOr)zReturns 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 rrU)rr)rrurrrrrrr8)F)NrrNFrN)r)8rZsympy.core.basicrZsympy.core.compatibilityrZsympy.core.symbolrZsympy.core.sympifyrZ(sympy.functions.elementary.trigonometricrrZsympy.simplify.simplifyrrHZsympy.utilities.decoratorr Zsympy.utilities.exceptionsr commonr Zdecompositionsr rZmatricesrZ repmatrixrrZsolversrrrrrErFZ MutableMatrixr)rarSrWr[r\r]rgrsrtrzr}rrrrrrrrrrrsN         G  +++  L+ % )M L