o 8Va5@sddlmZmZddlmZddlmZddlmZm Z ddl m Z m Z m Z mZmZmZddZGdd d eZd d Zd d ZddZGdddeZdS))Mulsympify) ShapeError) MatrixExpr) ZeroMatrix OneMatrix)unpackflatten conditionexhaustrm_idsortcGsB|stdt|t|dkr|dSdd|D}t|S)a~ Return the elementwise (aka Hadamard) product of matrices. Examples ======== >>> from sympy.matrices import hadamard_product, MatrixSymbol >>> A = MatrixSymbol('A', 2, 3) >>> B = MatrixSymbol('B', 2, 3) >>> hadamard_product(A) A >>> hadamard_product(A, B) HadamardProduct(A, B) >>> hadamard_product(A, B)[0, 1] A[0, 1]*B[0, 1] z#Empty Hadamard product is undefinedrcSsg|]}|js|qS)Z is_Identity.0irrE/usr/lib/python3/dist-packages/sympy/matrices/expressions/hadamard.py !sz$hadamard_product..) TypeErrorvalidatelenHadamardProductdoit)Zmatricesrrrhadamard_product s  rcs`eZdZdZdZdddfdd ZeddZd d Zd d Z d dZ ddZ ddZ Z S)ra1 Elementwise product of matrix expressions Examples ======== Hadamard product for matrix symbols: >>> from sympy.matrices import hadamard_product, HadamardProduct, MatrixSymbol >>> A = MatrixSymbol('A', 5, 5) >>> B = MatrixSymbol('B', 5, 5) >>> isinstance(hadamard_product(A, B), HadamardProduct) True Notes ===== This is a symbolic object that simply stores its argument without evaluating it. To actually compute the product, use the function ``hadamard_product()`` or ``HadamardProduct.doit`` TF)evaluatecheckcsBttt|}|r t|tj|g|R}|r|jdd}|S)NF)Zdeep)listmaprrsuper__new__r)clsrrargsobj __class__rrr =s zHadamardProduct.__new__cCs |jdjSNr)r"shapeselfrrrr'Gs zHadamardProduct.shapec stfdd|jDS)Ncs g|] }|jfiqSr)_entryrargrjkwargsrrrLs z*HadamardProduct._entry..)rr")r)rr.r/rr-rr*KszHadamardProduct._entrycCs ddlm}ttt||jSNr) transpose)$sympy.matrices.expressions.transposer1rrrr"r)r1rrr_eval_transposeNs zHadamardProduct._eval_transposec s|jfdd|jD}ddlmddlm}fdd|jDrEfdd|jD}|ddtDj|j}t |g|}t |S) Ncsg|] }|jdiqS)r)rr)ignoredrrrSsz(HadamardProduct.doit..r MatrixBase)ImmutableMatrixcsg|] }t|r|qSr) isinstancerr6rrrWscsg|]}|vr|qSrrr)explicitrrrYscSsg|]}t|qSr)rfromiterrrrrrZs ) funcr"sympyr7Zsympy.matrices.immutabler8zipZreshaper'r canonicalize)r)r5exprr8Z remainderZexpl_matr)r7r:r5rrRs  zHadamardProduct.doitcCspddlm}g}t|j}tt|D]}|d||||g||dd}|t|q| |S)Nr)Addr) r=rArr"rangerdiffappendrr;)r)xrAZtermsr"rZfactorsrrr_eval_derivativeas  , z HadamardProduct._eval_derivativec sHddlm}ddlm}ddlm}ddlm}fddtjD}g}|D]y}jd|} j|dd} j| } t | | } d d g} fd dt| D} | D]G}|j |j }|j |j }|||||||g| |||ggg| }|jdjdj|_d|_|jdjd j|_d|_|g|_ ||qYq(|S) Nr ExprBuilder ArrayDiagonalArrayTensorProduct _make_matrixcsg|] \}}|r|qSr)Zhas)rrr,rErrrpszAHadamardProduct._eval_derivative_matrix_lines..r)rcs"g|] \}}j|dkr|qSr)r'rr.er(rrrys"rP)sympy.core.exprrH0sympy.tensor.array.expressions.array_expressionsrJrL"sympy.matrices.expressions.matexprrN enumerater"_eval_derivative_matrix_linesr_lines_first_line_index_second_line_index_first_pointer_parent_first_pointer_index_second_pointer_parent_second_pointer_indexrD)r)rErHrJrLrNZ with_x_indlinesZindZ left_argsZ right_argsdZhadamdiagonalrl1l2subexprr)r)rErr[jsJ           z-HadamardProduct._eval_derivative_matrix_lines)__name__ __module__ __qualname____doc__Zis_HadamardProductr propertyr'r*r4rrFr[ __classcell__rrr$rr%s  rcGsTtdd|Ds td|d}|ddD]}|j|jkr'td||fqdS)Ncss|]}|jVqdSN) is_Matrixr+rrr szvalidate..z Mix of Matrix and Scalar symbolsrrz"Matrices %s and %s are not aligned)allrr'r)r"ABrrrrs rc Csddlm}tddt}t|}||}tddtdd}||}dd}td d|}||}t|trddd lm }||j }g}| D]\}} | d krW| |qI| t || qIt|}td dt|}||}t|}|S) aCanonicalize the Hadamard product ``x`` with mathematical properties. Examples ======== >>> from sympy.matrices.expressions import MatrixSymbol, HadamardProduct >>> from sympy.matrices.expressions import OneMatrix, ZeroMatrix >>> from sympy.matrices.expressions.hadamard import canonicalize >>> from sympy import init_printing >>> init_printing(use_unicode=False) >>> A = MatrixSymbol('A', 2, 2) >>> B = MatrixSymbol('B', 2, 2) >>> C = MatrixSymbol('C', 2, 2) Hadamard product associativity: >>> X = HadamardProduct(A, HadamardProduct(B, C)) >>> X A.*(B.*C) >>> canonicalize(X) A.*B.*C Hadamard product commutativity: >>> X = HadamardProduct(A, B) >>> Y = HadamardProduct(B, A) >>> X A.*B >>> Y B.*A >>> canonicalize(X) A.*B >>> canonicalize(Y) A.*B Hadamard product identity: >>> X = HadamardProduct(A, OneMatrix(2, 2)) >>> X A.*1 >>> canonicalize(X) A Absorbing element of Hadamard product: >>> X = HadamardProduct(A, ZeroMatrix(2, 2)) >>> X A.*0 >>> canonicalize(X) 0 Rewriting to Hadamard Power >>> X = HadamardProduct(A, A, A) >>> X A.*A.*A >>> canonicalize(X) .3 A Notes ===== As the Hadamard product is associative, nested products can be flattened. The Hadamard product is commutative so that factors can be sorted for canonical form. A matrix of only ones is an identity for Hadamard product, so every matrices of only ones can be removed. Any zero matrix will make the whole product a zero matrix. Duplicate elements can be collected and rewritten as HadamardPower References ========== .. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices) r)default_sort_keycS t|tSror9rrOrrr zcanonicalize..cSrvrorwrOrrrrxrycSrvro)r9rrOrrrrxrycSs"tdd|jDrt|jS|S)Ncss|]}t|tVqdSro)r9r)rcrrrrqsz/canonicalize..absorb..)anyr"rr'rOrrrabsorbs zcanonicalize..absorbcSrvrorwrOrrrrx ry)CounterrcSrvrorwrOrrrrxry)Zsympy.core.compatibilityrur r r r r9r collectionsr}r"itemsrD HadamardPowerr r) rEruZruleZfunr|r}ZtallyZnew_argbaseexprrrr?sD R     r?cCsBt|}t|}|dkr|S|js||S|jrtdt||S)Nrz#cannot raise expression to a matrix)rrp ValueErrorr)rrrrrhadamard_power)s rcsdeZdZdZfddZeddZeddZedd Zd d Z d d Z ddZ ddZ Z S)ra Elementwise power of matrix expressions Parameters ========== base : scalar or matrix exp : scalar or matrix Notes ===== There are four definitions for the hadamard power which can be used. Let's consider `A, B` as `(m, n)` matrices, and `a, b` as scalars. Matrix raised to a scalar exponent: .. math:: A^{\circ b} = \begin{bmatrix} A_{0, 0}^b & A_{0, 1}^b & \cdots & A_{0, n-1}^b \\ A_{1, 0}^b & A_{1, 1}^b & \cdots & A_{1, n-1}^b \\ \vdots & \vdots & \ddots & \vdots \\ A_{m-1, 0}^b & A_{m-1, 1}^b & \cdots & A_{m-1, n-1}^b \end{bmatrix} Scalar raised to a matrix exponent: .. math:: a^{\circ B} = \begin{bmatrix} a^{B_{0, 0}} & a^{B_{0, 1}} & \cdots & a^{B_{0, n-1}} \\ a^{B_{1, 0}} & a^{B_{1, 1}} & \cdots & a^{B_{1, n-1}} \\ \vdots & \vdots & \ddots & \vdots \\ a^{B_{m-1, 0}} & a^{B_{m-1, 1}} & \cdots & a^{B_{m-1, n-1}} \end{bmatrix} Matrix raised to a matrix exponent: .. math:: A^{\circ B} = \begin{bmatrix} A_{0, 0}^{B_{0, 0}} & A_{0, 1}^{B_{0, 1}} & \cdots & A_{0, n-1}^{B_{0, n-1}} \\ A_{1, 0}^{B_{1, 0}} & A_{1, 1}^{B_{1, 1}} & \cdots & A_{1, n-1}^{B_{1, n-1}} \\ \vdots & \vdots & \ddots & \vdots \\ A_{m-1, 0}^{B_{m-1, 0}} & A_{m-1, 1}^{B_{m-1, 1}} & \cdots & A_{m-1, n-1}^{B_{m-1, n-1}} \end{bmatrix} Scalar raised to a scalar exponent: .. math:: a^{\circ b} = a^b csdt|}t|}|jr|jr||S|jr(|jr(|j|jkr(td|j|jt|||}|S)NzGThe shape of the base {} and the shape of the exponent {} do not match.)r is_scalarrpr'rformatrr )r!rrr#r$rrr ns  zHadamardPower.__new__cC |jdSr&Z_argsr(rrrr~ zHadamardPower.basecCr)Nrrr(rrrrrzHadamardPower.expcCs|jjr|jjS|jjSro)rrpr'rr(rrrr'szHadamardPower.shapecKs|j}|j}|jr|j||fi|}n |jr|}ntd||jr2|j||fi|}||S|jr;|}||Std|)Nz)The base {} must be a scalar or a matrix.z-The exponent {} must be a scalar or a matrix.)rrrpr*rrr)r)rr.r/rrabrrrr*s$zHadamardPower._entrycCsddlm}t||j|jSr0)r2r1rrrr3rrrr4s zHadamardPower._eval_transposecCsFddlm}|j|}|j|}||}t|||j||S)Nr)log)r=rrrCrZ applyfuncr)r)rErZdexpZlogbaseZdlbaserrrrFs    zHadamardPower._eval_derivativec sddlm}ddlm}ddlm}ddlm}j|}|D]d}ddg}fdd t |D}|j |j } |j |j } ||||||| gj tjj d ||| ggg||jd } | jdjdj|_d|_d|_ | jdjd j|_d|_d|_ | g|_ q |S) NrrKrIrGrM)rrPrQcs$g|]\}}jj|dkr|qSrT)rr'rUr(rrrs$z?HadamardPower._eval_derivative_matrix_lines..r)Z validatorrP)rXrLrJrWrHrYrNrr[rZr\r]r^rrZ _validater"r_r`rarb) r)rErLrJrHrNZlrrrerfrgrhrr(rr[s@            z+HadamardPower._eval_derivative_matrix_lines)rirjrkrlr rmrrr'r*r4rFr[rnrrr$rr5s 8    rN)Z sympy.corerrZsympy.matrices.commonrrYrZ"sympy.matrices.expressions.specialrrZsympy.strategiesrr r r r r rrrr?rrrrrrs   p