a RG5dN@sddlZddlmZddlmZddlmZddlmZddl m Z m Z ddl m Z ddlmZdd lmZd d lmZd d lmZmZd d lmZd dlmZmZd dlmZmZddZGdddeZ ddZ!Gddde eZ"e"Z#Z$e%fddZ&e%fddZ'ddZ(ddZ)d d!Z*e d"d#d$d%Z+dCd'd(Z,d)d*Z-d&d+d,d-d.Z.dDd/d0Z/dEd2d3Z0d4d5Z1d6d7Z2d8d9Z3dFd>> 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_mutableTcCs t||dSN) hermitian)rr r1rrrcholeskyOszDenseMatrix.choleskycCs t||dSr0)rr2rrrLDLdecompositionRszDenseMatrix.LDLdecompositioncCs t||SN)rr rhsrrrlower_triangular_solveUsz"DenseMatrix.lower_triangular_solvecCs t||Sr5)rr6rrrupper_triangular_solveXsz"DenseMatrix.upper_triangular_solveN)T)T)__name__ __module__ __qualname____doc__ is_MatrixExpr _op_priority_class_prioritypropertyr!r(r-r/r3r4r8r9rrrrrrrrrs"   rcCsVt|ddr|St|tr"|St|drR|}t|jdkrJt|St |S|S)z0Return a matrix as a Matrix, otherwise return x. is_MatrixF __array__r) getattrr/ isinstancerhasattrrClenshaperr.)rarrr_force_mutableas   rJc@seZdZddZdS)MutableDenseMatrixcKsBddlm}|D]$\\}}}||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 r)simplifyN)sympy.simplify.simplifyrLtodokitems)r r' _simplifyijelementrrrrLqs zMutableDenseMatrix.simplifyN)r:r;r<rLrrrrrKosrKcCs8ddlm}|t||}t|D]\}}|||<q"|S)zmConverts Python list of SymPy expressions to a NumPy array. See Also ======== matrix2numpy rempty)numpyrUrG enumerate)ldtyperUrIrQsrrr list2numpys   r[cCsPddlm}||j|}t|jD](}t|jD]}|||f|||f<q0q"|S)zYConverts SymPy's matrix to a NumPy array. See Also ======== list2numpy rrT)rVrUrHrangerowscols)mrYrUrIrQrRrrr matrix2numpys   r`cCs0t|}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, 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.thetactstZlilrrr rot_axis3s" rfcCs0t|}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, 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 rrarbrrr rot_axis2s" rgcCs0t|}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, 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 rrrrarbrrr rot_axis1s" rh)rV)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)rUndindex)rYz%s_%s_)rVrUrjobjectrjoinmapstr)prefixrHr'rUrjarrindexrrrsymarray%sA   rsTcsHttt|s"fdd}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||Sr5subsrQrRnseqsrrzcasoratian..cs||Sr5rtrvrwrrrzr{)listrnrrGr.det)ryrxzerofkrrwr casoratianrs rcOstj|i|S)z`Create square identity matrix n x n See Also ======== diag zeros ones )r.eyeargsr'rrrrs rFstrictunpackcOstj|||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 r)r.diag)rrvaluesr'rrrrs"rcCstj||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)rKZ orthogonalize)vlistZ orthonormalrrr GramSchmidts$rrc Cs`t|tr8d|jvrtd|jdkr,|j}|d}t|rVt|}|s^tdnt dt |dstt d|t|}||}t |}t |D]F\}}t |dst d|t |D]} ||| ||| |f<qqt |D]<} t | |D],} ||| || || || |f<qqt |D]0} t | d|D]} || | f|| | f<q<q*|S)a 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 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)rErrHr r^Ttolistr rG ValueErrorrDzerosrWr\r) rvarlist constraintsrxr_NoutrgrQrRrrrhessians8,           , rcCstj||dS)z 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]]) )size eigenvalue)r. jordan_block)eigenvalrxrrr jordan_cellFsrcCs ||S)aReturn 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__ )multiply_elementwise)ABrrrmatrix_multiply_elementwiseZsrcOs&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 cr^)popr.onesrrrrrms rcdc Cs|p t|}|dur|}|r6||kr6td||ft||}|dkrf||tt||d}t||} |s|D]&} t| |\} } | ||| | | f<qxn@|D]:} t| |\} } | | kr| ||| | | f<| | | f<q| 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 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) randomRandomrr\sampleintrGrdivmodrandint) rrminmaxseed symmetricpercentprngijr_ijkrQrRrrr randMatrixs$1   "rbareisscsDddDt}|dkr$tjSt||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 cSsg|] }t|qSrr).0rrrr r{zwronskian..rcs||Sr5)rrv functionsvarrrrzr{zwronskian..)rGrOner.r})rrr"rxWrrr wronskians rcOs&d|vr|d|d<tj|i|S)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 rr^)rr.rrrrrrs r)T)F)r)NrrNFrN)r)7rsympy.core.basicrZsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr(sympy.functions.elementary.trigonometricrrsympy.utilities.decoratorr sympy.utilities.exceptionsr sympy.utilities.iterablesr commonr ZdecompositionsrrmatricesrZ repmatrixrrsolversrrrrrJrK MutableMatrixr.rlr[r`rfrgrhrsrrrrrrrrrrrrrrrsL         I  *** L +% ) M L