EdNddlZddlmZddlmZddlmZddlmZddl m Z m Z ddl m Z ddlmZdd lmZd d lmZd d lmZmZd d lmZd dlmZmZd dlmZmZdZGddeZ dZ!Gdde eZ"e"xZ#Z$e%fdZ&e%fdZ'dZ(dZ)dZ*e ddZ+d0dZ,d Z-dd!d"d#Z.d1d$Z/d2d&Z0d'Z1d(Z2d)Z3 d3d,Z4d4d.Z5d/Z6dS)5N)Basic)S)Symbolsympify)cossin)doctest_depends_on)sympy_deprecation_warning) is_sequence) ShapeError) _cholesky_LDLdecomposition) MatrixBase)MutableRepMatrix RepMatrix)_lower_triangular_solve_upper_triangular_solvec|jS)zReturns True if x is zero.)is_zero)xs 4lib/python3.11/site-packages/sympy/matrices/dense.py_iszerors 9ceZdZdZdZdZdZedZdZ dZ dZ dd Z dd Z d Zd Zeje _eje _eje_eje_dS) DenseMatrixzJMatrix implementation based on DomainMatrix as the internal representationFgQ$@cNtddd|S)Nzy The private _mat attribute of Matrix is deprecated. Use the .flat() method instead. z1.9z$deprecated-private-matrix-attributes)deprecated_since_versionactive_deprecations_target)r flatselfs r_matzDenseMatrix._mat&s6! &+'M     yy{{rc ||dd|dt|ddS)NmethodGE iszerofunctry_block_diagF)r'r)r*)invgetr)r$kwargss r _eval_inversezDenseMatrix._eval_inverse3sSxxvzz(D99#)::lG#D#D'-zz2BE'J'JLL Lrcfddlm}||jS)z4Returns an Immutable version of this Matrix r )ImmutableDenseMatrix) immutabler0_fromrep_repcopy)r$clss r as_immutablezDenseMatrix.as_immutable8s4 ;:::::||DINN,,---rc t|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]]) )Matrixr#s r as_mutablezDenseMatrix.as_mutable>sd||rTc$t||SN) hermitian)rr$r<s rcholeskyzDenseMatrix.choleskyOs3333rc$t||Sr;)rr=s rLDLdecompositionzDenseMatrix.LDLdecompositionRs ;;;;rc"t||SN)rr$rhss rlower_triangular_solvez"DenseMatrix.lower_triangular_solveU&tS111rc"t||SrB)rrCs rupper_triangular_solvez"DenseMatrix.upper_triangular_solveXrFrNT)__name__ __module__ __qualname____doc__ is_MatrixExpr _op_priority_class_prioritypropertyr%r.r6r9r>r@rErHrrrrrrrrsTTMLO   X LLL ... "4444<<<<222222&/%6H%6%>%<%D"%<%D"""rrc2t|ddr|St|tr|St |drJ|}t |jdkrt|St|S|S)z0Return a matrix as a Matrix, otherwise return x. is_MatrixF __array__r) getattrr9 isinstancerhasattrrUlenshaperr8)ras r_force_mutabler\asq+u%%||~~ Au   K  KKMM qw<<1  1:: ayy HrceZdZdZdS)MutableDenseMatrixc ddlm}|D]\\}}}||fi||||f<dS)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.simplifyr`todokitems)r$r- _simplifyijelements rr`zMutableDenseMatrix.simplifyqsm BAAAAA#zz||1133 6 6OFQG"755f55DAJJ 6 6rN)rJrKrLr`rRrrr^r^os# 6 6 6 6 6rr^cxddlm}|t||}t|D] \}}|||< |S)zmConverts Python list of SymPy expressions to a NumPy array. See Also ======== matrix2numpy rempty)numpyrjrY enumerate)ldtyperjr[ress r list2numpyrpsV c!ffeA! 1! Hrcddlm}||j|}t|jD](}t|jD]}|||f|||f<)|S)zYConverts SymPy's matrix to a NumPy array. See Also ======== list2numpy rri)rkrjrZrangerowscols)mrnrjr[rerfs r matrix2numpyrvsx aguA 16]]qv  A1gAadGG  Hrctt|}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 rr r8thetactstlils r rot_axis3r~sGD UB UB A; CQ<  C #;;rctt|}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 rrxrys r rot_axis2rsGD UB UB B3<  2; C #;;rctt|}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 rrrrxrys r rot_axis1rsGD UB UB  r2; sB< C #;;r)rk)modulesc ddlm}m}||t}||D]=}t |ddt t|fi|||<>|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)rjndindex)rn_)rkrjrobjectrjoinmapstr)prefixrZr-rjrarrindexs rsymarrayr%sB%$$$$$$$ %V $ $ $C&&vvvsxxC/H/H/HI&&$&&E JrTcttt|sfd}nfd}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 cB||zSrBsubsrerfnseqss rzcasoratian..saaQ//rc<||SrBrrs rrzcasoratian..saa++r)listrrrYr8det)rrzerofks`` r casoratianrrsp> GT"" # #D , / / / / / + + + + + D A !Q??    rc$tj|i|S)z`Create square identity matrix n x n See Also ======== diag zeros ones )r8eyeargsr-s rrrs :t &v & &&rFstrictunpackc*tj|||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)r8diag)rrvaluesr-s rrrs!D ;vf G G G GGrc&tj||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) normalize rankcheck)r^ orthogonalize)vlist orthonormals r GramSchmidtrs&H  + +   rrRct|trDd|jvrtd|jdkr|j}|d}t|r!t|}|stdntdt|dstd|zt|}||z}t|}t|D]\\}}t|dstd|zt|D]%} ||| ||| |zf<&]t|D]T} t| |D]A} ||| || || |z| |zf<BUt|D]'} t| dz|D]} || | f|| | f<(|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)rWrrZrrtTtolistr rY ValueErrorrVzerosrlrrr) rvarlist constraintsrruNoutrgrerfs rhessianrs4X':&&& GM ! JHII I <1  iG..""1%7G LL A?@@ @ AEFFF 1f  HBQFGGG KA AA ((C+&&//1q&!! LFJKK Kq / /AFF71:..C1q5MM / 1XXDDq! D DA !wqz 2 2 7 7 C CCAq1u   D 1XX""q1ua " "AAqD C1II " Jrc:t||S)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)r8 jordan_block)eigenvalrs r jordan_cellrFs"   A(  ; ;;rc,||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)ABs rmatrix_multiply_elementwiserZs ! !! $ $$rc\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 crt)popr8onesrs rrrms8 f})Cv ; ' ' ''rcdc|ptj|}||}|r||krtd||fzt||z}|dkr6||t t ||zdz}t||} |s4|D]0} t| |\} } | ||| | | f<1n@|D]=} t| |\} } | | kr"| ||x| | | f<| | | f<>| 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) randomRandomrrrsampleintrYrdivmodrandint) rrminmaxseed symmetricpercentprngijruijkrerfs r randMatrixrsNb  &6=&&D ZQ!VZOSTVWRXXYYY q1uB#~: [[SRC!788 9 9 a A ; - -C#q>>DAqll3,,AadGG - ; ;C#q>>DAqAv ;$(LLc$:$::!Q$!AqD' HrbareisscdDt}|dkr tjSt||fd}||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 c,g|]}t|SrRr).0rs r zwronskian..s//////rrc<||SrB)r)rerf functionsvars rrzwronskian..s)A,"3"3C";";r)rYrOner8r)rrr'rWs`` r wronskianrse.0/Y///I IAAvu q!;;;;;<rs& """"""""""""$$$$$$&&&&&&========888888@@@@@@11111188888888 22222222EEEEEEEE FEFEFEFEFE)FEFEFER    66666&6666",+      !     '''T'''T'''TJ'''EE('EX(!(!(!(!V ' ' 'e"H"H"H"H"HJ&&&&RJJJJZ<<<(%%%&((($?D!%I I I I X>)))))r