ó ¡¼™\c@s‰ddlZddlmZmZmZddlmZddlmZddl m Z d„Z d„Z d„Z d „Zd „ZdS( iÿÿÿÿN(tStTupletdiff(tIterable(tImmutableDenseNDimArray(t NDimArraycCsLddlm}t|tƒr#|St||tttfƒrHt|ƒS|S(Niÿÿÿÿ(t MatrixBase(tsympy.matricesRt isinstanceRtlistttupleRR(taR((s9lib/python2.7/site-packages/sympy/tensor/array/arrayop.pyt_arrayfy s  cGst|ƒdkrtjSt|ƒdkr9t|dƒSt|ƒdkrmtt|d|dƒ|dŒStt|ƒ\}}t|tƒ s¢t|tƒ rª||St|ƒ}t|ƒ}g|D]}|D]}||^qÓqÉ}t ||j |j ƒS(s§ Tensor product among scalars or array-like objects. Examples ======== >>> from sympy.tensor.array import tensorproduct, Array >>> from sympy.abc import x, y, z, t >>> A = Array([[1, 2], [3, 4]]) >>> B = Array([x, y]) >>> tensorproduct(A, B) [[[x, y], [2*x, 2*y]], [[3*x, 3*y], [4*x, 4*y]]] >>> tensorproduct(A, x) [[x, 2*x], [3*x, 4*x]] >>> tensorproduct(A, B, B) [[[[x**2, x*y], [x*y, y**2]], [[2*x**2, 2*x*y], [2*x*y, 2*y**2]]], [[[3*x**2, 3*x*y], [3*x*y, 3*y**2]], [[4*x**2, 4*x*y], [4*x*y, 4*y**2]]]] Applying this function on two matrices will result in a rank 4 array. >>> from sympy import Matrix, eye >>> m = Matrix([[x, y], [z, t]]) >>> p = tensorproduct(eye(3), m) >>> p [[[[x, y], [z, t]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[x, y], [z, t]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[x, y], [z, t]]]] iii( tlenRtOneR t tensorproducttmapRRR Rtshape(targsR tbtaltbltitjt product_list((s9lib/python2.7/site-packages/sympy/tensor/array/arrayop.pyRs"   *cGsÔt|ƒ}tgƒ}x˜|D]}t|tƒsCtdƒ‚n|j|d}xX|D]P}||kr|tdƒ‚n||j|kržtdƒ‚n|j|ƒq[WqW|jƒ}gt|jƒD]\}}||krÏ|^qÏ}dg|} d} xEt |ƒD]7}| | ||d<| t |j||dƒ9} qWgt |ƒD]@}||kr[gt |j|ƒD]} | || ^q^q[} g} xq|D]i}g}xMt |j|dƒD]4}|j t g|D]}| ||^q胃qÒW| j |ƒq®Wg}xkt j| ŒD]Z}t |ƒ}tj}x/t j| ŒD]}|||t |ƒ7}q\W|j |ƒq1Wt| ƒdkrÁt|ƒdks¹t‚|dSt|ƒ||ƒS(s Contraction of an array-like object on the specified axes. Examples ======== >>> from sympy import Array, tensorcontraction >>> from sympy import Matrix, eye >>> tensorcontraction(eye(3), (0, 1)) 3 >>> A = Array(range(18), (3, 2, 3)) >>> A [[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]] >>> tensorcontraction(A, (0, 2)) [21, 30] Matrix multiplication may be emulated with a proper combination of ``tensorcontraction`` and ``tensorproduct`` >>> from sympy import tensorproduct >>> from sympy.abc import a,b,c,d,e,f,g,h >>> m1 = Matrix([[a, b], [c, d]]) >>> m2 = Matrix([[e, f], [g, h]]) >>> p = tensorproduct(m1, m2) >>> p [[[[a*e, a*f], [a*g, a*h]], [[b*e, b*f], [b*g, b*h]]], [[[c*e, c*f], [c*g, c*h]], [[d*e, d*f], [d*g, d*h]]]] >>> tensorcontraction(p, (1, 2)) [[a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]] >>> m1*m2 Matrix([ [a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]]) s(collections of contraction axes expectedis"dimension specified more than onces3cannot contract between axes of different dimensioni(R tsetRRt ValueErrorRtaddtrankt enumeratetrangetinttappendtsumt itertoolstproductRtZeroR tAssertionErrorttype(tarraytcontraction_axest taken_dimst axes_grouptdimtdRRtremaining_shapet cum_shapet_cumulRtremaining_indicest summed_deltastlidxtjstigtcontracted_arrayticontribtindex_base_positiontisumt sum_to_index((s9lib/python2.7/site-packages/sympy/tensor/array/arrayop.pyttensorcontractionAsL"      4 # P 2   cCs>ddlm}t|tf}t||ƒrft|ƒ}x)|D]}|jsAtdƒ‚qAqAWnt||ƒròt|ƒ}t||ƒrâg|D](}g|D]}|j|ƒ^q¤^q—}t |ƒ||j |j ƒS|j|ƒSnHt||ƒr-tg|D]}|j|ƒ^q |j ƒSt||ƒSdS(s¶ Derivative by arrays. Supports both arrays and scalars. Given the array `A_{i_1, \ldots, i_N}` and the array `X_{j_1, \ldots, j_M}` this function will return a new array `B` defined by `B_{j_1,\ldots,j_M,i_1,\ldots,i_N} := \frac{\partial A_{i_1,\ldots,i_N}}{\partial X_{j_1,\ldots,j_M}}` Examples ======== >>> from sympy import derive_by_array >>> from sympy.abc import x, y, z, t >>> from sympy import cos >>> derive_by_array(cos(x*t), x) -t*sin(t*x) >>> derive_by_array(cos(x*t), [x, y, z, t]) [-t*sin(t*x), 0, 0, -x*sin(t*x)] >>> derive_by_array([x, y**2*z], [[x, y], [z, t]]) [[[1, 0], [0, 2*y*z]], [[0, y**2], [0, 0]]] iÿÿÿÿ(Rscannot derive by this arrayN( RRRRRRt _diff_wrtRRR&R(texprtdxRt array_typesRtxtyt new_array((s9lib/python2.7/site-packages/sympy/tensor/array/arrayop.pytderive_by_array§s     5,c Cst|tƒstdƒ‚nddlm}t||ƒsR|t|ƒƒ}n|j|jƒkrvtdƒ‚n|}|g|j D]}t |ƒ^qŠƒ}dgt |ƒ}x=t tj|ŒƒD]&\}}||ƒ}||||>> from sympy.abc import x, y, z, t >>> from sympy import sin >>> from sympy import Array, permutedims >>> a = Array([[x, y, z], [t, sin(x), 0]]) >>> a [[x, y, z], [t, sin(x), 0]] >>> permutedims(a, (1, 0)) [[x, t], [y, sin(x)], [z, 0]] If the array is of second order, ``transpose`` can be used: >>> from sympy import transpose >>> transpose(a) [[x, t], [y, sin(x)], [z, 0]] Examples on higher dimensions: >>> b = Array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]]) >>> permutedims(b, (2, 1, 0)) [[[1, 5], [3, 7]], [[2, 6], [4, 8]]] >>> permutedims(b, (1, 2, 0)) [[[1, 5], [2, 6]], [[3, 7], [4, 8]]] ``Permutation`` objects are also allowed: >>> from sympy.combinatorics import Permutation >>> permutedims(b, Permutation([1, 2, 0])) [[[1, 5], [2, 6]], [[3, 7], [4, 8]]] s#expression has to be an N-dim arrayiÿÿÿÿ(t Permutationswrong permutation sizeN(RRt TypeErrortsympy.combinatoricsRCR tsizeRRRRtNoneR RR"R#R&( R<tpermRCtipermRt indices_spanRAtidxttt new_shape((s9lib/python2.7/site-packages/sympy/tensor/array/arrayop.pyt permutedimsÕs'(" (R"tsympyRRRtsympy.core.compatibilityRtsympy.tensor.arrayRtsympy.tensor.array.ndim_arrayRR RR:RBRN(((s9lib/python2.7/site-packages/sympy/tensor/array/arrayop.pyts  . f .