+ iv^RIHt^RIt^RIHt^RIHt^RIHt^RI H t ^RI H t H t ^RIHt^R IHt^R IHt^R IHt^R IHtHt^R IHt^RIHtHt^RIHtH t R1R./t!Rt"!RR]4t#Rt$!RR]#]4t%]%;t&t'](3Rlt)](3Rlt*R2Rlt+Rt,Rt-Rt.Rt/Rt0Rt1]!R3R 7R!4t2R4R#lt3R$t4R%R"R&R'/R(lt5R5R)lt6R6R*lt7R+t8R,t9R-t:R7R.lt;R8R/lt\RRRR7VP4#)zy The private _mat attribute of Matrix is deprecated. Use the .flat() method instead. z1.9z$deprecated-private-matrix-attributes)deprecated_since_versionactive_deprecations_target)r flatselfs&r_matDenseMatrix._mat*s%! &+'M  yy{rc VPVPRR4VPR\4VPRR4R7#)methodGE iszerofunctry_block_diagF)r)r+r,)invgetr)r%kwargss&,r _eval_inverseDenseMatrix._eval_inverse7sDxxvzz(D9#)::lG#D'-zz2BE'JL Lrc `^RIHpVPVPP 44#)z4Returns an Immutable version of this Matrix )ImmutableDenseMatrix) immutabler3_fromrep_repcopy)r%clss& r as_immutableDenseMatrix.as_immutable<s! ;||DIINN,--rc \V4#)zReturns a mutable version of this matrix Examples ======== >>> from sympy import ImmutableMatrix >>> X = ImmutableMatrix([[1, 2], [3, 4]]) >>> Y = X.as_mutable() >>> Y[1, 1] = 5 # Can set values in Y >>> Y Matrix([ [1, 2], [3, 5]]) )Matrixr$s&r as_mutableDenseMatrix.as_mutableBsd|rc \WR7#) hermitian)rr%rAs&&rcholeskyDenseMatrix.choleskySs 33rc \WR7#r@)rrBs&&rLDLdecompositionDenseMatrix.LDLdecompositionVs  ;;rc \W4#N)rr%rhss&&rlower_triangular_solve"DenseMatrix.lower_triangular_solveY &t11rc \W4#rI)rrJs&&rupper_triangular_solve"DenseMatrix.upper_triangular_solve\rNrNT)__name__ __module__ __qualname____firstlineno____doc__r__annotations__ _op_priority_class_prioritypropertyr&r0r9r=rCrFrLrPrrrr__static_attributes__rRrrrrsT M4LO   L . "4<22&/%6%6H%6%>%>%<%D%D"%<%D%D""rrc \VRR4'dVP4#\V\4'dV#\ VR4'dAVP 4p\ VP4^8Xd \V4#\V4#V#)z0Return a matrix as a Matrix, otherwise return x. is_MatrixF __array__) getattrr= isinstancerhasattrr`lenshaperr<)ras& r_force_mutablergespq+u%%||~ Au   K KKM qww<1 1: ay Hrc]tRt^stRtRtR#)MutableDenseMatrixc ^RIHpVP4P4Fwwr4pV!V3/VBWV3&K R#)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 )simplifyN)sympy.simplify.simplifyrktodokitems)r%r/ _simplifyijelements&, rrkMutableDenseMatrix.simplifyus< B#zz|113OFQG"75f5DAJ 4rrRN)rTrUrVrWrkr]rRrrririss 6rricf^RIHpV!\V4V4p\V4F wrEWSV&K V#)z]Converts Python list of SymPy expressions to a NumPy array. See Also ======== matrix2numpy empty)rrvrd enumerate)ldtypervrfrpss&& r list2numpyr{s3 c!feA! ! Hrc^RIHpV!VPV4p\VP4F-p\VP 4FpWV3,W4V3&K K/ V#)zIConverts SymPy's matrix to a NumPy array. See Also ======== list2numpy ru)rrvrerangerowscols)mryrvrfrprqs&& r matrix2numpyrsS agguA 166]qvvA1gAdG Hrc\V\4'dV^8d\RPV44hW8Xd\RPW44hW3FOp\V\4'dV^8gWC^, 8gK/\RPV^, W44h \ V4p\ V4p\ V4p\V4pWWW3&WWW3&WgW3&V)WqV3&V#)a 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 dimensions. 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) z/dim must be an integer biggen than one, got {}.z'i and j must be different, got ({}, {})z=i and j must be integers between 0 and {}, got i={} and j={}.)rbint ValueErrorformatrrr eye)rprqthetadimijcrzMs&&&& r rot_givensrs~ c3  37##)6#;0 0 v((.q 6 6f"c""b1f1W 66>> 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) rrrs&r rot_axis3r(h aEq ))rc \^^V^R7#)aReturns 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) rrrs&r rot_axis2r_rrc \^^V^R7#)aReturns 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) rrrs&r rot_axis1rrrc \^^V^R7#)aReturns 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) rrrs&r rot_ccw_axis3rrrc \^^V^R7#)aReturns 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) rrrs&r rot_ccw_axis2rrrc \^^V^R7#)aReturns 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) rrrs&r rot_ccw_axis1r;rr)modulesc ^RIHpHpV!V\R7pV!V4F5p\ V: RRP \ \V44: 23/VBWV&K7 V#)aCreate 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] )rvndindex)ry_)rrvrobjectrjoinmapstr)prefixrer/rvrarrindexs&&, rsymarrayrrsSB% V $CvsxxC/HI&$&   JrTcaa\\\S44oV'g VV3RlpMVV3Rlp\S4p\ WDV4P 4#)aGiven 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 cD<SV,PSSV,4#rIsubsrprqnseqss&&rcasoratian..saaQ/rc6<SV,PSV4#rIrrs&&rrrsaa+r)listrrrdr<det)rrzerofksff& r casoratianrsA> GT" #D  / + D A !?    rc.\P!V/VB#)zHCreate square identity matrix n x n See Also ======== diag zeros ones )r<rargsr/s*,rrrs ::t &v &&rstrictunpackFc6\P!VRVRV/VB#)aReturns 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 ======== .matrixbase.MatrixBase.eye .matrixbase.MatrixBase.diagonal .matrixbase.MatrixBase.diag .expressions.blockmatrix.BlockMatrix rr)r<diag)rrvaluesr/s$$*,rrrs#D ;; Gv Gf G GGrc2\P!VRVRR/#)aApply 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 ======== .matrixbase.MatrixBase.orthogonalize References ========== .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process normalize rankcheckT)ri orthogonalize)vlist orthonormals&&r GramSchmidtrs+H  + +  % 15 rc\V\4'dQ^VP9d \R4hVP^8Xd VP pVP 4^,p\V4'd \V4pV'g \R4hM \R4h\VR4'g\RV,4h\V4pWC,p\V4p\V4F\wrx\VR4'g\RV,4h\V4F$p VPW,4WgW,3&K& K^ \V4FRp \W4F@p VPW,4PW,4WiV,W,3&KB KT \V4F+p \V ^,V4Fp WiV 3,WjV 3&K K- V#)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.matrixbase.MatrixBase.jacobian wronskian z)`varlist` must be a column or row vector.z `len(varlist)` must not be zero.z*Improper variable list in hessian functiondiffz'Function `f` (%s) is not differentiable)rbrrer rTtolistr rdrrazerosrwr}r) rvarlist constraintsrrNoutrgrprqs &&& rhessianrFsX':&& GMM !HI I <<1 iiG.."1%7 L?@ @EFF 1f  BQFGG KA A (C+&q&!!FJK KqAFF7:.C15M ' 1XqA !wz 2 7 7 CCAqu 1Xq1uaAqD C1I! Jrc.\PWR7#)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)eigenvalrs&&r jordan_cellrs"   A  ;;rc$VPV4#)a]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.matrixbase.MatrixBase.__mul__ )multiply_elementwise)ABs&&rmatrix_multiply_elementwisers ! !! $$rcdRV9dVPR4VR&\P!V/VB#)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 rr)popr<onesrs*,rrrs0 f}Cv ;; ' ''rcT;'g\P!V4pVfTpV'dW8wd\RW3,4h\W,4pV^d8wd3VP V\ \ V4V,^d,44p\W4p V'g.VF%p \W4wrVPW#4WV 3&K' V #VF3p \W4wrW8:gKVPW#4;WV 3&WV 3&K5 V #)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] z4For symmetric matrices, r must equal c, but %i != %i) randomRandomrr}samplerrdrdivmodrandint) rrminmaxseed symmetricpercentprngrrijkrprqs &&&&&&&& r randMatrixrsb  & &6==&Dy QVOSTRXXYY quB#~ [[SRC!78 9 a A C#>DAll3,AdG H C#>DAv$(LL$::Q$!qD' HrcaaSUu.uFp\V4NK upo\S4pV^8Xd\P#\ WDVV3Rl4pVP V4#uupi)a: Compute Wronskian for [] of functions :: | f1 f2 ... fn | | f1' f2' ... fn' | | . . . . | W(f1, ..., fn) = | . . . . | | . . . . | | (n) (n) (n) | | D (f1) D (f2) ... D (fn) | see: https://en.wikipedia.org/wiki/Wronskian See Also ======== sympy.matrices.matrixbase.MatrixBase.jacobian hessian c6<SV,PSV4#rI)r)rprq functionsvars&&rrwronskian..3s)A,"3"3C";r)rrdrOner<r)rrr)rrWsff& r wronskianrsV.&//YY/I IAAvuu q; __future__rrsympy.core.basicrsympy.core.singletonrsympy.core.symbolrsympy.core.sympifyr(sympy.functions.elementary.trigonometricrr sympy.utilities.decoratorr sympy.utilities.exceptionsr sympy.utilities.iterablesr exceptionsr decompositionsrr matrixbaser repmatrixrrsolversrr__doctest_requires__rrrgri MutableMatrixr<rr{rrrrrrrrrrrrrrrrrrrrrRrrr s!" ""$&=8@1"8"2E&y1 FE)FER  6&66",+   !  ,t n4*n4*n4*n4*n4*n4*nJ'E(EX(!V '"H"He"HJ&RJZ<(%&($I X>)r